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Equiv.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>Equiv: Program Equivalence</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">Equiv<span class="subtitle">Program Equivalence</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Imp</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab456"></a><h3 class="section">Some general advice for working on exercises:</h3>
<div class="paragraph"> </div>
<ul class="doclist">
<li> Most of the Coq proofs we ask you to do are similar to proofs
that we've provided. Before starting to work on the homework
problems, take the time to work through our proofs (both
informally, on paper, and in Coq) and make sure you understand
them in detail. This will save you a lot of time.
<div class="paragraph"> </div>
</li>
<li> The Coq proofs we're doing now are sufficiently complicated that
it is more or less impossible to complete them simply by random
experimentation or "following your nose." You need to start
with an idea about why the property is true and how the proof is
going to go. The best way to do this is to write out at least a
sketch of an informal proof on paper — one that intuitively
convinces you of the truth of the theorem — before starting to
work on the formal one. Alternately, grab a friend and try to
convince them that the theorem is true; then try to formalize
your explanation.
<div class="paragraph"> </div>
</li>
<li> Use automation to save work! Some of the proofs in this
chapter's exercises are pretty long if you try to write out all
the cases explicitly.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab457"></a><h1 class="section">Behavioral Equivalence</h1>
<div class="paragraph"> </div>
In the last chapter, we investigated the correctness of a very
simple program transformation: the <span class="inlinecode"><span class="id" type="var">optimize_0plus</span></span> function. The
programming language we were considering was the first version of
the language of arithmetic expressions — with no variables — so
in that setting it was very easy to define what it <i>means</i> for a
program transformation to be correct: it should always yield a
program that evaluates to the same number as the original.
<div class="paragraph"> </div>
To go further and talk about the correctness of program
transformations in the full Imp language, we need to consider the
role of variables and state.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab458"></a><h2 class="section">Definitions</h2>
<div class="paragraph"> </div>
For <span class="inlinecode"><span class="id" type="var">aexp</span></span>s and <span class="inlinecode"><span class="id" type="var">bexp</span></span>s with variables, the definition we want is
clear. We say
that two <span class="inlinecode"><span class="id" type="var">aexp</span></span>s or <span class="inlinecode"><span class="id" type="var">bexp</span></span>s are <i>behaviorally equivalent</i> if they
evaluate to the same result <i>in every state</i>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">aequiv</span> (<span class="id" type="var">a1</span> <span class="id" type="var">a2</span> : <span class="id" type="var">aexp</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">st</span>:<span class="id" type="var">state</span>), <br/>
<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> <span class="id" type="var">a1</span> = <span class="id" type="var">aeval</span> <span class="id" type="var">st</span> <span class="id" type="var">a2</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">bequiv</span> (<span class="id" type="var">b1</span> <span class="id" type="var">b2</span> : <span class="id" type="var">bexp</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">st</span>:<span class="id" type="var">state</span>), <br/>
<span class="id" type="var">beval</span> <span class="id" type="var">st</span> <span class="id" type="var">b1</span> = <span class="id" type="var">beval</span> <span class="id" type="var">st</span> <span class="id" type="var">b2</span>.<br/>
<br/>
</div>
<div class="doc">
For commands, the situation is a little more subtle. We can't
simply say "two commands are behaviorally equivalent if they
evaluate to the same ending state whenever they are started in the
same initial state," because some commands (in some starting
states) don't terminate in any final state at all! What we need
instead is this: two commands are behaviorally equivalent if, for
any given starting state, they either both diverge or both
terminate in the same final state. A compact way to express this
is "if the first one terminates in a particular state then so does
the second, and vice versa."
