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ImpCEvalFun.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>ImpCEvalFun: Evaluation Function for Imp</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">ImpCEvalFun<span class="subtitle">Evaluation Function for Imp</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
<div class="paragraph"> </div>
We saw in the <a href="Imp.html"><span class="inlineref">Imp</span></a> chapter how a naive approach to defining a
function representing evaluation for Imp runs into difficulties.
There, we adopted the solution of changing from a functional to a
relational definition of evaluation. In this optional chapter, we
consider strategies for getting the functional approach to
work.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab355"></a><h1 class="section">A Broken Evaluator</h1>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library/Coq.omega.Omega.html#"><span class="id" type="library">Coq.omega.Omega</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Arith.Arith.html#"><span class="id" type="library">Coq.Arith.Arith</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Imp.html#"><span class="id" type="library">Imp</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Maps.html#"><span class="id" type="library">Maps</span></a>.<br/>
<br/>
</div>
<div class="doc">
Here was our first try at an evaluation function for commands,
omitting <span class="inlinecode"><span class="id" type="var">WHILE</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="ceval_step1"><span class="id" type="definition">ceval_step1</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">c</span> : <a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a>) : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Imp.html#::'SKIP'"><span class="id" type="notation">SKIP</span></a> ⇒<br/>
<a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
| <span class="id" type="var">l</span> ::= <span class="id" type="var">a<sub>1</sub></span> ⇒<br/>
<a class="idref" href="Maps.html#t_update"><span class="id" type="definition">t_update</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">l</span> (<a class="idref" href="Imp.html#aeval"><span class="id" type="definition">aeval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">a<sub>1</sub></span>)<br/>
| <span class="id" type="var">c<sub>1</sub></span> ;; <span class="id" type="var">c<sub>2</sub></span> ⇒<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">st'</span> := <a class="idref" href="ImpCEvalFun.html#ceval_step1"><span class="id" type="definition">ceval_step1</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="keyword">in</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step1"><span class="id" type="definition">ceval_step1</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a> <span class="id" type="var">c<sub>2</sub></span><br/>
| <a class="idref" href="Imp.html#::'IFB'_x_'THEN'_x_'ELSE'_x_'FI'"><span class="id" type="notation">IFB</span></a> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">FI</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b</span>)<br/>
<span class="id" type="keyword">then</span> <a class="idref" href="ImpCEvalFun.html#ceval_step1"><span class="id" type="definition">ceval_step1</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="ImpCEvalFun.html#ceval_step1"><span class="id" type="definition">ceval_step1</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>2</sub></span><br/>
| <a class="idref" href="Imp.html#::'WHILE'_x_'DO'_x_'END'"><span class="id" type="notation">WHILE</span></a> <span class="id" type="var">b<sub>1</sub></span> <span class="id" type="var">DO</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">END</span> ⇒<br/>
<a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="comment">(* bogus *)</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
As we remarked in chapter <a href="Imp.html"><span class="inlineref">Imp</span></a>, in a traditional functional
programming language like ML or Haskell we could write the WHILE
case as follows:
<pre>
| WHILE b<sub>1</sub> DO c<sub>1</sub> END => if (beval st b<sub>1</sub>) then ceval_step1 st (c<sub>1</sub>;;
WHILE b<sub>1</sub> DO c<sub>1</sub> END) else st
</pre>
Coq doesn't accept such a definition (<span class="inlinecode"><span class="id" type="var">Error</span>:</span> <span class="inlinecode"><span class="id" type="var">Cannot</span></span> <span class="inlinecode"><span class="id" type="var">guess</span></span>
<span class="inlinecode"><span class="id" type="var">decreasing</span></span> <span class="inlinecode"><span class="id" type="var">argument</span></span> <span class="inlinecode"><span class="id" type="var">of</span></span> <span class="inlinecode"><span class="id" type="var">fix</span></span>) because the function we want to
define is not guaranteed to terminate. Indeed, the changed
<span class="inlinecode"><span class="id" type="var">ceval_step1</span></span> function applied to the <span class="inlinecode"><span class="id" type="var">loop</span></span> program from <span class="inlinecode"><span class="id" type="var">Imp.v</span></span>
would never terminate. Since Coq is not just a functional
programming language, but also a consistent logic, any potentially
non-terminating function needs to be rejected. Here is an
invalid(!) Coq program showing what would go wrong if Coq allowed
non-terminating recursive functions:
<pre>
Fixpoint loop_false (n : nat) : False := loop_false n.
