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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>ImpParser: Lexing and Parsing in Coq</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">ImpParser<span class="subtitle">Lexing and Parsing in Coq</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* $Date: 2013-07-01 18:48:47 -0400 (Mon, 01 Jul 2013) $ *)</span><br/>
<br/>
</div>
<div class="doc">
The development of the <span class="inlinecode"><span class="id" type="var">Imp</span></span> language in Imp.v completely ignores
issues of concrete syntax — how an ascii string that a programmer
might write gets translated into the abstract syntax trees defined
by the datatypes <span class="inlinecode"><span class="id" type="var">aexp</span></span>, <span class="inlinecode"><span class="id" type="var">bexp</span></span>, and <span class="inlinecode"><span class="id" type="var">com</span></span>. In this file we
illustrate how the rest of the story can be filled in by building
a simple lexical analyzer and parser using Coq's functional
programming facilities.
<div class="paragraph"> </div>
This development is not intended to be understood in detail: the
explanations are fairly terse and there are no exercises. The
main point is simply to demonstrate that it can be done. You are
invited to look through the code — most of it is not very
complicated, though the parser relies on some "monadic"
programming idioms that may require a little work to make out —
but most readers will probably want to just skip down to the
Examples section at the very end to get the punchline.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab437"></a><h1 class="section">Internals</h1>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">SfLib</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Imp</span>.<br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">String</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Ascii</span>.<br/>
<br/>
<span class="id" type="keyword">Open</span> <span class="id" type="keyword">Scope</span> <span class="id" type="var">list_scope</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab438"></a><h2 class="section">Lexical Analysis</h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">isWhite</span> (<span class="id" type="var">c</span> : <span class="id" type="var">ascii</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">n</span> := <span class="id" type="var">nat_of_ascii</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">orb</span> (<span class="id" type="var">orb</span> (<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 32) <span class="comment">(* space *)</span><br/>
(<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 9)) <span class="comment">(* tab *)</span><br/>
(<span class="id" type="var">orb</span> (<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 10) <span class="comment">(* linefeed *)</span><br/>
(<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 13)). <span class="comment">(* Carriage return. *)</span><br/>
<br/>
<span class="id" type="keyword">Notation</span> "x '<=?' y" := (<span class="id" type="var">ble_nat</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>)<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 70, <span class="id" type="var">no</span> <span class="id" type="var">associativity</span>) : <span class="id" type="var">nat_scope</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">isLowerAlpha</span> (<span class="id" type="var">c</span> : <span class="id" type="var">ascii</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">n</span> := <span class="id" type="var">nat_of_ascii</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">andb</span> (97 <=? <span class="id" type="var">n</span>) (<span class="id" type="var">n</span> <=? 122).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">isAlpha</span> (<span class="id" type="var">c</span> : <span class="id" type="var">ascii</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">n</span> := <span class="id" type="var">nat_of_ascii</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">orb</span> (<span class="id" type="var">andb</span> (65 <=? <span class="id" type="var">n</span>) (<span class="id" type="var">n</span> <=? 90))<br/>
(<span class="id" type="var">andb</span> (97 <=? <span class="id" type="var">n</span>) (<span class="id" type="var">n</span> <=? 122)).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">isDigit</span> (<span class="id" type="var">c</span> : <span class="id" type="var">ascii</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">n</span> := <span class="id" type="var">nat_of_ascii</span> <span class="id" type="var">c</span> <span class="id" type="keyword">in</span> <br/>
