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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>StlcProp: Properties of STLC</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">StlcProp<span class="subtitle">Properties of STLC</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">Stlc</span>.<br/>
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">STLCProp</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
</div>
<div class="doc">
In this chapter, we develop the fundamental theory of the Simply
Typed Lambda Calculus — in particular, the type safety
theorem.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab680"></a><h1 class="section">Canonical Forms</h1>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">canonical_forms_bool</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">TBool</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">t</span> = <span class="id" type="var">ttrue</span>) <span style="font-family: arial;">∨</span> (<span class="id" type="var">t</span> = <span class="id" type="var">tfalse</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">HT</span> <span class="id" type="var">HVal</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HVal</span>; <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">HT</span>; <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">canonical_forms_fun</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>,<br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ (<span class="id" type="var">TArrow</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">x</span> <span class="id" type="var">u</span>, <span class="id" type="var">t</span> = <span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">u</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="var">HT</span> <span class="id" type="var">HVal</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HVal</span>; <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">HT</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">auto</span>.<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">x0</span>. <span style="font-family: arial;">∃</span><span class="id" type="var">t0</span>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab681"></a><h1 class="section">Progress</h1>
<div class="paragraph"> </div>
As before, the <i>progress</i> theorem tells us that closed, well-typed
terms are not stuck: either a well-typed term is a value, or it
can take an evaluation step. The proof is a relatively
straightforward extension of the progress proof we saw in the
<span class="inlinecode"><span class="id" type="keyword">Types</span></span> chapter.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="tactic">progress</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T</span>, <br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span>, <span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Proof</i>: by induction on the derivation of <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> The last rule of the derivation cannot be <span class="inlinecode"><span class="id" type="var">T_Var</span></span>, since a
variable is never well typed in an empty context.
<div class="paragraph"> </div>
</li>
<li> The <span class="inlinecode"><span class="id" type="var">T_True</span></span>, <span class="inlinecode"><span class="id" type="var">T_False</span></span>, and <span class="inlinecode"><span class="id" type="var">T_Abs</span></span> cases are trivial, since in
each of these cases we know immediately that <span class="inlinecode"><span class="id" type="var">t</span></span> is a value.
<div class="paragraph"> </div>
</li>
<li> If the last rule of the derivation was <span class="inlinecode"><span class="id" type="var">T_App</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, and we know that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> are also well typed in the
empty context; in particular, there exists a type <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> such that
<span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span>. By the induction
hypothesis, either <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value or it can take an evaluation
step.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value, we now consider <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, which by the other
induction hypothesis must also either be a value or take an
evaluation step.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> is a value. Since <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value with an
arrow type, it must be a lambda abstraction; hence <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> can take a step by <span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> can take a step, and hence so can <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" type="var">ST_App2</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> can take a step, then so can <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" type="var">ST_App1</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If the last rule of the derivation was <span class="inlinecode"><span class="id" type="var">T_If</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, where <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> has type <span class="inlinecode"><span class="id" type="var">Bool</span></span>. By the IH, <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
either is a value or takes a step.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value, then since it has type <span class="inlinecode"><span class="id" type="var">Bool</span></span> it must be
either <span class="inlinecode"><span class="id" type="var">true</span></span> or <span class="inlinecode"><span class="id" type="var">false</span></span>. If it is <span class="inlinecode"><span class="id" type="var">true</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> steps
to <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>; otherwise it steps to <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> takes a step, and therefore so does <span class="inlinecode"><span class="id" type="var">t</span></span> (by
<span class="inlinecode"><span class="id" type="var">ST_If</span></span>).