</pre>
That is, propositions like <span class="inlinecode"><span class="id" type="var">False</span></span> would become
provable (e.g., <span class="inlinecode"><span class="id" type="var">loop_false</span></span> <span class="inlinecode">0</span> would be a proof of <span class="inlinecode"><span class="id" type="var">False</span></span>), which
would be a disaster for Coq's logical consistency.
<div class="paragraph"> </div>
Thus, because it doesn't terminate on all inputs, the full version
of <span class="inlinecode"><span class="id" type="var">ceval_step1</span></span> cannot be written in Coq — at least not without
one additional trick...
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab356"></a><h1 class="section">A Step-Indexed Evaluator</h1>
<div class="paragraph"> </div>
The trick we need is to pass an <i>additional</i> parameter to the
evaluation function that tells it how long to run. Informally, we
start the evaluator with a certain amount of "gas" in its tank,
and we allow it to run until either it terminates in the usual way
<i>or</i> it runs out of gas, at which point we simply stop evaluating
and say that the final result is the empty memory. (We could also
say that the result is the current state at the point where the
evaluator runs out fo gas — it doesn't really matter because the
result is going to be wrong in either case!)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="ceval_step2"><span class="id" type="definition">ceval_step2</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">c</span> : <a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a>) (<span class="id" type="var">i</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> ⇒ <a class="idref" href="Imp.html#empty_state"><span class="id" type="definition">empty_state</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">i'</span> ⇒<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Imp.html#::'SKIP'"><span class="id" type="notation">SKIP</span></a> ⇒<br/>
<a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
| <span class="id" type="var">l</span> ::= <span class="id" type="var">a<sub>1</sub></span> ⇒<br/>
<a class="idref" href="Maps.html#t_update"><span class="id" type="definition">t_update</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">l</span> (<a class="idref" href="Imp.html#aeval"><span class="id" type="definition">aeval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">a<sub>1</sub></span>)<br/>
| <span class="id" type="var">c<sub>1</sub></span> ;; <span class="id" type="var">c<sub>2</sub></span> ⇒<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">st'</span> := <a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span> <span class="id" type="keyword">in</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="Imp.html#::'IFB'_x_'THEN'_x_'ELSE'_x_'FI'"><span class="id" type="notation">IFB</span></a> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">FI</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b</span>)<br/>
<span class="id" type="keyword">then</span> <a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="Imp.html#::'WHILE'_x_'DO'_x_'END'"><span class="id" type="notation">WHILE</span></a> <span class="id" type="var">b<sub>1</sub></span> <span class="id" type="var">DO</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">END</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b<sub>1</sub></span>)<br/>
<span class="id" type="keyword">then</span> <span class="id" type="keyword">let</span> <span class="id" type="var">st'</span> := <a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span> <span class="id" type="keyword">in</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step2"><span class="id" type="definition">ceval_step2</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="var">i'</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Note</i>: It is tempting to think that the index <span class="inlinecode"><span class="id" type="var">i</span></span> here is
counting the "number of steps of evaluation." But if you look
closely you'll see that this is not the case: for example, in the
rule for sequencing, the same <span class="inlinecode"><span class="id" type="var">i</span></span> is passed to both recursive
calls. Understanding the exact way that <span class="inlinecode"><span class="id" type="var">i</span></span> is treated will be
important in the proof of <span class="inlinecode"><span class="id" type="var">ceval__ceval_step</span></span>, which is given as
an exercise below.