<span class="id" type="var">andb</span> (48 <=? <span class="id" type="var">n</span>) (<span class="id" type="var">n</span> <=? 57).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">chartype</span> := <span class="id" type="var">white</span> | <span class="id" type="var">alpha</span> | <span class="id" type="var">digit</span> | <span class="id" type="var">other</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">classifyChar</span> (<span class="id" type="var">c</span> : <span class="id" type="var">ascii</span>) : <span class="id" type="var">chartype</span> :=<br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">isWhite</span> <span class="id" type="var">c</span> <span class="id" type="keyword">then</span> <br/>
<span class="id" type="var">white</span><br/>
<span class="id" type="keyword">else</span> <span class="id" type="keyword">if</span> <span class="id" type="var">isAlpha</span> <span class="id" type="var">c</span> <span class="id" type="keyword">then</span><br/>
<span class="id" type="var">alpha</span><br/>
<span class="id" type="keyword">else</span> <span class="id" type="keyword">if</span> <span class="id" type="var">isDigit</span> <span class="id" type="var">c</span> <span class="id" type="keyword">then</span><br/>
<span class="id" type="var">digit</span><br/>
<span class="id" type="keyword">else</span><br/>
<span class="id" type="var">other</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">list_of_string</span> (<span class="id" type="var">s</span> : <span class="id" type="var">string</span>) : <span class="id" type="var">list</span> <span class="id" type="var">ascii</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">s</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">EmptyString</span> ⇒ []<br/>
| <span class="id" type="var">String</span> <span class="id" type="var">c</span> <span class="id" type="var">s</span> ⇒ <span class="id" type="var">c</span> :: (<span class="id" type="var">list_of_string</span> <span class="id" type="var">s</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">string_of_list</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">ascii</span>) : <span class="id" type="var">string</span> :=<br/>
<span class="id" type="var">fold_right</span> <span class="id" type="var">String</span> <span class="id" type="var">EmptyString</span> <span class="id" type="var">xs</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">token</span> := <span class="id" type="var">string</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">tokenize_helper</span> (<span class="id" type="var">cls</span> : <span class="id" type="var">chartype</span>) (<span class="id" type="var">acc</span> <span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">ascii</span>) <br/>
: <span class="id" type="var">list</span> (<span class="id" type="var">list</span> <span class="id" type="var">ascii</span>) :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">tk</span> := <span class="id" type="keyword">match</span> <span class="id" type="var">acc</span> <span class="id" type="keyword">with</span> [] ⇒ [] | <span class="id" type="var">_</span>::_ ⇒ [<span class="id" type="var">rev</span> <span class="id" type="var">acc</span>] <span class="id" type="keyword">end</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ <span class="id" type="var">tk</span><br/>
| (<span class="id" type="var">x</span>::<span class="id" type="var">xs'</span>) ⇒<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">cls</span>, <span class="id" type="var">classifyChar</span> <span class="id" type="var">x</span>, <span class="id" type="var">x</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">_</span>, <span class="id" type="var">_</span>, "(" ⇒ <span class="id" type="var">tk</span> ++ ["("]::(<span class="id" type="var">tokenize_helper</span> <span class="id" type="var">other</span> [] <span class="id" type="var">xs'</span>) <br/>
| <span class="id" type="var">_</span>, <span class="id" type="var">_</span>, ")" ⇒ <span class="id" type="var">tk</span> ++ [")"]::(<span class="id" type="var">tokenize_helper</span> <span class="id" type="var">other</span> [] <span class="id" type="var">xs'</span>) <br/>
| <span class="id" type="var">_</span>, <span class="id" type="var">white</span>, <span class="id" type="var">_</span> ⇒ <span class="id" type="var">tk</span> ++ (<span class="id" type="var">tokenize_helper</span> <span class="id" type="var">white</span> [] <span class="id" type="var">xs'</span>)<br/>
| <span class="id" type="var">alpha</span>,<span class="id" type="var">alpha</span>,<span class="id" type="var">x</span> ⇒ <span class="id" type="var">tokenize_helper</span> <span class="id" type="var">alpha</span> (<span class="id" type="var">x</span>::<span class="id" type="var">acc</span>) <span class="id" type="var">xs'</span><br/>
| <span class="id" type="var">digit</span>,<span class="id" type="var">digit</span>,<span class="id" type="var">x</span> ⇒ <span class="id" type="var">tokenize_helper</span> <span class="id" type="var">digit</span> (<span class="id" type="var">x</span>::<span class="id" type="var">acc</span>) <span class="id" type="var">xs'</span><br/>