</li>
</ul>
</li>
</ul>
</div>
<div class="code code-tight">
<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> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">Ht</span>.<br/>
<span class="id" type="var">remember</span> (@<span class="id" type="var">empty</span> <span class="id" type="var">ty</span>) <span class="id" type="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Ht</span>; <span class="id" type="tactic">subst</span> <span style="font-family: serif; font-size:85%;">Γ</span>...<br/>
- <span class="comment">(* T_Var *)</span><br/>
<span class="comment">(* contradictory: variables cannot be typed in an <br/>
empty context *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="comment">(* <span class="inlinecode"><span class="id" type="var">t</span></span> = <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>. Proceed by cases on whether <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a <br/>
value or steps... *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>...<br/>
+ <span class="comment">(* t<sub>1</sub> is a value *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span>...<br/>
× <span class="comment">(* t<sub>2</sub> is also a value *)</span><br/>
<span class="id" type="tactic">assert</span> (<span style="font-family: arial;">∃</span><span class="id" type="var">x0</span> <span class="id" type="var">t0</span>, <span class="id" type="var">t<sub>1</sub></span> = <span class="id" type="var">tabs</span> <span class="id" type="var">x0</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t0</span>).<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">canonical_forms_fun</span>; <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H1</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">x0</span> [<span class="id" type="var">t0</span> <span class="id" type="var">Heq</span>]]. <span class="id" type="tactic">subst</span>.<br/>
<span style="font-family: arial;">∃</span>([<span class="id" type="var">x0</span>:=<span class="id" type="var">t<sub>2</sub></span>]<span class="id" type="var">t0</span>)...<br/>
<br/>
× <span class="comment">(* t<sub>2</sub> steps *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t<sub>2</sub>'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>)...<br/>
<br/>
+ <span class="comment">(* t<sub>1</sub> steps *)</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">t<sub>1</sub>'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span>)...<br/>
<br/>
- <span class="comment">(* T_If *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>...<br/>
<br/>
+ <span class="comment">(* t<sub>1</sub> is a value *)</span><br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">canonical_forms_bool</span> <span class="id" type="var">t<sub>1</sub></span>); <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">eauto</span>.<br/>
<br/>
+ <span class="comment">(* t<sub>1</sub> also steps *)</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">t<sub>1</sub>'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>)...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab682"></a><h4 class="section">Exercise: 3 stars, optional (progress_from_term_ind)</h4>
Show that progress can also be proved by induction on terms
instead of induction on typing derivations.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">progress'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span>, <span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">t</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">T</span> <span class="id" type="var">Ht</span>; <span class="id" type="tactic">auto</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="lab683"></a><h1 class="section">Preservation</h1>
<div class="paragraph"> </div>
The other half of the type soundness property is the preservation
of types during reduction. For this, we need to develop some
technical machinery for reasoning about variables and
substitution. Working from top to bottom (the high-level property
we are actually interested in to the lowest-level technical lemmas
that are needed by various cases of the more interesting proofs),
the story goes like this:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The <i>preservation theorem</i> is proved by induction on a typing
derivation, pretty much as we did in the <span class="inlinecode"><span class="id" type="keyword">Types</span></span> chapter. The
one case that is significantly different is the one for the
<span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span> rule, which is defined using the substitution
operation. To see that this step preserves typing, we need to
know that the substitution itself does. So we prove a...
<div class="paragraph"> </div>
</li>
<li> <i>substitution lemma</i>, stating that substituting a (closed)
term <span class="inlinecode"><span class="id" type="var">s</span></span> for a variable <span class="inlinecode"><span class="id" type="var">x</span></span> in a term <span class="inlinecode"><span class="id" type="var">t</span></span> preserves the type
of <span class="inlinecode"><span class="id" type="var">t</span></span>. The proof goes by induction on the form of <span class="inlinecode"><span class="id" type="var">t</span></span> and
requires looking at all the different cases in the definition
of substitition. This time, the tricky cases are the ones for
variables and for function abstractions. In both cases, we
discover that we need to take a term <span class="inlinecode"><span class="id" type="var">s</span></span> that has been shown
to be well-typed in some context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> and consider the same
term <span class="inlinecode"><span class="id" type="var">s</span></span> in a slightly different context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>. For this
we prove a...