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">cequiv</span> (<span class="id" type="var">c1</span> <span class="id" type="var">c2</span> : <span class="id" type="var">com</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">st</span> <span class="id" type="var">st'</span> : <span class="id" type="var">state</span>), <br/>
(<span class="id" type="var">c1</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st'</span>) <span style="font-family: arial;">↔</span> (<span class="id" type="var">c2</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st'</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab459"></a><h4 class="section">Exercise: 2 stars (equiv_classes)</h4>
<div class="paragraph"> </div>
Given the following programs, group together those that are
equivalent in <span class="inlinecode"><span class="id" type="var">Imp</span></span>. For example, if you think programs (a)
through (h) are all equivalent to each other, but not to (i), your
answer should look like this: {a,b,c,d,e,f,g,h} {i}.
<div class="paragraph"> </div>
(a)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> > 0 <span class="id" type="var">DO</span><br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1<br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(b)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">IFB</span> <span class="id" type="var">X</span> = 0 <span class="id" type="var">THEN</span><br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1;;<br/>
<span class="id" type="var">Y</span> ::= 1<br/>
<span class="id" type="var">ELSE</span><br/>
<span class="id" type="var">Y</span> ::= 0<br/>
<span class="id" type="var">FI</span>;;<br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> - <span class="id" type="var">Y</span>;;<br/>
<span class="id" type="var">Y</span> ::= 0
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(c)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">SKIP</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(d)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> ≠ 0 <span class="id" type="var">DO</span><br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> × <span class="id" type="var">Y</span> + 1<br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(e)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Y</span> ::= 0
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(f)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Y</span> ::= <span class="id" type="var">X</span> + 1;;<br/>
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> ≠ <span class="id" type="var">Y</span> <span class="id" type="var">DO</span><br/>
<span class="id" type="var">Y</span> ::= <span class="id" type="var">X</span> + 1<br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(g)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">WHILE</span> <span class="id" type="var">TRUE</span> <span class="id" type="var">DO</span><br/>
<span class="id" type="var">SKIP</span><br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(h)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> ≠ <span class="id" type="var">X</span> <span class="id" type="var">DO</span><br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1<br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
(i)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> ≠ <span class="id" type="var">Y</span> <span class="id" type="var">DO</span><br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">Y</span> + 1<br/>
<span class="id" type="var">END</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab460"></a><h2 class="section">Examples</h2>
<div class="paragraph"> </div>
Here are some simple examples of equivalences of arithmetic
and boolean expressions.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">aequiv_example</span>:<br/>
<span class="id" type="var">aequiv</span> (<span class="id" type="var">AMinus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>)) (<span class="id" type="var">ANum</span> 0).<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">bequiv_example</span>:<br/>
<span class="id" type="var">bequiv</span> (<span class="id" type="var">BEq</span> (<span class="id" type="var">AMinus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>)) (<span class="id" type="var">ANum</span> 0)) <span class="id" type="var">BTrue</span>.<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">beval</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">aequiv_example</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
For examples of command equivalence, let's start by looking at
some trivial program transformations involving <span class="inlinecode"><span class="id" type="var">SKIP</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">skip_left</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span>,<br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">SKIP</span>;; <span class="id" type="var">c</span>) <br/>
<span class="id" type="var">c</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</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">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>. <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_Seq</span> <span class="id" type="keyword">with</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_Skip</span>.<br/>
<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab461"></a><h4 class="section">Exercise: 2 stars (skip_right)</h4>
Prove that adding a SKIP after a command results in an equivalent
program
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">skip_right</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span>,<br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">c</span>;; <span class="id" type="var">SKIP</span>) <br/>
<span class="id" type="var">c</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
Similarly, here is a simple transformations that simplifies <span class="inlinecode"><span class="id" type="var">IFB</span></span>
commands:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">IFB_true_simple</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c1</span> <span class="id" type="var">c2</span>,<br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">BTrue</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c1</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c2</span> <span class="id" type="var">FI</span>) <br/>
<span class="id" type="var">c1</span>.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span>.<br/>
<span class="id" type="tactic">split</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<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">assumption</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_IfTrue</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
Of course, few programmers would be tempted to write a conditional
whose guard is literally <span class="inlinecode"><span class="id" type="var">BTrue</span></span>. A more interesting case is when
the guard is <i>equivalent</i> to true:
<div class="paragraph"> </div>
<i>Theorem</i>: If <span class="inlinecode"><span class="id" type="var">b</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span>, then <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span> <span class="inlinecode"><span class="id" type="var">c1</span></span>
<span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">c1</span></span>.