| <span class="id" type="var">other</span>,<span class="id" type="var">other</span>,<span class="id" type="var">x</span> ⇒ <span class="id" type="var">tokenize_helper</span> <span class="id" type="var">other</span> (<span class="id" type="var">x</span>::<span class="id" type="var">acc</span>) <span class="id" type="var">xs'</span><br/>
| <span class="id" type="var">_</span>,<span class="id" type="var">tp</span>,<span class="id" type="var">x</span> ⇒ <span class="id" type="var">tk</span> ++ (<span class="id" type="var">tokenize_helper</span> <span class="id" type="var">tp</span> [<span class="id" type="var">x</span>] <span class="id" type="var">xs'</span>)<br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span> %<span class="id" type="var">char</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">tokenize</span> (<span class="id" type="var">s</span> : <span class="id" type="var">string</span>) : <span class="id" type="var">list</span> <span class="id" type="var">string</span> :=<br/>
<span class="id" type="var">map</span> <span class="id" type="var">string_of_list</span> (<span class="id" type="var">tokenize_helper</span> <span class="id" type="var">white</span> [] (<span class="id" type="var">list_of_string</span> <span class="id" type="var">s</span>)).<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">tokenize_ex1</span> : <br/>
<span class="id" type="var">tokenize</span> "abc12==3 223*(3+(a+c))" %<span class="id" type="var">string</span><br/>
= ["abc"; "12"; "=="; "3"; "223";<br/>
"×"; "("; "3"; "+"; "(";<br/>
"a"; "+"; "c"; ")"; ")"]%<span class="id" type="var">string</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab439"></a><h2 class="section">Parsing</h2>
</div>
<div class="code code-space">
<br/>
</div>
<div class="doc">
<a name="lab440"></a><h3 class="section">Options with Errors</h3>
</div>
<div class="code code-space">
<br/>
<span class="comment">(* An option with error messages. *)</span><br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">optionE</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">SomeE</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">optionE</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">NoneE</span> : <span class="id" type="var">string</span> <span style="font-family: arial;">→</span> <span class="id" type="var">optionE</span> <span class="id" type="var">X</span>.<br/>
<br/>
<span class="id" type="keyword">Implicit Arguments</span> <span class="id" type="var">SomeE</span> [[<span class="id" type="var">X</span>]].<br/>
<span class="id" type="keyword">Implicit Arguments</span> <span class="id" type="var">NoneE</span> [[<span class="id" type="var">X</span>]].<br/>
<br/>
<span class="comment">(* Some syntactic sugar to make writing nested match-expressions on<br/>
optionE more convenient. *)</span><br/>
<br/>
<span class="id" type="keyword">Notation</span> "'DO' ( x , y ) <== e1 ; e2" <br/>
:= (<span class="id" type="keyword">match</span> <span class="id" type="var">e1</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">SomeE</span> (<span class="id" type="var">x</span>,<span class="id" type="var">y</span>) ⇒ <span class="id" type="var">e2</span><br/>
| <span class="id" type="var">NoneE</span> <span class="id" type="var">err</span> ⇒ <span class="id" type="var">NoneE</span> <span class="id" type="var">err</span><br/>
<span class="id" type="keyword">end</span>)<br/>
(<span class="id" type="var">right</span> <span class="id" type="var">associativity</span>, <span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 60).<br/>
<br/>
<span class="id" type="keyword">Notation</span> "'DO' ( x , y ) <-- e1 ; e2 'OR' e3" <br/>
:= (<span class="id" type="keyword">match</span> <span class="id" type="var">e1</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">SomeE</span> (<span class="id" type="var">x</span>,<span class="id" type="var">y</span>) ⇒ <span class="id" type="var">e2</span><br/>
| <span class="id" type="var">NoneE</span> <span class="id" type="var">err</span> ⇒ <span class="id" type="var">e3</span><br/>
<span class="id" type="keyword">end</span>)<br/>
(<span class="id" type="var">right</span> <span class="id" type="var">associativity</span>, <span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 60, <span class="id" type="var">e2</span> <span class="id" type="tactic">at</span> <span class="id" type="var">next</span> <span class="id" type="var">level</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab441"></a><h3 class="section">Symbol Table</h3>
</div>
<div class="code code-space">
<br/>
<span class="comment">(* Build a mapping from <span class="inlinecode"><span class="id" type="var">tokens</span></span> to <span class="inlinecode"><span class="id" type="var">nats</span></span>. A real parser would do<br/>
this incrementally as it encountered new symbols, but passing<br/>
around the symbol table inside the parsing functions is a bit<br/>
