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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<html xmlns="http://www.w3.org/1999/xhtml">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<div class='booktitleinheader'><a href='index.html'>Volume 1: Logical Foundations</a></div>
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<li class='section_name'><a href='toc.html'>Table of Contents</a></li>
<li class='section_name'><a href='coqindex.html'>Index</a></li>
<li class='section_name'><a href='deps.html'>Roadmap</a></li>
</ul>
</div>
</div>
<div id="main">
<div id="toc">
<div id="accordion">
<h2><a href="Preface.html">Preface</a></h2>
<div>
<ul class="doclist">
<li><a href="Preface.html#lab1">Welcome</a>
</li>
<li><a href="Preface.html#lab2">Overview</a>
<div>
<ul class="doclist">
<li><a href="Preface.html#lab3">Logic</a>
</li>
<li><a href="Preface.html#lab4">Proof Assistants</a>
</li>
<li><a href="Preface.html#lab5">Functional Programming</a>
</li>
<li><a href="Preface.html#lab6">Further Reading</a>
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<li><a href="Preface.html#lab7">Practicalities</a>
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<ul class="doclist">
<li><a href="Preface.html#lab8">System Requirements</a>
</li>
<li><a href="Preface.html#lab10">Exercises</a>
</li>
<li><a href="Preface.html#lab11">Downloading the Coq Files</a>
</li>
<li><a href="Preface.html#lab12">Chapter Dependencies</a>
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<li><a href="Preface.html#lab13">Recommended Citation Format</a>
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</ul>
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</li>
<li><a href="Preface.html#lab14">Resources</a>
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<li><a href="Preface.html#lab15">Sample Exams</a>
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<li><a href="Preface.html#lab16">Lecture Videos</a>
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<li><a href="Preface.html#lab17">Note for Instructors and Contributors</a>
</li>
<li><a href="Preface.html#lab18">Translations</a>
</li>
<li><a href="Preface.html#lab19">Thanks</a>
</li>
</ul>
</div>
<h2><a href="Basics.html">Functional Programming in Coq (<i>Basics</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Basics.html#lab20">Introduction</a>
</li>
<li><a href="Basics.html#lab21">Data and Functions</a>
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<li><a href="Basics.html#lab22">Enumerated Types</a>
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<li><a href="Basics.html#lab23">Days of the Week</a>
</li>
<li><a href="Basics.html#lab24">Homework Submission Guidelines</a>
</li>
<li><a href="Basics.html#lab25">Booleans</a>
</li>
<li><a href="Basics.html#lab28">Types</a>
</li>
<li><a href="Basics.html#lab29">New Types from Old</a>
</li>
<li><a href="Basics.html#lab30">Modules</a>
</li>
<li><a href="Basics.html#lab31">Tuples</a>
</li>
<li><a href="Basics.html#lab32">Numbers</a>
</li>
</ul>
</div>
</li>
<li><a href="Basics.html#lab35">Proof by Simplification</a>
</li>
<li><a href="Basics.html#lab36">Proof by Rewriting</a>
</li>
<li><a href="Basics.html#lab39">Proof by Case Analysis</a>
<div>
<ul class="doclist">
<li><a href="Basics.html#lab42">More on Notation (Optional)</a>
</li>
<li><a href="Basics.html#lab43">Fixpoints and Structural Recursion (Optional)</a>
</li>
</ul>
</div>
</li>
<li><a href="Basics.html#lab45">More Exercises</a>
<div>
<ul class="doclist">
<li><a href="Basics.html#lab46">Warmups</a>
</li>
<li><a href="Basics.html#lab50">Course Late Policies, Formalized</a>
</li>
<li><a href="Basics.html#lab58">Binary Numerals</a>
</li>
</ul>
</div>
</li>
<li><a href="Basics.html#lab60">Testing Your Solutions</a>
</li>
</ul>
</div>
<h2><a href="Induction.html">Proof by Induction (<i>Induction</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Induction.html#lab61">Separate Compilation</a>
</li>
<li><a href="Induction.html#lab62">Proof by Induction</a>
</li>
<li><a href="Induction.html#lab68">Proofs Within Proofs</a>
</li>
<li><a href="Induction.html#lab69">Formal vs. Informal Proof</a>