<div class="paragraph"> </div>
One thing that is not so nice about this evaluator is that we
can't tell, from its result, whether it stopped because the
program terminated normally or because it ran out of gas. Our
next version returns an <span class="inlinecode"><span class="id" type="var">option</span></span> <span class="inlinecode"><span class="id" type="var">state</span></span> instead of just a <span class="inlinecode"><span class="id" type="var">state</span></span>,
so that we can distinguish between normal and abnormal
termination.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="ceval_step3"><span class="id" type="definition">ceval_step3</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">c</span> : <a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a>) (<span class="id" type="var">i</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>)<br/>
: <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#option"><span class="id" type="inductive">option</span></a> <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">i'</span> ⇒<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Imp.html#::'SKIP'"><span class="id" type="notation">SKIP</span></a> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
| <span class="id" type="var">l</span> ::= <span class="id" type="var">a<sub>1</sub></span> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> (<a class="idref" href="Maps.html#t_update"><span class="id" type="definition">t_update</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">l</span> (<a class="idref" href="Imp.html#aeval"><span class="id" type="definition">aeval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">a<sub>1</sub></span>))<br/>
| <span class="id" type="var">c<sub>1</sub></span> ;; <span class="id" type="var">c<sub>2</sub></span> ⇒<br/>
<span class="id" type="keyword">match</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span>) <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">st'</span> ⇒ <a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <span class="id" type="var">st'</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
| <a class="idref" href="Imp.html#::'IFB'_x_'THEN'_x_'ELSE'_x_'FI'"><span class="id" type="notation">IFB</span></a> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">FI</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b</span>)<br/>
<span class="id" type="keyword">then</span> <a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="Imp.html#::'WHILE'_x_'DO'_x_'END'"><span class="id" type="notation">WHILE</span></a> <span class="id" type="var">b<sub>1</sub></span> <span class="id" type="var">DO</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">END</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b<sub>1</sub></span>)<br/>
<span class="id" type="keyword">then</span> <span class="id" type="keyword">match</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span>) <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">st'</span> ⇒ <a class="idref" href="ImpCEvalFun.html#ceval_step3"><span class="id" type="definition">ceval_step3</span></a> <span class="id" type="var">st'</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
We can improve the readability of this version by introducing a
bit of auxiliary notation to hide the plumbing involved in
repeatedly matching against optional states.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "'LETOPT' x <== e<sub>1</sub> 'IN' e<sub>2</sub>"<br/>
:= (<span class="id" type="keyword">match</span> <span class="id" type="var">e<sub>1</sub></span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">x</span> ⇒ <span class="id" type="var">e<sub>2</sub></span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><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">Fixpoint</span> <a name="ceval_step"><span class="id" type="definition">ceval_step</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">c</span> : <a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a>) (<span class="id" type="var">i</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>)<br/>
: <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#option"><span class="id" type="inductive">option</span></a> <a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">i'</span> ⇒<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Imp.html#::'SKIP'"><span class="id" type="notation">SKIP</span></a> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
| <span class="id" type="var">l</span> ::= <span class="id" type="var">a<sub>1</sub></span> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> (<a class="idref" href="Maps.html#t_update"><span class="id" type="definition">t_update</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">l</span> (<a class="idref" href="Imp.html#aeval"><span class="id" type="definition">aeval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">a<sub>1</sub></span>))<br/>
| <span class="id" type="var">c<sub>1</sub></span> ;; <span class="id" type="var">c<sub>2</sub></span> ⇒<br/>
<a class="idref" href="ImpCEvalFun.html#::'LETOPT'_x_'<=='_x_'IN'_x"><span class="id" type="notation">LETOPT</span></a> <span class="id" type="var">st'</span> <== <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span> <span class="id" type="var">IN</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st'</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="Imp.html#::'IFB'_x_'THEN'_x_'ELSE'_x_'FI'"><span class="id" type="notation">IFB</span></a> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">FI</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b</span>)<br/>
<span class="id" type="keyword">then</span> <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>2</sub></span> <span class="id" type="var">i'</span><br/>
| <a class="idref" href="Imp.html#::'WHILE'_x_'DO'_x_'END'"><span class="id" type="notation">WHILE</span></a> <span class="id" type="var">b<sub>1</sub></span> <span class="id" type="var">DO</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">END</span> ⇒<br/>
<span class="id" type="keyword">if</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">b<sub>1</sub></span>)<br/>
<span class="id" type="keyword">then</span> <a class="idref" href="ImpCEvalFun.html#::'LETOPT'_x_'<=='_x_'IN'_x"><span class="id" type="notation">LETOPT</span></a> <span class="id" type="var">st'</span> <== <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span> <span class="id" type="var">IN</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st'</span> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <span class="id" type="var">i'</span><br/>