inconvenient, so instead we do it as a first pass. *)</span><br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">build_symtable</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) : (<span class="id" type="var">token</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ (<span class="id" type="keyword">fun</span> <span class="id" type="var">s</span> ⇒ <span class="id" type="var">n</span>)<br/>
| <span class="id" type="var">x</span>::<span class="id" type="var">xs</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<span class="id" type="var">forallb</span> <span class="id" type="var">isLowerAlpha</span> (<span class="id" type="var">list_of_string</span> <span class="id" type="var">x</span>))<br/>
<span class="id" type="keyword">then</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">s</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">string_dec</span> <span class="id" type="var">s</span> <span class="id" type="var">x</span> <span class="id" type="keyword">then</span> <span class="id" type="var">n</span> <span class="id" type="keyword">else</span> (<span class="id" type="var">build_symtable</span> <span class="id" type="var">xs</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) <span class="id" type="var">s</span>))<br/>
<span class="id" type="keyword">else</span> <span class="id" type="var">build_symtable</span> <span class="id" type="var">xs</span> <span class="id" type="var">n</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab442"></a><h3 class="section">Generic Combinators for Building Parsers</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Open</span> <span class="id" type="keyword">Scope</span> <span class="id" type="var">string_scope</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parser</span> (<span class="id" type="var">T</span> : <span class="id" type="keyword">Type</span>) := <br/>
<span class="id" type="var">list</span> <span class="id" type="var">token</span> <span style="font-family: arial;">→</span> <span class="id" type="var">optionE</span> (<span class="id" type="var">T</span> × <span class="id" type="var">list</span> <span class="id" type="var">token</span>).<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">many_helper</span> {<span class="id" type="var">T</span>} (<span class="id" type="var">p</span> : <span class="id" type="var">parser</span> <span class="id" type="var">T</span>) <span class="id" type="var">acc</span> <span class="id" type="var">steps</span> <span class="id" type="var">xs</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span>, <span class="id" type="var">p</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span><br/>
| 0, <span class="id" type="var">_</span> ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">_</span>, <span class="id" type="var">NoneE</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">SomeE</span> ((<span class="id" type="var">rev</span> <span class="id" type="var">acc</span>), <span class="id" type="var">xs</span>)<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span>, <span class="id" type="var">SomeE</span> (<span class="id" type="var">t</span>, <span class="id" type="var">xs'</span>) ⇒ <span class="id" type="var">many_helper</span> <span class="id" type="var">p</span> (<span class="id" type="var">t</span>::<span class="id" type="var">acc</span>) <span class="id" type="var">steps'</span> <span class="id" type="var">xs'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="comment">(* A (step-indexed) parser which expects zero or more <span class="inlinecode"><span class="id" type="var">p</span></span>s *)</span><br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">many</span> {<span class="id" type="var">T</span>} (<span class="id" type="var">p</span> : <span class="id" type="var">parser</span> <span class="id" type="var">T</span>) (<span class="id" type="var">steps</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">parser</span> (<span class="id" type="var">list</span> <span class="id" type="var">T</span>) :=<br/>
<span class="id" type="var">many_helper</span> <span class="id" type="var">p</span> [] <span class="id" type="var">steps</span>.<br/>
<br/>
<span class="comment">(* A parser which expects a given token, followed by p *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">firstExpect</span> {<span class="id" type="var">T</span>} (<span class="id" type="var">t</span> : <span class="id" type="var">token</span>) (<span class="id" type="var">p</span> : <span class="id" type="var">parser</span> <span class="id" type="var">T</span>) : <span class="id" type="var">parser</span> <span class="id" type="var">T</span> :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">xs</span> ⇒ <span class="id" type="keyword">match</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">x</span>::<span class="id" type="var">xs'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">string_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <br/>
<span class="id" type="keyword">then</span> <span class="id" type="var">p</span> <span class="id" type="var">xs'</span> <br/>
<span class="id" type="keyword">else</span> <span class="id" type="var">NoneE</span> ("expected '" ++ <span class="id" type="var">t</span> ++ "'.")<br/>