</li>
<li><a href="Induction.html#lab72">More Exercises</a>
</li>
<li><a href="Induction.html#lab77">Nat to Bin and Back to Nat</a>
</li>
<li><a href="Induction.html#lab80">Bin to Nat and Back to Bin (Advanced)</a>
</li>
</ul>
</div>
<h2><a href="Lists.html">Working with Structured Data (<i>Lists</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Lists.html#lab83">Pairs of Numbers</a>
</li>
<li><a href="Lists.html#lab86">Lists of Numbers</a>
</li>
<li><a href="Lists.html#lab98">Reasoning About Lists</a>
<div>
<ul class="doclist">
<li><a href="Lists.html#lab99">Induction on Lists</a>
</li>
<li><a href="Lists.html#lab101"><span class="inlinecode"><span class="id" title="keyword">Search</span></span></a>
</li>
<li><a href="Lists.html#lab102">List Exercises, Part 1</a>
</li>
<li><a href="Lists.html#lab105">List Exercises, Part 2</a>
</li>
</ul>
</div>
</li>
<li><a href="Lists.html#lab111">Options</a>
</li>
<li><a href="Lists.html#lab114">Partial Maps</a>
</li>
</ul>
</div>
<h2><a href="Poly.html">Polymorphism and Higher-Order Functions (<i>Poly</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Poly.html#lab118">Polymorphism</a>
<div>
<ul class="doclist">
<li><a href="Poly.html#lab119">Polymorphic Lists</a>
</li>
<li><a href="Poly.html#lab128">Polymorphic Pairs</a>
</li>
<li><a href="Poly.html#lab131">Polymorphic Options</a>
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</ul>
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<li><a href="Poly.html#lab133">Functions as Data</a>
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<li><a href="Poly.html#lab134">Higher-Order Functions</a>
</li>
<li><a href="Poly.html#lab135">Filter</a>
</li>
<li><a href="Poly.html#lab136">Anonymous Functions</a>
</li>
<li><a href="Poly.html#lab139">Map</a>
</li>
<li><a href="Poly.html#lab144">Fold</a>
</li>
<li><a href="Poly.html#lab146">Functions That Construct Functions</a>
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</ul>
</div>
</li>
<li><a href="Poly.html#lab147">Additional Exercises</a>
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<li><a href="Poly.html#lab152">Church Numerals (Advanced)</a>
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</li>
</ul>
</div>
<h2><a href="Tactics.html">More Basic Tactics (<i>Tactics</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Tactics.html#lab157">The <span class="inlinecode"><span class="id" title="tactic">apply</span></span> Tactic</a>
</li>
<li><a href="Tactics.html#lab161">The <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="keyword">with</span></span> Tactic</a>
</li>
<li><a href="Tactics.html#lab163">The <span class="inlinecode"><span class="id" title="tactic">injection</span></span> and <span class="inlinecode"><span class="id" title="tactic">discriminate</span></span> Tactics</a>
</li>
<li><a href="Tactics.html#lab166">Using Tactics on Hypotheses</a>
</li>
<li><a href="Tactics.html#lab167">Specializing Hypotheses</a>
</li>
<li><a href="Tactics.html#lab168">Varying the Induction Hypothesis</a>
</li>
<li><a href="Tactics.html#lab173">Unfolding Definitions</a>
</li>
<li><a href="Tactics.html#lab174">Using <span class="inlinecode"><span class="id" title="tactic">destruct</span></span> on Compound Expressions</a>
</li>
<li><a href="Tactics.html#lab177">Review</a>
</li>
<li><a href="Tactics.html#lab178">Additional Exercises</a>
</li>
</ul>
</div>
<h2><a href="Logic.html">Logic in Coq (<i>Logic</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Logic.html#lab185">Logical Connectives</a>
<div>
<ul class="doclist">
<li><a href="Logic.html#lab186">Conjunction</a>
</li>
<li><a href="Logic.html#lab190">Disjunction</a>
</li>
<li><a href="Logic.html#lab193">Falsehood and Negation</a>
</li>
<li><a href="Logic.html#lab200">Truth</a>
</li>
<li><a href="Logic.html#lab201">Logical Equivalence</a>
</li>
<li><a href="Logic.html#lab204">Setoids and Logical Equivalence</a>
</li>
<li><a href="Logic.html#lab205">Existential Quantification</a>
</li>
</ul>
</div>
</li>
<li><a href="Logic.html#lab209">Programming with Propositions</a>
</li>