<div class="paragraph"> </div>
</li>
<li> <i>context invariance</i> lemma, showing that typing is preserved
under "inessential changes" to the context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> — in
particular, changes that do not affect any of the free
variables of the term. For this, we need a careful definition
of
<div class="paragraph"> </div>
</li>
<li> the <i>free variables</i> of a term — i.e., the variables occuring
in the term that are not in the scope of a function
abstraction that binds them.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab684"></a><h2 class="section">Free Occurrences</h2>
<div class="paragraph"> </div>
A variable <span class="inlinecode"><span class="id" type="var">x</span></span> <i>appears free in</i> a term <i>t</i> if <span class="inlinecode"><span class="id" type="var">t</span></span> contains some
occurrence of <span class="inlinecode"><span class="id" type="var">x</span></span> that is not under an abstraction labeled <span class="inlinecode"><span class="id" type="var">x</span></span>. For example:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">y</span></span> appears free, but <span class="inlinecode"><span class="id" type="var">x</span></span> does not, in <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T</span><span style="font-family: arial;">→</span><span class="id" type="var">U</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span>
</li>
<li> both <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">y</span></span> appear free in <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">T</span><span style="font-family: arial;">→</span><span class="id" type="var">U</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span>)</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
</li>
<li> no variables appear free in <span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T</span><span style="font-family: arial;">→</span><span class="id" type="var">U</span>.</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T</span>.</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">appears_free_in</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">afi_var</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)<br/>
| <span class="id" type="var">afi_app1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">afi_app2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">afi_abs</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="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>,<br/>
<span class="id" type="var">y</span> ≠ <span class="id" type="var">x</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>12</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>)<br/>
| <span class="id" type="var">afi_if1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>)<br/>
| <span class="id" type="var">afi_if2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>)<br/>
| <span class="id" type="var">afi_if3</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>3</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></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')">
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">appears_free_in</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
A term in which no variables appear free is said to be <i>closed</i>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">closed</span> (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, ¬ <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab685"></a><h2 class="section">Substitution</h2>
<div class="paragraph"> </div>
We first need a technical lemma connecting free variables and
typing contexts. If a variable <span class="inlinecode"><span class="id" type="var">x</span></span> appears free in a term <span class="inlinecode"><span class="id" type="var">t</span></span>,
and if we know <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed in context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, then it must
be the case that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> assigns a type to <span class="inlinecode"><span class="id" type="var">x</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">free_in_context</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span style="font-family: serif; font-size:85%;">Γ</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">T'</span>, <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> = <span class="id" type="var">Some</span> <span class="id" type="var">T'</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Proof</i>: We show, by induction on the proof that <span class="inlinecode"><span class="id" type="var">x</span></span> appears free
in <span class="inlinecode"><span class="id" type="var">t</span></span>, that, for all contexts <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, if <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed
under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, then <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> assigns some type to <span class="inlinecode"><span class="id" type="var">x</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last rule used was <span class="inlinecode"><span class="id" type="var">afi_var</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">x</span></span>, and from
the assumption that <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> we have
immediately that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> assigns a type to <span class="inlinecode"><span class="id" type="var">x</span></span>.
<div class="paragraph"> </div>
</li>
<li> If the last rule used was <span class="inlinecode"><span class="id" type="var">afi_app1</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> and <span class="inlinecode"><span class="id" type="var">x</span></span>
appears free in <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>. Since <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>,
we can see from the typing rules that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> must also be, and
the IH then tells us that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> assigns <span class="inlinecode"><span class="id" type="var">x</span></span> a type.
<div class="paragraph"> </div>
</li>
<li> Almost all the other cases are similar: <span class="inlinecode"><span class="id" type="var">x</span></span> appears free in a
subterm of <span class="inlinecode"><span class="id" type="var">t</span></span>, and since <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, we
know the subterm of <span class="inlinecode"><span class="id" type="var">t</span></span> in which <span class="inlinecode"><span class="id" type="var">x</span></span> appears is well typed
under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> as well, and the IH gives us exactly the
conclusion we want.
<div class="paragraph"> </div>
</li>
<li> The only remaining case is <span class="inlinecode"><span class="id" type="var">afi_abs</span></span>. In this case <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span>
<span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub>.t<sub>12</sub></span></span>, and <span class="inlinecode"><span class="id" type="var">x</span></span> appears free in <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>; we also know that
<span class="inlinecode"><span class="id" type="var">x</span></span> is different from <span class="inlinecode"><span class="id" type="var">y</span></span>. The difference from the previous
cases is that whereas <span class="inlinecode"><span class="id" type="var">t</span></span> is well typed under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, its
body <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> is well typed under <span class="inlinecode">(<span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>)</span>, so the IH
allows us to conclude that <span class="inlinecode"><span class="id" type="var">x</span></span> is assigned some type by the
extended context <span class="inlinecode">(<span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>)</span>. To conclude that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>
assigns a type to <span class="inlinecode"><span class="id" type="var">x</span></span>, we appeal to lemma <span class="inlinecode"><span class="id" type="var">extend_neq</span></span>, noting
that <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">y</span></span> are different variables.