| [] ⇒ <span class="id" type="var">NoneE</span> ("expected '" ++ <span class="id" type="var">t</span> ++ "'.")<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="comment">(* A parser which expects a particular token *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">expect</span> (<span class="id" type="var">t</span> : <span class="id" type="var">token</span>) : <span class="id" type="var">parser</span> <span class="id" type="var">unit</span> :=<br/>
<span class="id" type="var">firstExpect</span> <span class="id" type="var">t</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">xs</span> ⇒ <span class="id" type="var">SomeE</span>(<span class="id" type="var">tt</span>, <span class="id" type="var">xs</span>)).<br/>
<br/>
</div>
<div class="doc">
<a name="lab443"></a><h3 class="section">A Recursive-Descent Parser for Imp</h3>
</div>
<div class="code code-space">
<br/>
<span class="comment">(* Identifiers *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parseIdentifier</span> (<span class="id" type="var">symtable</span> :<span class="id" type="var">string</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>) (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) <br/>
: <span class="id" type="var">optionE</span> (<span class="id" type="var">id</span> × <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ <span class="id" type="var">NoneE</span> "Expected identifier"<br/>
| <span class="id" type="var">x</span>::<span class="id" type="var">xs'</span> ⇒ <br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">forallb</span> <span class="id" type="var">isLowerAlpha</span> (<span class="id" type="var">list_of_string</span> <span class="id" type="var">x</span>) <span class="id" type="keyword">then</span><br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">Id</span> (<span class="id" type="var">symtable</span> <span class="id" type="var">x</span>), <span class="id" type="var">xs'</span>)<br/>
<span class="id" type="keyword">else</span><br/>
<span class="id" type="var">NoneE</span> ("Illegal identifier:'" ++ <span class="id" type="var">x</span> ++ "'")<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="comment">(* Numbers *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parseNumber</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) : <span class="id" type="var">optionE</span> (<span class="id" type="var">nat</span> × <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">xs</span> <span class="id" type="keyword">with</span> <br/>
| [] ⇒ <span class="id" type="var">NoneE</span> "Expected number"<br/>
| <span class="id" type="var">x</span>::<span class="id" type="var">xs'</span> ⇒ <br/>
<span class="id" type="keyword">if</span> <span class="id" type="var">forallb</span> <span class="id" type="var">isDigit</span> (<span class="id" type="var">list_of_string</span> <span class="id" type="var">x</span>) <span class="id" type="keyword">then</span><br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">fold_left</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">n</span> <span class="id" type="var">d</span> ⇒<br/>
10 × <span class="id" type="var">n</span> + (<span class="id" type="var">nat_of_ascii</span> <span class="id" type="var">d</span> - <span class="id" type="var">nat_of_ascii</span> "0"%<span class="id" type="var">char</span>))<br/>
(<span class="id" type="var">list_of_string</span> <span class="id" type="var">x</span>)<br/>
0,<br/>
<span class="id" type="var">xs'</span>)<br/>
<span class="id" type="keyword">else</span> <br/>
<span class="id" type="var">NoneE</span> "Expected number"<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="comment">(* Parse arithmetic expressions *)</span><br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">parsePrimaryExp</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) <span class="id" type="var">symtable</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) <br/>
: <span class="id" type="var">optionE</span> (<span class="id" type="var">aexp</span> × <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">i</span>, <span class="id" type="var">rest</span>) <-- <span class="id" type="var">parseIdentifier</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span> ;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">AId</span> <span class="id" type="var">i</span>, <span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">n</span>, <span class="id" type="var">rest</span>) <-- <span class="id" type="var">parseNumber</span> <span class="id" type="var">xs</span> ;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">ANum</span> <span class="id" type="var">n</span>, <span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> (<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest</span>) <== <span class="id" type="var">firstExpect</span> "(" (<span class="id" type="var">parseSumExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>, <span class="id" type="var">rest'</span>) <== <span class="id" type="var">expect</span> ")" <span class="id" type="var">rest</span> ;<br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">e</span>,<span class="id" type="var">rest'</span>))<br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">with</span> <span class="id" type="var">parseProductExp</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) <span class="id" type="var">symtable</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒ <br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest</span>) <== <br/>