<a name="lab462"></a><h2 class="section"> </h2>
<div class="paragraph"> </div>
<i>Proof</i>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> (<span class="inlinecode"><span style="font-family: arial;">→</span></span>) We must show, for all <span class="inlinecode"><span class="id" type="var">st</span></span> and <span class="inlinecode"><span class="id" type="var">st'</span></span>, that if <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span>
<span class="inlinecode"><span class="id" type="var">THEN</span></span> <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> then <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>.
<div class="paragraph"> </div>
Proceed by cases on the rules that could possibly have been
used to show <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span> <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>, namely
<span class="inlinecode"><span class="id" type="var">E_IfTrue</span></span> and <span class="inlinecode"><span class="id" type="var">E_IfFalse</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose the final rule rule in the derivation of <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span>
<span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> was <span class="inlinecode"><span class="id" type="var">E_IfTrue</span></span>. We then have, by
the premises of <span class="inlinecode"><span class="id" type="var">E_IfTrue</span></span>, that <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>. This is
exactly what we set out to prove.
<div class="paragraph"> </div>
</li>
<li> On the other hand, suppose the final rule in the derivation
of <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span> <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> was <span class="inlinecode"><span class="id" type="var">E_IfFalse</span></span>.
We then know that <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">false</span></span> and <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>.
<div class="paragraph"> </div>
Recall that <span class="inlinecode"><span class="id" type="var">b</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span>, i.e. forall <span class="inlinecode"><span class="id" type="var">st</span></span>,
<span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">BTrue</span></span>. In particular, this means
that <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">true</span></span>, since <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">BTrue</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">true</span></span>. But
this is a contradiction, since <span class="inlinecode"><span class="id" type="var">E_IfFalse</span></span> requires that
<span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">false</span></span>. Thus, the final rule could not have
been <span class="inlinecode"><span class="id" type="var">E_IfFalse</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> (<span class="inlinecode"><span style="font-family: arial;">←</span></span>) We must show, for all <span class="inlinecode"><span class="id" type="var">st</span></span> and <span class="inlinecode"><span class="id" type="var">st'</span></span>, that if <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span>
<span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> then <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span> <span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>.
<div class="paragraph"> </div>
Since <span class="inlinecode"><span class="id" type="var">b</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span>, we know that <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> =
<span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">BTrue</span></span> = <span class="inlinecode"><span class="id" type="var">true</span></span>. Together with the assumption that
<span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>, we can apply <span class="inlinecode"><span class="id" type="var">E_IfTrue</span></span> to derive <span class="inlinecode"><span class="id" type="var">IFB</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">THEN</span></span>
<span class="inlinecode"><span class="id" type="var">c1</span></span> <span class="inlinecode"><span class="id" type="var">ELSE</span></span> <span class="inlinecode"><span class="id" type="var">c2</span></span> <span class="inlinecode"><span class="id" type="var">FI</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>. <font size=-2>☐</font>
</li>
</ul>
<div class="paragraph"> </div>
Here is the formal version of this proof:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">IFB_true</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span>,<br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">BTrue</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c1</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c2</span> <span class="id" type="var">FI</span>) <br/>
<span class="id" type="var">c1</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span> <span class="id" type="var">Hb</span>.<br/>
<span class="id" type="tactic">split</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">SCase</span> "b evaluates to true".<br/>