</li>
</ul>
</div>
<div class="code code-tight">
<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">x</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">H</span> <span class="id" type="var">H0</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">T</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>; <br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> [<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>; <span class="id" type="tactic">eauto</span>].<br/>
- <span class="comment">(* afi_abs *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H1</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHappears_free_in</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H7</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">extend_neq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H7</span>; <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
Next, we'll need the fact that any term <span class="inlinecode"><span class="id" type="var">t</span></span> which is well typed in
the empty context is closed — that is, it has no free variables.
<div class="paragraph"> </div>
<a name="lab686"></a><h4 class="section">Exercise: 2 stars, optional (typable_empty__closed)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Corollary</span> <span class="id" type="var">typable_empty__closed</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T</span>, <br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">closed</span> <span class="id" type="var">t</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>
Sometimes, when we have a proof <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T</span></span>, we will need to
replace <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> by a different context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>. When is it safe
to do this? Intuitively, it must at least be the case that
<span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> assigns the same types as <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> to all the variables
that appear free in <span class="inlinecode"><span class="id" type="var">t</span></span>. In fact, this is the only condition that
is needed.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">context_invariance</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: serif; font-size:85%;">Γ'</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> = <span style="font-family: serif; font-size:85%;">Γ'</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ'</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Proof</i>: By induction on the derivation of <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last rule in the derivation was <span class="inlinecode"><span class="id" type="var">T_Var</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">x</span></span>
and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">T</span></span>. By assumption, <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">T</span></span> as well, and
hence <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> by <span class="inlinecode"><span class="id" type="var">T_Var</span></span>.
<div class="paragraph"> </div>
</li>
<li> If the last rule was <span class="inlinecode"><span class="id" type="var">T_Abs</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, with <span class="inlinecode"><span class="id" type="var">T</span></span>
<span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span>. The induction
hypothesis is that for any context <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ''</span></span>, if <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span>
<span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ''</span></span> assign the same types to all the free
variables in <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, then <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> has type <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span> under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ''</span></span>.
Let <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> be a context which agrees with <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> on the
free variables in <span class="inlinecode"><span class="id" type="var">t</span></span>; we must show <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span>
<span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span>.
<div class="paragraph"> </div>
By <span class="inlinecode"><span class="id" type="var">T_Abs</span></span>, it suffices to show that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span>
<span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span>. By the IH (setting <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ''</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span>), it
suffices to show that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> agree
on all the variables that appear free in <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>.
<div class="paragraph"> </div>
Any variable occurring free in <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> must either be <span class="inlinecode"><span class="id" type="var">y</span></span>, or
some other variable. <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span>
clearly agree on <span class="inlinecode"><span class="id" type="var">y</span></span>. Otherwise, we note that any variable
other than <span class="inlinecode"><span class="id" type="var">y</span></span> which occurs free in <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> also occurs free in
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, and by assumption <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>
agree on all such variables, and hence so do <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span>
and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,</span> <span class="inlinecode"><span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span>.
<div class="paragraph"> </div>
</li>
<li> If the last rule was <span class="inlinecode"><span class="id" type="var">T_App</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, with <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span>
<span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span>. One induction
hypothesis states that for all contexts <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>, if <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>
agrees with <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> on the free variables in <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>, then <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
has type <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>; there is a similar IH for
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>. We must show that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> also has type <span class="inlinecode"><span class="id" type="var">T</span></span> under
<span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span>, given the assumption that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> agrees with
<span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> on all the free variables in <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>. By <span class="inlinecode"><span class="id" type="var">T_App</span></span>, it
suffices to show that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> each have the same type
under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> as under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>. However, we note that all
free variables in <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> are also free in <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, and similarly
for free variables in <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>; hence the desired result follows
by the two IHs.
