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Extraction.html
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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>Extraction: Extracting ML from Coq</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
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<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Extraction<span class="subtitle">Extracting ML from Coq</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* $Date: 2013-01-16 22:29:57 -0500 (Wed, 16 Jan 2013) $ *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab452"></a><h1 class="section">Basic Extraction</h1>
<div class="paragraph"> </div>
In its simplest form, program extraction from Coq is completely straightforward.
<div class="paragraph"> </div>
First we say what language we want to extract into. Options are OCaml (the
most mature), Haskell (which mostly works), and Scheme (a bit out
of date).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extraction</span> <span class="id" type="var">Language</span> <span class="id" type="var">Ocaml</span>.<br/>
<br/>
</div>
<div class="doc">
Now we load up the Coq environment with some definitions, either
directly or by importing them from other modules.
</div>
<div class="code code-tight">
<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">ImpCEvalFun</span>.<br/>
<br/>
</div>
<div class="doc">
Finally, we tell Coq the name of a definition to extract and the
name of a file to put the extracted code into.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extraction</span> "imp1.ml" <span class="id" type="var">ceval_step</span>.<br/>
<br/>
</div>
<div class="doc">
When Coq processes this command, it generates a file <span class="inlinecode"><span class="id" type="var">imp1.ml</span></span>
containing an extracted version of <span class="inlinecode"><span class="id" type="var">ceval_step</span></span>, together with
everything that it recursively depends on. Have a look at this
file now.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab453"></a><h1 class="section">Controlling Extraction of Specific Types</h1>
<div class="paragraph"> </div>
We can tell Coq to extract certain <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definitions to
specific OCaml types. For each one, we must say
<div class="paragraph"> </div>
<ul class="doclist">
<li> how the Coq type itself should be represented in OCaml, and
</li>
<li> how each constructor should be translated.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extract</span> <span class="id" type="keyword">Inductive</span> <span class="id" type="var">bool</span> ⇒ "bool" [ "true" "false" ].<br/>
<br/>
</div>
<div class="doc">
Also, for non-enumeration types (where the constructors take
arguments), we give an OCaml expression that can be used as a
"recursor" over elements of the type. (Think Church numerals.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extract</span> <span class="id" type="keyword">Inductive</span> <span class="id" type="var">nat</span> ⇒ "int"<br/>
[ "0" "(fun x <span style="font-family: arial;">→</span> x + 1)" ]<br/>
"(fun zero succ n <span style="font-family: arial;">→</span><br/>
if n=0 then zero () else succ (n-1))".<br/>
<br/>
</div>
<div class="doc">
We can also extract defined constants to specific OCaml terms or
operators.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">plus</span> ⇒ "( + )".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">mult</span> ⇒ "( × )".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">beq_nat</span> ⇒ "( = )".<br/>
<br/>
</div>
<div class="doc">
Important: It is entirely <i>your responsibility</i> to make sure that
the translations you're proving make sense. For example, it might
be tempting to include this one
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">minus</span> ⇒ "( - )".
<div class="paragraph"> </div>
</div>
but doing so could lead to serious confusion! (Why?)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extraction</span> "imp2.ml" <span class="id" type="var">ceval_step</span>.<br/>
<br/>
</div>
<div class="doc">
Have a look at the file <span class="inlinecode"><span class="id" type="var">imp2.ml</span></span>. Notice how the fundamental
definitions have changed from <span class="inlinecode"><span class="id" type="var">imp1.ml</span></span>.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab454"></a><h1 class="section">A Complete Example</h1>
<div class="paragraph"> </div>
To use our extracted evaluator to run Imp programs, all we need to
add is a tiny driver program that calls the evaluator and somehow
prints out the result.
<div class="paragraph"> </div>
For simplicity, we'll print results by dumping out the first four
memory locations in the final state.
<div class="paragraph"> </div>
Also, to make it easier to type in examples, let's extract a
parser from the <span class="inlinecode"><span class="id" type="var">ImpParser</span></span> Coq module. To do this, we need a few
more declarations to set up the right correspondence between Coq
strings and lists of OCaml characters.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Ascii</span> <span class="id" type="var">String</span>.<br/>
<span class="id" type="var">Extract</span> <span class="id" type="keyword">Inductive</span> <span class="id" type="var">ascii</span> ⇒ <span class="id" type="var">char</span><br/>
[<br/>
"(× If this appears, you're using Ascii internals. Please don't *) (fun (b0,b1,b2,b3,b4,b5,b6,b7) <span style="font-family: arial;">→</span> let f b i = if b then 1 lsl i else 0 in Char.chr (f b0 0 + f b1 1 + f b2 2 + f b3 3 + f b4 4 + f b5 5 + f b6 6 + f b7 7))"<br/>
]<br/>
"(× If this appears, you're using Ascii internals. Please don't *) (fun f c <span style="font-family: arial;">→</span> let n = Char.code c in let h i = (n land (1 lsl i)) ≠ 0 in f (h 0) (h 1) (h 2) (h 3) (h 4) (h 5) (h 6) (h 7))".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">zero</span> ⇒ "'\000'".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">one</span> ⇒ "'\001'".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Constant</span> <span class="id" type="var">shift</span> ⇒<br/>
"fun b c <span style="font-family: arial;">→</span> Char.chr (((Char.code c) lsl 1) land 255 + if b then 1 else 0)".<br/>
<span class="id" type="var">Extract</span> <span class="id" type="var">Inlined</span> <span class="id" type="var">Constant</span> <span class="id" type="var">ascii_dec</span> ⇒ "(=)".<br/>
<br/>
</div>
<div class="doc">
We also need one more variant of booleans.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Extract</span> <span class="id" type="keyword">Inductive</span> <span class="id" type="var">sumbool</span> ⇒ "bool" ["true" "false"].<br/>
<br/>
</div>
<div class="doc">
The extraction is the same as always.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Imp</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">ImpParser</span>.<br/>
<span class="id" type="var">Extraction</span> "imp.ml" <span class="id" type="var">empty_state</span> <span class="id" type="var">ceval_step</span> <span class="id" type="var">parse</span>.<br/>
<br/>
</div>
<div class="doc">
Now let's run our generated Imp evaluator. First, have a look at
<span class="inlinecode"><span class="id" type="var">impdriver.ml</span></span>. (This was written by hand, not extracted.)
<div class="paragraph"> </div>
Next, compile the driver together with the extracted code and
execute it, as follows.
<pre>
ocamlc -w -20 -w -26 -o impdriver imp.mli imp.ml impdriver.ml
./impdriver
</pre>
(The <span class="inlinecode">-<span class="id" type="var">w</span></span> flags to <span class="inlinecode"><span class="id" type="var">ocamlc</span></span> are just there to suppress a few
spurious warnings.)
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab455"></a><h1 class="section">Discussion</h1>
<div class="paragraph"> </div>
Since we've proved that the <span class="inlinecode"><span class="id" type="var">ceval_step</span></span> function behaves the same
as the <span class="inlinecode"><span class="id" type="var">ceval</span></span> relation in an appropriate sense, the extracted
program can be viewed as a <i>certified</i> Imp interpreter. (Of
course, the parser is not certified in any interesting sense,
since we didn't prove anything about it.)
</div>
<div class="code code-tight">
</div>
</div>
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