<li><a href="Logic.html#lab214">Applying Theorems to Arguments</a>
</li>
<li><a href="Logic.html#lab215">Working with Decidable Properties</a>
</li>
<li><a href="Logic.html#lab221">The Logic of Coq</a>
<div>
<ul class="doclist">
<li><a href="Logic.html#lab222">Functional Extensionality</a>
</li>
<li><a href="Logic.html#lab224">Classical vs. Constructive Logic</a>
</li>
</ul>
</div>
</li>
</ul>
</div>
<h2><a href="IndProp.html">Inductively Defined Propositions (<i>IndProp</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="IndProp.html#lab228">Inductively Defined Propositions</a>
<div>
<ul class="doclist">
<li><a href="IndProp.html#lab229">Example: The Collatz Conjecture</a>
</li>
<li><a href="IndProp.html#lab230">Example: Ordering</a>
</li>
<li><a href="IndProp.html#lab231">Example: Transitive Closure</a>
</li>
<li><a href="IndProp.html#lab233">Example: Permutations</a>
</li>
<li><a href="IndProp.html#lab235">Example: Evenness (yet again)</a>
</li>
</ul>
</div>
</li>
<li><a href="IndProp.html#lab237">Using Evidence in Proofs</a>
<div>
<ul class="doclist">
<li><a href="IndProp.html#lab238">Inversion on Evidence</a>
</li>
<li><a href="IndProp.html#lab241">Induction on Evidence</a>
</li>
</ul>
</div>
</li>
<li><a href="IndProp.html#lab246">Inductive Relations</a>
</li>
<li><a href="IndProp.html#lab255">A Digression on Notation</a>
</li>
<li><a href="IndProp.html#lab256">Case Study: Regular Expressions</a>
<div>
<ul class="doclist">
<li><a href="IndProp.html#lab259">The <span class="inlinecode"><span class="id" title="var">remember</span></span> Tactic</a>
</li>
</ul>
</div>
</li>
<li><a href="IndProp.html#lab263">Case Study: Improving Reflection</a>
</li>
<li><a href="IndProp.html#lab266">Additional Exercises</a>
<div>
<ul class="doclist">
<li><a href="IndProp.html#lab274">Extended Exercise: A Verified Regular-Expression Matcher</a>
</li>
</ul>
</div>
</li>
</ul>
</div>
<h2><a href="Maps.html">Total and Partial Maps (<i>Maps</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Maps.html#lab283">The Coq Standard Library</a>
</li>
<li><a href="Maps.html#lab284">Identifiers</a>
</li>
<li><a href="Maps.html#lab285">Total Maps</a>
</li>
<li><a href="Maps.html#lab292">Partial maps</a>
</li>
</ul>
</div>
<h2><a href="ProofObjects.html">The Curry-Howard Correspondence (<i>ProofObjects</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="ProofObjects.html#lab293">Proof Scripts</a>
</li>
<li><a href="ProofObjects.html#lab295">Quantifiers, Implications, Functions</a>
</li>
<li><a href="ProofObjects.html#lab296">Programming with Tactics</a>
</li>
<li><a href="ProofObjects.html#lab297">Logical Connectives as Inductive Types</a>
<div>
<ul class="doclist">
<li><a href="ProofObjects.html#lab298">Conjunction</a>
</li>
<li><a href="ProofObjects.html#lab300">Disjunction</a>
</li>
<li><a href="ProofObjects.html#lab302">Existential Quantification</a>
</li>
<li><a href="ProofObjects.html#lab304"><span class="inlinecode"><span class="id" title="var">True</span></span> and <span class="inlinecode"><span class="id" title="var">False</span></span></a>
</li>
</ul>
</div>
</li>
<li><a href="ProofObjects.html#lab307">Equality</a>
<div>
<ul class="doclist">
<li><a href="ProofObjects.html#lab312">Inversion, Again</a>
</li>
</ul>
</div>
</li>
<li><a href="ProofObjects.html#lab313">Coq's Trusted Computing Base</a>
</li>
<li><a href="ProofObjects.html#lab314">More Exercises</a>
</li>
<li><a href="ProofObjects.html#lab319">Proof Irrelevance (Advanced)</a>
</li>
</ul>
</div>
<h2><a href="IndPrinciples.html">Induction Principles (<i>IndPrinciples</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="IndPrinciples.html#lab323">Basics</a>
</li>
<li><a href="IndPrinciples.html#lab328">Polymorphism</a>
</li>
<li><a href="IndPrinciples.html#lab333">Induction Hypotheses</a>
</li>
<li><a href="IndPrinciples.html#lab334">More on the <span class="inlinecode"><span class="id" title="tactic">induction</span></span> Tactic</a>
</li>
<li><a href="IndPrinciples.html#lab336">Induction Principles for Propositions</a>
</li>
<li><a href="IndPrinciples.html#lab337">Another Form of Induction Principles on Propositions (Optional)</a>