</li>
</ul>
</div>
<div class="code code-tight">
<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> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span style="font-family: serif; font-size:85%;">Γ'</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">auto</span>.<br/>
- <span class="comment">(* T_Var *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Var</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">H0</span>...<br/>
- <span class="comment">(* T_Abs *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Abs</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHhas_type</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">x1</span> <span class="id" type="var">Hafi</span>.<br/>
<span class="comment">(* the only tricky step... the <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> we use to <br/>
instantiate is <span class="inlinecode"><span class="id" type="var">extend</span></span> <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> *)</span><br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">extend</span>. <span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x0</span> <span class="id" type="var">x1</span>)...<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_App</span> <span class="id" type="keyword">with</span> <span class="id" type="var">T<sub>11</sub></span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
Now we come to the conceptual heart of the proof that reduction
preserves types — namely, the observation that <i>substitution</i>
preserves types.
<div class="paragraph"> </div>
Formally, the so-called <i>Substitution Lemma</i> says this: suppose we
have a term <span class="inlinecode"><span class="id" type="var">t</span></span> with a free variable <span class="inlinecode"><span class="id" type="var">x</span></span>, and suppose we've been
able to assign a type <span class="inlinecode"><span class="id" type="var">T</span></span> to <span class="inlinecode"><span class="id" type="var">t</span></span> under the assumption that <span class="inlinecode"><span class="id" type="var">x</span></span> has
some type <span class="inlinecode"><span class="id" type="var">U</span></span>. Also, suppose that we have some other term <span class="inlinecode"><span class="id" type="var">v</span></span> and
that we've shown that <span class="inlinecode"><span class="id" type="var">v</span></span> has type <span class="inlinecode"><span class="id" type="var">U</span></span>. Then, since <span class="inlinecode"><span class="id" type="var">v</span></span> satisfies
the assumption we made about <span class="inlinecode"><span class="id" type="var">x</span></span> when typing <span class="inlinecode"><span class="id" type="var">t</span></span>, we should be
able to substitute <span class="inlinecode"><span class="id" type="var">v</span></span> for each of the occurrences of <span class="inlinecode"><span class="id" type="var">x</span></span> in <span class="inlinecode"><span class="id" type="var">t</span></span>
and obtain a new term that still has type <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<i>Lemma</i>: If <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">U</span></span>, then <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span>
<span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">substitution_preserves_typing</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span> <span class="id" type="var">t</span> <span class="id" type="var">v</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">extend</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">v</span> ∈ <span class="id" type="var">U</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span>.<br/>
<br/>
</div>
<div class="doc">
One technical subtlety in the statement of the lemma is that we
assign <span class="inlinecode"><span class="id" type="var">v</span></span> the type <span class="inlinecode"><span class="id" type="var">U</span></span> in the <i>empty</i> context — in other words,
we assume <span class="inlinecode"><span class="id" type="var">v</span></span> is closed. This assumption considerably simplifies
the <span class="inlinecode"><span class="id" type="var">T_Abs</span></span> case of the proof (compared to assuming <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> <span class="inlinecode">∈</span>
<span class="inlinecode"><span class="id" type="var">U</span></span>, which would be the other reasonable assumption at this point)
because the context invariance lemma then tells us that <span class="inlinecode"><span class="id" type="var">v</span></span> has
type <span class="inlinecode"><span class="id" type="var">U</span></span> in any context at all — we don't have to worry about
free variables in <span class="inlinecode"><span class="id" type="var">v</span></span> clashing with the variable being introduced
into the context by <span class="inlinecode"><span class="id" type="var">T_Abs</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: We prove, by induction on <span class="inlinecode"><span class="id" type="var">t</span></span>, that, for all <span class="inlinecode"><span class="id" type="var">T</span></span> and
<span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, if <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">U</span></span>, then <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span>
<span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> is a variable, there are two cases to consider, depending
on whether <span class="inlinecode"><span class="id" type="var">t</span></span> is <span class="inlinecode"><span class="id" type="var">x</span></span> or some other variable.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">x</span></span>, then from the fact that <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span> <span class="inlinecode"><span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> we
conclude that <span class="inlinecode"><span class="id" type="var">U</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">T</span></span>. We must show that <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">v</span></span> has
type <span class="inlinecode"><span class="id" type="var">T</span></span> under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>, given the assumption that <span class="inlinecode"><span class="id" type="var">v</span></span> has
type <span class="inlinecode"><span class="id" type="var">U</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">T</span></span> under the empty context. This follows from
context invariance: if a closed term has type <span class="inlinecode"><span class="id" type="var">T</span></span> in the
empty context, it has that type in any context.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> is some variable <span class="inlinecode"><span class="id" type="var">y</span></span> that is not equal to <span class="inlinecode"><span class="id" type="var">x</span></span>, then
we need only note that <span class="inlinecode"><span class="id" type="var">y</span></span> has the same type under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,</span>
<span class="inlinecode"><span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> as under <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> is an abstraction <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, then the IH tells us,
for all <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> and <span class="inlinecode"><span class="id" type="var">T'</span></span>, that if <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T'</span></span>
and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">U</span></span>, then <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ'</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T'</span></span>.