<span class="id" type="keyword">else</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="test_ceval"><span class="id" type="definition">test_ceval</span></a> (<span class="id" type="var">st</span>:<a class="idref" href="Imp.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">c</span>:<a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a>) :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> 500 <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">st</span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> (<a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="Imp.html#X"><span class="id" type="definition">X</span></a>, <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="Imp.html#Y"><span class="id" type="definition">Y</span></a>, <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="Imp.html#Z"><span class="id" type="definition">Z</span></a>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="comment">(* Compute<br/>
(test_ceval empty_state<br/>
(X ::= ANum 2;;<br/>
IFB BLe (AId X) (ANum 1)<br/>
THEN Y ::= ANum 3<br/>
ELSE Z ::= ANum 4<br/>
FI)).<br/>
====><br/>
Some (2, 0, 4) *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab357"></a><h4 class="section">Exercise: 2 stars, recommended (pup_to_n)</h4>
Write an Imp program that sums the numbers from <span class="inlinecode">1</span> to
<span class="inlinecode"><span class="id" type="var">X</span></span> (inclusive: <span class="inlinecode">1</span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode">+</span> <span class="inlinecode">...</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">X</span></span>) in the variable <span class="inlinecode"><span class="id" type="var">Y</span></span>. Make sure
your solution satisfies the test that follows.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="pup_to_n"><span class="id" type="definition">pup_to_n</span></a> : <a class="idref" href="Imp.html#com"><span class="id" type="inductive">com</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="comment">(* <br/>
Example pup_to_n_1 :<br/>
test_ceval (t_update empty_state X 5) pup_to_n<br/>
= Some (0, 15, 0).<br/>
Proof. reflexivity. Qed.<br/>
*)</span><br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab358"></a><h4 class="section">Exercise: 2 stars, optional (peven)</h4>
Write a <span class="inlinecode"><span class="id" type="var">While</span></span> program that sets <span class="inlinecode"><span class="id" type="var">Z</span></span> to <span class="inlinecode">0</span> if <span class="inlinecode"><span class="id" type="var">X</span></span> is even and
sets <span class="inlinecode"><span class="id" type="var">Z</span></span> to <span class="inlinecode">1</span> otherwise. Use <span class="inlinecode"><span class="id" type="var">ceval_test</span></span> to test your
program.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab359"></a><h1 class="section">Relational vs. Step-Indexed Evaluation</h1>
<div class="paragraph"> </div>
As for arithmetic and boolean expressions, we'd hope that
the two alternative definitions of evaluation would actually
amount to the same thing in the end. This section shows that this
is the case.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ceval_step__ceval"><span class="id" type="lemma">ceval_step__ceval</span></a>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span>,<br/>
(<span style="font-family: arial;">∃</span><span class="id" type="var">i</span>, <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a>) <span style="font-family: arial;">→</span><br/>
<a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> / <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span style="font-family: arial;">⇓</span> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">i</span> <span class="id" type="var">E</span>].<br/>
<span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">st'</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">c</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">i</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">i'</span> ].<br/>
<br/>
- <span class="comment">(* i = 0 -- contradictory *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<br/>
- <span class="comment">(* i = S i' *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">c</span>;<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
+ <span class="comment">(* SKIP *)</span> <span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_Skip"><span class="id" type="constructor">E_Skip</span></a>.<br/>
+ <span class="comment">(* ::= *)</span> <span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_Ass"><span class="id" type="constructor">E_Ass</span></a>. <span class="id" type="tactic">reflexivity</span>.<br/>
<br/>
+ <span class="comment">(* ;; *)</span><br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i'</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Heqr1</span>.<br/>
* <span class="comment">(* Evaluation of r<sub>1</sub> terminates normally *)</span><br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_Seq"><span class="id" type="constructor">E_Seq</span></a> <span class="id" type="keyword">with</span> <span class="id" type="var">s</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqr1</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H<sub>1</sub></span>. <span class="id" type="tactic">assumption</span>.<br/>
* <span class="comment">(* Otherwise -- contradiction *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>1</sub></span>.<br/>
<br/>
+ <span class="comment">(* IFB *)</span><br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <span class="id" type="var">st</span> <span class="id" type="var">b</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Heqr</span>.<br/>
* <span class="comment">(* r = true *)</span><br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_IfTrue"><span class="id" type="constructor">E_IfTrue</span></a>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqr</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">assumption</span>.<br/>