<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">SCase</span> "b evaluates to false (contradiction)".<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hb</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hb</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Hb</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_IfTrue</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hb</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hb</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Hb</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab463"></a><h4 class="section">Exercise: 2 stars (IFB_false)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">IFB_false</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span>,<br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">BFalse</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">c1</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">c2</span> <span class="id" type="var">FI</span>) <br/>
<span class="id" type="var">c2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab464"></a><h4 class="section">Exercise: 3 stars (swap_if_branches)</h4>
Show that we can swap the branches of an IF by negating its
condition
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">swap_if_branches</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">e1</span> <span class="id" type="var">e2</span>,<br/>
<span class="id" type="var">cequiv</span><br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">e1</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">e2</span> <span class="id" type="var">FI</span>)<br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">BNot</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> <span class="id" type="var">e2</span> <span class="id" type="var">ELSE</span> <span class="id" type="var">e1</span> <span class="id" type="var">FI</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab465"></a><h2 class="section"> </h2>
<div class="paragraph"> </div>
For <span class="inlinecode"><span class="id" type="var">WHILE</span></span> loops, we can give a similar pair of theorems. A loop
whose guard is equivalent to <span class="inlinecode"><span class="id" type="var">BFalse</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">SKIP</span></span>,
while a loop whose guard is equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span> is equivalent to
<span class="inlinecode"><span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">BTrue</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">SKIP</span></span> <span class="inlinecode"><span class="id" type="var">END</span></span> (or any other non-terminating program).
The first of these facts is easy.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">WHILE_false</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">BFalse</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">cequiv</span><br/>
(<span class="id" type="var">WHILE</span> <span class="id" type="var">b</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>)<br/>
<span class="id" type="var">SKIP</span>.<br/>
<div class="togglescript" id="proofcontrol4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')"><span class="show"></span></div>
<div class="proofscript" id="proof4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">Hb</span>. <span class="id" type="tactic">split</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">SCase</span> "E_WhileEnd".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_Skip</span>.<br/>
<span class="id" type="var">SCase</span> "E_WhileLoop".<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Hb</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H2</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_WhileEnd</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Hb</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab466"></a><h4 class="section">Exercise: 2 stars, advanced, optional (WHILE_false_informal)</h4>
Write an informal proof of <span class="inlinecode"><span class="id" type="var">WHILE_false</span></span>.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab467"></a><h2 class="section"> </h2>
To prove the second fact, we need an auxiliary lemma stating that
<span class="inlinecode"><span class="id" type="var">WHILE</span></span> loops whose guards are equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span> never
terminate:
<div class="paragraph"> </div>
<i>Lemma</i>: If <span class="inlinecode"><span class="id" type="var">b</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">BTrue</span></span>, then it cannot be the
case that <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: Suppose that <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>. We show,
by induction on a derivation of <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>,
that this assumption leads to a contradiction.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> is proved using rule
<span class="inlinecode"><span class="id" type="var">E_WhileEnd</span></span>. Then by assumption <span class="inlinecode"><span class="id" type="var">beval</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">false</span></span>. But
this contradicts the assumption that <span class="inlinecode"><span class="id" type="var">b</span></span> is equivalent to
<span class="inlinecode"><span class="id" type="var">BTrue</span></span>.
<div class="paragraph"> </div>
</li>
<li> Suppose <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> is proved using rule
<span class="inlinecode"><span class="id" type="var">E_WhileLoop</span></span>. Then we are given the induction hypothesis
that <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span> is contradictory, which
is exactly what we are trying to prove!