<span class="id" type="var">parsePrimaryExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span> ;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">es</span>, <span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">many</span> (<span class="id" type="var">firstExpect</span> "×" (<span class="id" type="var">parsePrimaryExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>)) <span class="id" type="var">steps'</span> <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">fold_left</span> <span class="id" type="var">AMult</span> <span class="id" type="var">es</span> <span class="id" type="var">e</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">with</span> <span class="id" type="var">parseSumExp</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) <span class="id" type="var">symtable</span> (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒ <br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest</span>) <== <br/>
<span class="id" type="var">parseProductExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span> ;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">es</span>, <span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">many</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">xs</span> ⇒<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest'</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> "+" (<span class="id" type="var">parseProductExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> ( (<span class="id" type="var">true</span>, <span class="id" type="var">e</span>), <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">firstExpect</span> "-" (<span class="id" type="var">parseProductExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> ( (<span class="id" type="var">false</span>, <span class="id" type="var">e</span>), <span class="id" type="var">rest'</span>))<br/>
<span class="id" type="var">steps'</span> <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">fold_left</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">e0</span> <span class="id" type="var">term</span> ⇒ <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">term</span> <span class="id" type="keyword">with</span> <br/>
(<span class="id" type="var">true</span>, <span class="id" type="var">e</span>) ⇒ <span class="id" type="var">APlus</span> <span class="id" type="var">e0</span> <span class="id" type="var">e</span><br/>
| (<span class="id" type="var">false</span>, <span class="id" type="var">e</span>) ⇒ <span class="id" type="var">AMinus</span> <span class="id" type="var">e0</span> <span class="id" type="var">e</span><br/>
<span class="id" type="keyword">end</span>)<br/>
<span class="id" type="var">es</span> <span class="id" type="var">e</span>, <br/>
<span class="id" type="var">rest'</span>) <br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parseAExp</span> := <span class="id" type="var">parseSumExp</span>.<br/>
<br/>
<span class="comment">(* Parsing boolean expressions. *)</span><br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">parseAtomicExp</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">symtable</span> : <span class="id" type="var">string</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>) (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>,<span class="id" type="var">rest</span>) <-- <span class="id" type="var">expect</span> "true" <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">BTrue</span>,<span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">u</span>,<span class="id" type="var">rest</span>) <-- <span class="id" type="var">expect</span> "false" <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">BFalse</span>,<span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest</span>) <-- <span class="id" type="var">firstExpect</span> "not" (<span class="id" type="var">parseAtomicExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">BNot</span> <span class="id" type="var">e</span>, <span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest</span>) <-- <span class="id" type="var">firstExpect</span> "(" (<span class="id" type="var">parseConjunctionExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
(<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>,<span class="id" type="var">rest'</span>) <== <span class="id" type="var">expect</span> ")" <span class="id" type="var">rest</span>; <span class="id" type="var">SomeE</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest'</span>))<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest</span>) <== <span class="id" type="var">parseProductExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span> ;<br/>