* <span class="comment">(* r = false *)</span><br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_IfFalse"><span class="id" type="constructor">E_IfFalse</span></a>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqr</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">assumption</span>.<br/>
<br/>
+ <span class="comment">(* WHILE *)</span> <span class="id" type="tactic">destruct</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <span class="id" type="var">st</span> <span class="id" type="var">b</span>) <span class="id" type="var">eqn</span> :<span class="id" type="var">Heqr</span>.<br/>
* <span class="comment">(* r = true *)</span><br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st</span> <span class="id" type="var">c</span> <span class="id" type="var">i'</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Heqr1</span>.<br/>
{ <span class="comment">(* r<sub>1</sub> = Some s *)</span><br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_WhileLoop"><span class="id" type="constructor">E_WhileLoop</span></a> <span class="id" type="keyword">with</span> <span class="id" type="var">s</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqr</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqr1</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHi'</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H<sub>1</sub></span>. <span class="id" type="tactic">assumption</span>. }<br/>
{ <span class="comment">(* r<sub>1</sub> = None *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>1</sub></span>. }<br/>
* <span class="comment">(* r = false *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>1</sub></span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Imp.html#E_WhileEnd"><span class="id" type="constructor">E_WhileEnd</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">Heqr</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab360"></a><h4 class="section">Exercise: 4 stars (ceval_step__ceval_inf)</h4>
Write an informal proof of <span class="inlinecode"><span class="id" type="var">ceval_step__ceval</span></span>, following the
usual template. (The template for case analysis on an inductively
defined value should look the same as for induction, except that
there is no induction hypothesis.) Make your proof communicate
the main ideas to a human reader; do not simply transcribe the
steps of the formal proof.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ceval_step_more"><span class="id" type="lemma">ceval_step_more</span></a>: <span style="font-family: arial;">∀</span><span class="id" type="var">i<sub>1</sub></span> <span class="id" type="var">i<sub>2</sub></span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">c</span>,<br/>
<a class="idref" href="ImpCEvalFun.html#i<sub>1</sub>"><span class="id" type="variable">i<sub>1</sub></span></a> ≤ <a class="idref" href="ImpCEvalFun.html#i<sub>2</sub>"><span class="id" type="variable">i<sub>2</sub></span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <a class="idref" href="ImpCEvalFun.html#i<sub>1</sub>"><span class="id" type="variable">i<sub>1</sub></span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <a class="idref" href="ImpCEvalFun.html#i<sub>2</sub>"><span class="id" type="variable">i<sub>2</sub></span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">i<sub>1</sub></span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">i<sub>1</sub>'</span>]; <span class="id" type="tactic">intros</span> <span class="id" type="var">i<sub>2</sub></span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">c</span> <span class="id" type="var">Hle</span> <span class="id" type="var">Hceval</span>.<br/>
- <span class="comment">(* i<sub>1</sub> = 0 *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hceval</span>.<br/>
- <span class="comment">(* i<sub>1</sub> = S i<sub>1</sub>' *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">i<sub>2</sub></span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">i<sub>2</sub>'</span>]. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hle</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">Hle'</span>: <span class="id" type="var">i<sub>1</sub>'</span> ≤ <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="tactic">by</span> <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">c</span>.<br/>
+ <span class="comment">(* SKIP *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hceval</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* ::= *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hceval</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* ;; *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st</span> <span class="id" type="var">c<sub>1</sub></span> <span class="id" type="var">i<sub>1</sub>'</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Heqst1'o</span>.<br/>
* <span class="comment">(* st<sub>1</sub>'o = Some *)</span><br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">IHi1'</span> <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">Heqst1'o</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Heqst1'o</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>.<br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">IHi1'</span> <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
* <span class="comment">(* st<sub>1</sub>'o = None *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hceval</span>.<br/>
<br/>
+ <span class="comment">(* IFB *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <span class="id" type="var">st</span> <span class="id" type="var">b</span>); <span class="id" type="tactic">apply</span> (<span class="id" type="var">IHi1'</span> <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>;<br/>
<span class="id" type="tactic">assumption</span>.<br/>
<br/>
+ <span class="comment">(* WHILE *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="Imp.html#beval"><span class="id" type="definition">beval</span></a> <span class="id" type="var">st</span> <span class="id" type="var">b</span>); <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <span class="id" type="var">st</span> <span class="id" type="var">c</span> <span class="id" type="var">i<sub>1</sub>'</span>) <span class="id" type="var">eqn</span>: <span class="id" type="var">Heqst1'o</span>.<br/>