</li>
<li><a href="IndPrinciples.html#lab338">Formal vs. Informal Proofs by Induction</a>
<div>
<ul class="doclist">
<li><a href="IndPrinciples.html#lab339">Induction Over an Inductively Defined Set</a>
</li>
<li><a href="IndPrinciples.html#lab340">Induction Over an Inductively Defined Proposition</a>
</li>
</ul>
</div>
</li>
<li><a href="IndPrinciples.html#lab341">Explicit Proof Objects for Induction (Optional)</a>
</li>
</ul>
</div>
<h2><a href="Rel.html">Properties of Relations (<i>Rel</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Rel.html#lab343">Relations</a>
</li>
<li><a href="Rel.html#lab344">Basic Properties</a>
</li>
<li><a href="Rel.html#lab361">Reflexive, Transitive Closure</a>
</li>
</ul>
</div>
<h2><a href="Imp.html">Simple Imperative Programs (<i>Imp</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab364">Arithmetic and Boolean Expressions</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab365">Syntax</a>
</li>
<li><a href="Imp.html#lab366">Evaluation</a>
</li>
<li><a href="Imp.html#lab367">Optimization</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab368">Coq Automation</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab369">Tacticals</a>
</li>
<li><a href="Imp.html#lab376">Defining New Tactics</a>
</li>
<li><a href="Imp.html#lab377">The <span class="inlinecode"><span class="id" title="var">lia</span></span> Tactic</a>
</li>
<li><a href="Imp.html#lab378">A Few More Handy Tactics</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab379">Evaluation as a Relation</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab380">Inference Rule Notation</a>
</li>
<li><a href="Imp.html#lab382">Equivalence of the Definitions</a>
</li>
<li><a href="Imp.html#lab384">Computational vs. Relational Definitions</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab385">Expressions With Variables</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab386">States</a>
</li>
<li><a href="Imp.html#lab387">Syntax</a>
</li>
<li><a href="Imp.html#lab388">Notations</a>
</li>
<li><a href="Imp.html#lab389">Evaluation</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab390">Commands</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab391">Syntax</a>
</li>
<li><a href="Imp.html#lab392">Desugaring Notations</a>
</li>
<li><a href="Imp.html#lab393"><span class="inlinecode"><span class="id" title="keyword">Locate</span></span> Again</a>
</li>
<li><a href="Imp.html#lab396">More Examples</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab400">Evaluating Commands</a>
<div>
<ul class="doclist">
<li><a href="Imp.html#lab401">Evaluation as a Function (Failed Attempt)</a>
</li>
<li><a href="Imp.html#lab402">Evaluation as a Relation</a>
</li>
<li><a href="Imp.html#lab406">Determinism of Evaluation</a>
</li>
</ul>
</div>
</li>
<li><a href="Imp.html#lab407">Reasoning About Imp Programs</a>
</li>
<li><a href="Imp.html#lab412">Additional Exercises</a>
</li>
</ul>
</div>
<h2><a href="ImpParser.html">Lexing and Parsing in Coq (<i>ImpParser</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="ImpParser.html#lab421">Internals</a>
<div>
<ul class="doclist">
<li><a href="ImpParser.html#lab422">Lexical Analysis</a>
</li>
<li><a href="ImpParser.html#lab423">Parsing</a>
</li>
</ul>
</div>
</li>
<li><a href="ImpParser.html#lab427">Examples</a>
</li>
</ul>
</div>
<h2><a href="ImpCEvalFun.html">An Evaluation Function for Imp (<i>ImpCEvalFun</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="ImpCEvalFun.html#lab428">A Broken Evaluator</a>
</li>
<li><a href="ImpCEvalFun.html#lab429">A Step-Indexed Evaluator</a>
</li>
<li><a href="ImpCEvalFun.html#lab432">Relational vs. Step-Indexed Evaluation</a>
</li>
<li><a href="ImpCEvalFun.html#lab435">Determinism of Evaluation Again</a>
</li>
</ul>
</div>
<h2><a href="Extraction.html">Extracting OCaml from Coq (<i>Extraction</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Extraction.html#lab436">Basic Extraction</a>
</li>
<li><a href="Extraction.html#lab437">Controlling Extraction of Specific Types</a>
</li>
<li><a href="Extraction.html#lab438">A Complete Example</a>