(<span class="id" type="var">DO</span> (<span class="id" type="var">e'</span>, <span class="id" type="var">rest'</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> "==" (<span class="id" type="var">parseAExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span> ;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">BEq</span> <span class="id" type="var">e</span> <span class="id" type="var">e'</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e'</span>, <span class="id" type="var">rest'</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> "≤" (<span class="id" type="var">parseAExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span> ;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">BLe</span> <span class="id" type="var">e</span> <span class="id" type="var">e'</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="var">OR</span><br/>
<span class="id" type="var">NoneE</span> "Expected '==' or '≤' after arithmetic expression")<br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">with</span> <span class="id" type="var">parseConjunctionExp</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">symtable</span> : <span class="id" type="var">string</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>) (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest</span>) <== <br/>
<span class="id" type="var">parseAtomicExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span> ;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">es</span>, <span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">many</span> (<span class="id" type="var">firstExpect</span> "&&" (<span class="id" type="var">parseAtomicExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>)) <span class="id" type="var">steps'</span> <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">fold_left</span> <span class="id" type="var">BAnd</span> <span class="id" type="var">es</span> <span class="id" type="var">e</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parseBExp</span> := <span class="id" type="var">parseConjunctionExp</span>.<br/>
<br/>
<span class="comment">(* <br/>
Eval compute in <br/>
(parseProductExp 100 (tokenize "x*y*(x*x)*x")).<br/>
<br/>
Eval compute in <br/>
(parseDisjunctionExp 100 (tokenize "not((x==x||x*x<=(x*x)*x)&&x==x)")). <br/>
*)</span><br/>
<br/>
<span class="comment">(* Parsing commands *)</span><br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">parseSimpleCommand</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">symtable</span>:<span class="id" type="var">string</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>) (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒ <br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>, <span class="id" type="var">rest</span>) <-- <span class="id" type="var">expect</span> "SKIP" <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">SomeE</span> (<span class="id" type="var">SKIP</span>, <span class="id" type="var">rest</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> "IF" (<span class="id" type="var">parseBExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">c</span>,<span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">firstExpect</span> "THEN" (<span class="id" type="var">parseSequencedCommand</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">c'</span>,<span class="id" type="var">rest''</span>) <== <br/>
<span class="id" type="var">firstExpect</span> "ELSE" (<span class="id" type="var">parseSequencedCommand</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest'</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>,<span class="id" type="var">rest'''</span>) <== <br/>
<span class="id" type="var">expect</span> "END" <span class="id" type="var">rest''</span>;<br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">IFB</span> <span class="id" type="var">e</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c'</span> <span class="id" type="var">FI</span>, <span class="id" type="var">rest'''</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">e</span>,<span class="id" type="var">rest</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> "WHILE" (<span class="id" type="var">parseBExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">c</span>,<span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">firstExpect</span> "DO" (<span class="id" type="var">parseSequencedCommand</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">u</span>,<span class="id" type="var">rest''</span>) <== <br/>
<span class="id" type="var">expect</span> "END" <span class="id" type="var">rest'</span>;<br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">WHILE</span> <span class="id" type="var">e</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>, <span class="id" type="var">rest''</span>)<br/>