* <span class="comment">(* st<sub>1</sub>'o = Some *)</span><br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">IHi1'</span> <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">Heqst1'o</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Heqst1'o</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>.<br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">IHi1'</span> <span class="id" type="var">i<sub>2</sub>'</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
* <span class="comment">(* i<sub>1</sub>'o = None *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hceval</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hceval</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab361"></a><h4 class="section">Exercise: 3 stars, recommended (ceval__ceval_step)</h4>
Finish the following proof. You'll need <span class="inlinecode"><span class="id" type="var">ceval_step_more</span></span> in a
few places, as well as some basic facts about <span class="inlinecode">≤</span> and <span class="inlinecode"><span class="id" type="var">plus</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ceval__ceval_step"><span class="id" type="lemma">ceval__ceval_step</span></a>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span>,<br/>
<a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> / <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span style="font-family: arial;">⇓</span> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">i</span>, <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">Hce</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Hce</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ceval_and_ceval_step_coincide"><span class="id" type="lemma">ceval_and_ceval_step_coincide</span></a>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span>,<br/>
<a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> / <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span style="font-family: arial;">⇓</span> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a><br/>
<span style="font-family: arial;">↔</span> <span style="font-family: arial;">∃</span><span class="id" type="var">i</span>, <a class="idref" href="ImpCEvalFun.html#ceval_step"><span class="id" type="definition">ceval_step</span></a> <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> <a class="idref" href="ImpCEvalFun.html#i"><span class="id" type="variable">i</span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="ImpCEvalFun.html#st'"><span class="id" type="variable">st'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span>.<br/>
<span class="id" type="tactic">split</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval__ceval_step"><span class="id" type="axiom">ceval__ceval_step</span></a>. <span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval_step__ceval"><span class="id" type="lemma">ceval_step__ceval</span></a>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab362"></a><h1 class="section">Determinism of Evaluation Again</h1>
<div class="paragraph"> </div>
Using the fact that the relational and step-indexed definition of
evaluation are the same, we can give a slicker proof that the
evaluation <i>relation</i> is deterministic.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ceval_deterministic'"><span class="id" type="lemma">ceval_deterministic'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st<sub>1</sub></span> <span class="id" type="var">st<sub>2</sub></span>,<br/>
<a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> / <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span style="font-family: arial;">⇓</span> <a class="idref" href="ImpCEvalFun.html#st<sub>1</sub>"><span class="id" type="variable">st<sub>1</sub></span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="ImpCEvalFun.html#c"><span class="id" type="variable">c</span></a> / <a class="idref" href="ImpCEvalFun.html#st"><span class="id" type="variable">st</span></a> <span style="font-family: arial;">⇓</span> <a class="idref" href="ImpCEvalFun.html#st<sub>2</sub>"><span class="id" type="variable">st<sub>2</sub></span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="ImpCEvalFun.html#st<sub>1</sub>"><span class="id" type="variable">st<sub>1</sub></span></a> = <a class="idref" href="ImpCEvalFun.html#st<sub>2</sub>"><span class="id" type="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st<sub>1</sub></span> <span class="id" type="var">st<sub>2</sub></span> <span class="id" type="var">He<sub>1</sub></span> <span class="id" type="var">He<sub>2</sub></span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval__ceval_step"><span class="id" type="axiom">ceval__ceval_step</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">He<sub>1</sub></span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval__ceval_step"><span class="id" type="axiom">ceval__ceval_step</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">He<sub>2</sub></span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">He<sub>1</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">i<sub>1</sub></span> <span class="id" type="var">E<sub>1</sub></span>].<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">He<sub>2</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">i<sub>2</sub></span> <span class="id" type="var">E<sub>2</sub></span>].<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval_step_more"><span class="id" type="lemma">ceval_step_more</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">i<sub>2</sub></span> := <span class="id" type="var">i<sub>1</sub></span> + <span class="id" type="var">i<sub>2</sub></span>) <span class="id" type="keyword">in</span> <span class="id" type="var">E<sub>1</sub></span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="ImpCEvalFun.html#ceval_step_more"><span class="id" type="lemma">ceval_step_more</span></a> <span class="id" type="keyword">with</span> (<span class="id" type="var">i<sub>2</sub></span> := <span class="id" type="var">i<sub>1</sub></span> + <span class="id" type="var">i<sub>2</sub></span>) <span class="id" type="keyword">in</span> <span class="id" type="var">E<sub>2</sub></span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">E<sub>1</sub></span> <span class="id" type="keyword">in</span> <span class="id" type="var">E<sub>2</sub></span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">E<sub>2</sub></span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">omega</span>. <span class="id" type="tactic">omega</span>. <span class="id" type="keyword">Qed</span>.<br/>
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