<div class="paragraph"> </div>
</li>
<li> Since these are the only rules that could have been used to
prove <span class="inlinecode">(<span class="id" type="var">WHILE</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">DO</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">END</span>)</span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇓</span></span> <span class="inlinecode"><span class="id" type="var">st'</span></span>, the other cases of
the induction are immediately contradictory. <font size=-2>☐</font>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">WHILE_true_nonterm</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</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="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">BTrue</span> <span style="font-family: arial;">→</span><br/>
~( (<span class="id" type="var">WHILE</span> <span class="id" type="var">b</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">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st'</span> ).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</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">Hb</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">remember</span> (<span class="id" type="var">WHILE</span> <span class="id" type="var">b</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="keyword">as</span> <span class="id" type="var">cw</span> <span class="id" type="var">eqn</span>:<span class="id" type="var">Heqcw</span>.<br/>
<span class="id" type="var">ceval_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>) <span class="id" type="var">Case</span>;<br/>
<span class="comment">(* Most rules don't apply, and we can rule them out <br/>
by inversion *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Heqcw</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">Heqcw</span>.<br/>
<span class="comment">(* The two interesting cases are the ones for WHILE loops: *)</span><br/>
<span class="id" type="var">Case</span> "E_WhileEnd". <span class="comment">(* contradictory -- b is always true! *)</span><br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hb</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> is able to instantiate the quantifier in <span class="inlinecode"><span class="id" type="var">st</span></span> *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Hb</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>.<br/>
<span class="id" type="var">Case</span> "E_WhileLoop". <span class="comment">(* immediate from the IH *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHceval2</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab468"></a><h4 class="section">Exercise: 2 stars, optional (WHILE_true_nonterm_informal)</h4>
Explain what the lemma <span class="inlinecode"><span class="id" type="var">WHILE_true_nonterm</span></span> means in English.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab469"></a><h4 class="section">Exercise: 2 stars (WHILE_true)</h4>
Prove the following theorem. <i>Hint</i>: You'll want to use
<span class="inlinecode"><span class="id" type="var">WHILE_true_nonterm</span></span> here.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">WHILE_true</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">BTrue</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">cequiv</span> <br/>
(<span class="id" type="var">WHILE</span> <span class="id" type="var">b</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>)<br/>
(<span class="id" type="var">WHILE</span> <span class="id" type="var">BTrue</span> <span class="id" type="var">DO</span> <span class="id" type="var">SKIP</span> <span class="id" type="var">END</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">loop_unrolling</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span>,<br/>
<span class="id" type="var">cequiv</span><br/>
(<span class="id" type="var">WHILE</span> <span class="id" type="var">b</span> <span class="id" type="var">DO</span> <span class="id" type="var">c</span> <span class="id" type="var">END</span>)<br/>
(<span class="id" type="var">IFB</span> <span class="id" type="var">b</span> <span class="id" type="var">THEN</span> (<span class="id" type="var">c</span>;; <span class="id" type="var">WHILE</span> <span class="id" type="var">b</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">ELSE</span> <span class="id" type="var">SKIP</span> <span class="id" type="var">FI</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<div class="togglescript" id="proofcontrol5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')"><span class="show"></span></div>
<div class="proofscript" id="proof5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')">
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</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">intros</span> <span class="id" type="var">Hce</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hce</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">SCase</span> "loop doesn't run".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_IfFalse</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E_Skip</span>.<br/>
<span class="id" type="var">SCase</span> "loop runs".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_IfTrue</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_Seq</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">st'</span> := <span class="id" type="var">st'0</span>). <span class="id" type="tactic">assumption</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hce</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">SCase</span> "loop runs".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E_WhileLoop</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">st'</span> := <span class="id" type="var">st'0</span>).<br/>
<span class="id" type="tactic">assumption</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">SCase</span> "loop doesn't run".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>; <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E_WhileEnd</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab470"></a><h4 class="section">Exercise: 2 stars, optional (seq_assoc)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">seq_assoc</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">c1</span> <span class="id" type="var">c2</span> <span class="id" type="var">c3</span>,<br/>
<span class="id" type="var">cequiv</span> ((<span class="id" type="var">c1</span>;;<span class="id" type="var">c2</span>);;<span class="id" type="var">c3</span>) (<span class="id" type="var">c1</span>;;(<span class="id" type="var">c2</span>;;<span class="id" type="var">c3</span>)).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab471"></a><h2 class="section">The Functional Equivalence Axiom</h2>
<div class="paragraph"> </div>
Finally, let's look at simple equivalences involving assignments.