</li>
<li><a href="Extraction.html#lab439">Discussion</a>
</li>
<li><a href="Extraction.html#lab440">Going Further</a>
</li>
</ul>
</div>
<h2><a href="Auto.html">More Automation (<i>Auto</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Auto.html#lab441">The <span class="inlinecode"><span class="id" title="tactic">auto</span></span> Tactic</a>
</li>
<li><a href="Auto.html#lab442">Searching For Hypotheses</a>
</li>
<li><a href="Auto.html#lab443">The <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> and <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> tactics</a>
</li>
<li><a href="Auto.html#lab444">Constraints on Existential Variables</a>
</li>
</ul>
</div>
<h2><a href="AltAuto.html">A Streamlined Treatment of Automation (<i>AltAuto</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab445">Tacticals</a>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab446">The <span class="inlinecode"><span class="id" title="tactic">try</span></span> Tactical</a>
</li>
<li><a href="AltAuto.html#lab447">The Sequence Tactical <span class="inlinecode">;</span> (Simple Form)</a>
</li>
<li><a href="AltAuto.html#lab449">The Sequence Tactical <span class="inlinecode">;</span> (Local Form)</a>
</li>
<li><a href="AltAuto.html#lab451">The <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> Tactical</a>
</li>
<li><a href="AltAuto.html#lab453">An Optimization Exercise</a>
</li>
</ul>
</div>
</li>
<li><a href="AltAuto.html#lab455">Tactics that Make Mentioning Names Unnecessary</a>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab456">The <span class="inlinecode"><span class="id" title="tactic">assumption</span></span> tactic</a>
</li>
<li><a href="AltAuto.html#lab457">The <span class="inlinecode"><span class="id" title="var">contradiction</span></span> tactic</a>
</li>
<li><a href="AltAuto.html#lab458">The <span class="inlinecode"><span class="id" title="tactic">subst</span></span> tactic</a>
</li>
<li><a href="AltAuto.html#lab459">The <span class="inlinecode"><span class="id" title="tactic">constructor</span></span> tactic</a>
</li>
</ul>
</div>
</li>
<li><a href="AltAuto.html#lab460">Automatic Solvers</a>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab461">Linear Integer Arithmetic: The <span class="inlinecode"><span class="id" title="var">lia</span></span> Tactic</a>
</li>
<li><a href="AltAuto.html#lab462">Equalities: The <span class="inlinecode"><span class="id" title="tactic">congruence</span></span> Tactic</a>
</li>
<li><a href="AltAuto.html#lab463">Propositions: The <span class="inlinecode"><span class="id" title="tactic">intuition</span></span> Tactic</a>
</li>
<li><a href="AltAuto.html#lab464">Exercises with Automatic Solvers</a>
</li>
</ul>
</div>
</li>
<li><a href="AltAuto.html#lab466">Search Tactics</a>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab467">The <span class="inlinecode"><span class="id" title="tactic">auto</span></span> Tactic</a>
</li>
<li><a href="AltAuto.html#lab471">The <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> variant</a>
</li>
</ul>
</div>
</li>
<li><a href="AltAuto.html#lab472">Ltac: The Tactic Language</a>
<div>
<ul class="doclist">
<li><a href="AltAuto.html#lab473">Ltac Functions</a>
</li>
<li><a href="AltAuto.html#lab476">Ltac Pattern Matching</a>
</li>
<li><a href="AltAuto.html#lab477">Using <span class="inlinecode"><span class="id" title="keyword">match</span></span> <span class="inlinecode"><span class="id" title="keyword">goal</span></span> to Prove Tautologies</a>
</li>
</ul>
</div>
</li>
<li><a href="AltAuto.html#lab479">Review</a>
</li>
</ul>
</div>
<h2><a href="Postscript.html">Postscript</a></h2>
<div>
<ul class="doclist">
<li><a href="Postscript.html#lab480">Looking Back</a>
</li>
<li><a href="Postscript.html#lab481">Looking Forward</a>
</li>
<li><a href="Postscript.html#lab482">Resources</a>
</li>
</ul>
</div>
<h2><a href="Bib.html">Bibliography (<i>Bib</i>)</a></h2>
<div>
<ul class="doclist">
<li><a href="Bib.html#lab483">Resources cited in this volume</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</body>
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