<span class="id" type="var">OR</span> <span class="id" type="var">DO</span> (<span class="id" type="var">i</span>, <span class="id" type="var">rest</span>) <== <br/>
<span class="id" type="var">parseIdentifier</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">e</span>, <span class="id" type="var">rest'</span>) <== <br/>
<span class="id" type="var">firstExpect</span> ":=" (<span class="id" type="var">parseAExp</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">i</span> ::= <span class="id" type="var">e</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="keyword">end</span><br/>
<br/>
<span class="id" type="keyword">with</span> <span class="id" type="var">parseSequencedCommand</span> (<span class="id" type="var">steps</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">symtable</span>:<span class="id" type="var">string</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>) (<span class="id" type="var">xs</span> : <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">steps</span> <span class="id" type="keyword">with</span><br/>
| 0 ⇒ <span class="id" type="var">NoneE</span> "Too many recursive calls"<br/>
| <span class="id" type="var">S</span> <span class="id" type="var">steps'</span> ⇒ <br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">c</span>, <span class="id" type="var">rest</span>) <== <br/>
<span class="id" type="var">parseSimpleCommand</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span> <span class="id" type="var">xs</span>;<br/>
<span class="id" type="var">DO</span> (<span class="id" type="var">c'</span>, <span class="id" type="var">rest'</span>) <-- <br/>
<span class="id" type="var">firstExpect</span> ";;" (<span class="id" type="var">parseSequencedCommand</span> <span class="id" type="var">steps'</span> <span class="id" type="var">symtable</span>) <span class="id" type="var">rest</span>;<br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">c</span> ;; <span class="id" type="var">c'</span>, <span class="id" type="var">rest'</span>)<br/>
<span class="id" type="var">OR</span><br/>
<span class="id" type="var">SomeE</span>(<span class="id" type="var">c</span>, <span class="id" type="var">rest</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">bignumber</span> := 1000.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">parse</span> (<span class="id" type="var">str</span> : <span class="id" type="var">string</span>) : <span class="id" type="var">optionE</span> (<span class="id" type="var">com</span> × <span class="id" type="var">list</span> <span class="id" type="var">token</span>) :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">tokens</span> := <span class="id" type="var">tokenize</span> <span class="id" type="var">str</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="var">parseSequencedCommand</span> <span class="id" type="var">bignumber</span> (<span class="id" type="var">build_symtable</span> <span class="id" type="var">tokens</span> 0) <span class="id" type="var">tokens</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab444"></a><h1 class="section">Examples</h1>
</div>
<div class="code code-space">
<br/>
<span class="comment">(*<br/>
Eval compute in parse "<br/>
IF x == y + 1 + 2 - y * 6 + 3 THEN<br/>
x := x * 1;;<br/>
y := 0<br/>
ELSE<br/>
SKIP<br/>
END ".<br/>
====><br/>
SomeE<br/>
(IFB BEq (AId (Id 0))<br/>
(APlus<br/>
(AMinus (APlus (APlus (AId (Id 1)) (ANum 1)) (ANum 2))<br/>
(AMult (AId (Id 1)) (ANum 6))) <br/>
(ANum 3))<br/>
THEN Id 0 ::= AMult (AId (Id 0)) (ANum 1);; Id 1 ::= ANum 0<br/>
ELSE SKIP FI, <span class="inlinecode"></span>)<br/>
*)</span><br/>
<br/>
<span class="comment">(*<br/>
Eval compute in parse "<br/>
SKIP;;<br/>
z:=x*y*(x*x);;<br/>
WHILE x==x DO<br/>
IF z <= z*z && not x == 2 THEN<br/>
x := z;;<br/>
y := z<br/>
ELSE<br/>
SKIP<br/>
END;;<br/>
SKIP<br/>
END;;<br/>
x:=z ".<br/>
====> <br/>
SomeE<br/>
(SKIP;;<br/>
Id 0 ::= AMult (AMult (AId (Id 1)) (AId (Id 2)))<br/>
(AMult (AId (Id 1)) (AId (Id 1)));;<br/>
WHILE BEq (AId (Id 1)) (AId (Id 1)) DO <br/>
IFB BAnd (BLe (AId (Id 0)) (AMult (AId (Id 0)) (AId (Id 0))))<br/>
(BNot (BEq (AId (Id 1)) (ANum 2)))<br/>
THEN Id 1 ::= AId (Id 0);; Id 2 ::= AId (Id 0) <br/>
ELSE SKIP FI;; <br/>
SKIP <br/>
END;; <br/>
Id 1 ::= AId (Id 0), <br/>
<span class="inlinecode"></span>) <br/>
*)</span><br/>
<br/>
<span class="comment">(*<br/>
Eval compute in parse "<br/>
SKIP;;<br/>
z:=x*y*(x*x);;<br/>
WHILE x==x DO<br/>
IF z <= z*z && not x == 2 THEN<br/>
x := z;;<br/>
y := z<br/>
ELSE<br/>
SKIP<br/>
END;;<br/>
SKIP<br/>
END;;<br/>
x:=z ".<br/>
=====> <br/>
SomeE<br/>
(SKIP;;<br/>
Id 0 ::= AMult (AMult (AId (Id 1)) (AId (Id 2)))<br/>
(AMult (AId (Id 1)) (AId (Id 1)));;<br/>
WHILE BEq (AId (Id 1)) (AId (Id 1)) DO <br/>
IFB BAnd (BLe (AId (Id 0)) (AMult (AId (Id 0)) (AId (Id 0))))<br/>
(BNot (BEq (AId (Id 1)) (ANum 2)))<br/>
THEN Id 1 ::= AId (Id 0);; <br/>
Id 2 ::= AId (Id 0) <br/>
ELSE SKIP <br/>
FI;; <br/>
SKIP <br/>
END;;<br/>
Id 1 ::= AId (Id 0), <br/>
<span class="inlinecode"></span>).<br/>
*)</span><br/>
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