For example, we might expect to be able to show that <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode">::=</span> <span class="inlinecode"><span class="id" type="var">AId</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>
is equivalent to <span class="inlinecode"><span class="id" type="var">SKIP</span></span>. However, when we try to show it, we get
stuck in an interesting way.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">identity_assignment_first_try</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span>:<span class="id" type="var">id</span>),<br/>
<span class="id" type="var">cequiv</span> (<span class="id" type="var">X</span> ::= <span class="id" type="var">AId</span> <span class="id" type="var">X</span>) <span class="id" type="var">SKIP</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">split</span>; <span class="id" type="tactic">intro</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<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">simpl</span>.<br/>
<span class="id" type="tactic">replace</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">st</span> <span class="id" type="var">X</span>)) <span class="id" type="keyword">with</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="var">constructor</span>.<br/>
<span class="comment">(* Stuck... *)</span> <span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
Here we're stuck. The goal looks reasonable, but in fact it is not
provable! If we look back at the set of lemmas we proved about
<span class="inlinecode"><span class="id" type="var">update</span></span> in the last chapter, we can see that lemma <span class="inlinecode"><span class="id" type="var">update_same</span></span>
almost does the job, but not quite: it says that the original and
updated states agree at all values, but this is not the same thing
as saying that they are <span class="inlinecode">=</span> in Coq's sense!
<div class="paragraph"> </div>
What is going on here? Recall that our states are just
functions from identifiers to values. For Coq, functions are only
equal when their definitions are syntactically the same, modulo
simplification. (This is the only way we can legally apply the
<span class="inlinecode"><span class="id" type="var">refl_equal</span></span> constructor of the inductively defined proposition
<span class="inlinecode"><span class="id" type="var">eq</span></span>!) In practice, for functions built up by repeated uses of the
<span class="inlinecode"><span class="id" type="var">update</span></span> operation, this means that two functions can be proven
equal only if they were constructed using the <i>same</i> <span class="inlinecode"><span class="id" type="var">update</span></span>
operations, applied in the same order. In the theorem above, the
sequence of updates on the first parameter <span class="inlinecode"><span class="id" type="var">cequiv</span></span> is one longer
than for the second parameter, so it is no wonder that the
equality doesn't hold.
<div class="paragraph"> </div>
<a name="lab472"></a><h2 class="section"> </h2>
This problem is actually quite general. If we try to prove other
simple facts, such as
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">cequiv</span> (<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1;;<br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1)<br/>
(<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 2)
<div class="paragraph"> </div>
</div>
or
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">cequiv</span> (<span class="id" type="var">X</span> ::= 1;; <span class="id" type="var">Y</span> ::= 2)<br/>
(<span class="id" type="var">y</span> ::= 2;; <span class="id" type="var">X</span> ::= 1)<br/>
<div class="paragraph"> </div>
</div>
we'll get stuck in the same way: we'll have two functions that
behave the same way on all inputs, but cannot be proven to be <span class="inlinecode"><span class="id" type="var">eq</span></span>
to each other.
<div class="paragraph"> </div>
The reasoning principle we would like to use in these situations
is called <i>functional extensionality</i>:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">∀</span>x, f x = g x</td>
<td class="infrulenamecol" rowspan="3">
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">f = g</td>
<td></td>
</td>
</table></center> Although this principle is not derivable in Coq's built-in logic,
it is safe to add it as an additional <i>axiom</i>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Axiom</span> <span class="id" type="var">functional_extensionality</span> : <span style="font-family: arial;">∀</span>{<span class="id" type="var">X</span> <span class="id" type="var">Y</span>: <span class="id" type="keyword">Type</span>} {<span class="id" type="var">f</span> <span class="id" type="var">g</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Y</span>},<br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">x</span>: <span class="id" type="var">X</span>), <span class="id" type="var">f</span> <span class="id" type="var">x</span> = <span class="id" type="var">g</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">f</span> = <span class="id" type="var">g</span>.<br/>
<br/>
</div>
<div class="doc">
It can be shown that adding this axiom doesn't introduce any
inconsistencies into Coq. (In this way, it is similar to adding
one of the classical logic axioms, such as <span class="inlinecode"><span class="id" type="var">excluded_middle</span></span>.)
<div class="paragraph"> </div>
With the benefit of this axiom we can prove our theorem.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">identity_assignment</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span>:<span class="id" type="var">id</span>),<br/>
<span class="id" type="var">cequiv</span><br/>
(<span class="id" type="var">X</span> ::= <span class="id" type="var">AId</span> <span class="id" type="var">X</span>)<br/>
<span class="id" type="var">SKIP</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">split</span>; <span class="id" type="tactic">intro</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">→</span>".<br/>
<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">simpl</span>.<br/>
<span class="id" type="tactic">replace</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">st</span> <span class="id" type="var">X</span>)) <span class="id" type="keyword">with</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="var">constructor</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">functional_extensionality</span>. <span class="id" type="tactic">intro</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">update_same</span>; <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "<span style="font-family: arial;">←</span>".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">st'</span> = (<span class="id" type="var">update</span> <span class="id" type="var">st'</span> <span class="id" type="var">X</span> (<span class="id" type="var">st'</span> <span class="id" type="var">X</span>))).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">functional_extensionality</span>. <span class="id" type="tactic">intro</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">update_same</span>; <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H0</span> <span class="id" type="tactic">at</span> 2.<br/>
<span class="id" type="var">constructor</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab473"></a><h4 class="section">Exercise: 2 stars (assign_aequiv)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">assign_aequiv</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">X</span> <span class="id" type="var">e</span>,<br/>
<span class="id" type="var">aequiv</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) <span class="id" type="var">e</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">cequiv</span> <span class="id" type="var">SKIP</span> (<span class="id" type="var">X</span> ::= <span class="id" type="var">e</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab474"></a><h1 class="section">Properties of Behavioral Equivalence</h1>
<div class="paragraph"> </div>
We now turn to developing some of the properties of the program
equivalences we have defined.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab475"></a><h2 class="section">Behavioral Equivalence is an Equivalence</h2>
<div class="paragraph"> </div>
First, we verify that the equivalences on <span class="inlinecode"><span class="id" type="var">aexps</span></span>, <span class="inlinecode"><span class="id" type="var">bexps</span></span>, and
<span class="inlinecode"><span class="id" type="var">com</span></span>s really are <i>equivalences</i> — i.e., that they are reflexive,
symmetric, and transitive. The proofs are all easy.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">refl_aequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a</span> : <span class="id" type="var">aexp</span>), <span class="id" type="var">aequiv</span> <span class="id" type="var">a</span> <span class="id" type="var">a</span>.<br/>
<div class="togglescript" id="proofcontrol6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')"><span class="show"></span></div>
<div class="proofscript" id="proof6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">a</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">sym_aequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a1</span> <span class="id" type="var">a2</span> : <span class="id" type="var">aexp</span>), <br/>
<span class="id" type="var">aequiv</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">aequiv</span> <span class="id" type="var">a2</span> <span class="id" type="var">a1</span>.<br/>
<div class="togglescript" id="proofcontrol7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')"><span class="show"></span></div>
<div class="proofscript" id="proof7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">symmetry</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">trans_aequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a1</span> <span class="id" type="var">a2</span> <span class="id" type="var">a3</span> : <span class="id" type="var">aexp</span>), <br/>
<span class="id" type="var">aequiv</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">aequiv</span> <span class="id" type="var">a2</span> <span class="id" type="var">a3</span> <span style="font-family: arial;">→</span> <span class="id" type="var">aequiv</span> <span class="id" type="var">a1</span> <span class="id" type="var">a3</span>.<br/>
<div class="togglescript" id="proofcontrol8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')"><span class="show"></span></div>
<div class="proofscript" id="proof8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">aequiv</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> <span class="id" type="var">a3</span> <span class="id" type="var">H12</span> <span class="id" type="var">H23</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="tactic">rewrite</span> (<span class="id" type="var">H12</span> <span class="id" type="var">st</span>). <span class="id" type="tactic">rewrite</span> (<span class="id" type="var">H23</span> <span class="id" type="var">st</span>). <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">refl_bequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">b</span> : <span class="id" type="var">bexp</span>), <span class="id" type="var">bequiv</span> <span class="id" type="var">b</span> <span class="id" type="var">b</span>.<br/>
<div class="togglescript" id="proofcontrol9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')"><span class="show"></span></div>
<div class="proofscript" id="proof9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">sym_bequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">b1</span> <span class="id" type="var">b2</span> : <span class="id" type="var">bexp</span>), <br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bequiv</span> <span class="id" type="var">b2</span> <span class="id" type="var">b1</span>.<br/>
<div class="togglescript" id="proofcontrol10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')"><span class="show"></span></div>
<div class="proofscript" id="proof10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">st</span>. <span class="id" type="tactic">symmetry</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">trans_bequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span class="id" type="var">b3</span> : <span class="id" type="var">bexp</span>), <br/>
<span class="id" type="var">bequiv</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bequiv</span> <span class="id" type="var">b2</span> <span class="id" type="var">b3</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bequiv</span> <span class="id" type="var">b1</span> <span class="id" type="var">b3</span>.<br/>
<div class="togglescript" id="proofcontrol11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')"><span class="show"></span></div>
<div class="proofscript" id="proof11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bequiv</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span class="id" type="var">b3</span> <span class="id" type="var">H12</span> <span class="id" type="var">H23</span> <span class="id" type="var">st</span>.<br/>
<span class="id" type="tactic">rewrite</span> (<span class="id" type="var">H12</span> <span class="id" type="var">st</span>). <span class="id" type="tactic">rewrite</span> (<span class="id" type="var">H23</span> <span class="id" type="var">st</span>). <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">refl_cequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">c</span> : <span class="id" type="var">com</span>), <span class="id" type="var">cequiv</span> <span class="id" type="var">c</span> <span class="id" type="var">c</span>.<br/>
<div class="togglescript" id="proofcontrol12" onclick="toggleDisplay('proof12');toggleDisplay('proofcontrol12')"><span class="show"></span></div>
<div class="proofscript" id="proof12" onclick="toggleDisplay('proof12');toggleDisplay('proofcontrol12')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">cequiv</span>. <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="tactic">apply</span> <span class="id" type="var">iff_refl</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">sym_cequiv</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">c1</span> <span class="id" type="var">c2</span> : <span class="id" type="var">com</span>), <br/>
<span class="id" type="var">cequiv</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">cequiv</span> <span class="id" type="var">c2</span> <span class="id" type="var">c1</span>.<br/>
<div class="togglescript" id="proofcontrol13" onclick="toggleDisplay('proof13');toggleDisplay('proofcontrol13')"><span class="show"></span></div>
<div class="proofscript" id="proof13" onclick="toggleDisplay('proof13');toggleDisplay('proofcontrol13')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">cequiv</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">c1</span> <span class="id" type="var">c2</span> <span class="id" type="var">H</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">c1</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st'</span> <span style="font-family: arial;">↔</span> <span class="id" type="var">c2</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st'</span>) <span class="id" type="keyword">as</span> <span class="id" type="var">H'</span>.<br/>
<span class="id" type="var">SCase</span> "Proof of assertion". <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">iff_sym</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>