<div class="paragraph"> </div>
The substitution in the conclusion behaves differently,
depending on whether <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">y</span></span> are the same variable name.
<div class="paragraph"> </div>
First, suppose <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">y</span></span>. Then, by the definition of
substitution, <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t</span></span>, so we just need to show <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span>
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>. But we know <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T</span></span>, and since the
variable <span class="inlinecode"><span class="id" type="var">y</span></span> does not appear free in <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, the
context invariance lemma yields <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
Second, suppose <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">≠</span> <span class="inlinecode"><span class="id" type="var">y</span></span>. We know <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span>,<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span>
<span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span> by inversion of the typing relation, and <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>,<span class="id" type="var">x</span>:<span class="id" type="var">U</span></span>
<span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span> follows from this by the context invariance
lemma, so the IH applies, giving us <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span>,<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span>
<span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span>. By <span class="inlinecode"><span class="id" type="var">T_Abs</span></span>, <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span><span style="font-family: arial;">→</span><span class="id" type="var">T<sub>12</sub></span></span>, and
by the definition of substitution (noting that <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">≠</span> <span class="inlinecode"><span class="id" type="var">y</span></span>),
<span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode">\<span class="id" type="var">y</span>:<span class="id" type="var">T<sub>11</sub></span>.</span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t<sub>12</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span><span style="font-family: arial;">→</span><span class="id" type="var">T<sub>12</sub></span></span> as required.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> is an application <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, the result follows
straightforwardly from the definition of substitution and the
induction hypotheses.
<div class="paragraph"> </div>
</li>
<li> The remaining cases are similar to the application case.
</li>
</ul>
<div class="paragraph"> </div>
Another technical note: This proof is a rare case where an
induction on terms, rather than typing derivations, yields a
simpler argument. The reason for this is that the assumption
<span class="inlinecode"><span class="id" type="var">extend</span></span> <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">U</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> is not completely generic, in
the sense that one of the "slots" in the typing relation — namely
the context — is not just a variable, and this means that Coq's
native induction tactic does not give us the induction hypothesis
that we want. It is possible to work around this, but the needed
generalization is a little tricky. The term <span class="inlinecode"><span class="id" type="var">t</span></span>, on the other
hand, <i>is</i> completely generic.
</div>
<div class="code code-tight">
<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="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span> <span class="id" type="var">t</span> <span class="id" type="var">v</span> <span class="id" type="var">T</span> <span class="id" type="var">Ht</span> <span class="id" type="var">Ht'</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span style="font-family: serif; font-size:85%;">Γ</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">T</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">t</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">T</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">H</span>;<br/>
<span class="comment">(* in each case, we'll want to get at the derivation of H *)</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="comment">(* tvar *)</span><br/>
<span class="id" type="tactic">rename</span> <span class="id" type="var">i</span> <span class="id" type="var">into</span> <span class="id" type="var">y</span>. <span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>).<br/>
+ <span class="comment">(* x=y *)</span><br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">extend_eq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>; <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">context_invariance</span>... <span class="id" type="tactic">intros</span> <span class="id" type="var">x</span> <span class="id" type="var">Hcontra</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">free_in_context</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">T</span> <span class="id" type="var">empty</span> <span class="id" type="var">Hcontra</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">T'</span> <span class="id" type="var">HT'</span>]...<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HT'</span>.<br/>
+ <span class="comment">(* x<>y *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Var</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">extend_neq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H2</span>...<br/>
- <span class="comment">(* tabs *)</span><br/>
<span class="id" type="tactic">rename</span> <span class="id" type="var">i</span> <span class="id" type="var">into</span> <span class="id" type="var">y</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">T_Abs</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>).<br/>
+ <span class="comment">(* x=y *)</span><br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">context_invariance</span>...<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">x</span> <span class="id" type="var">Hafi</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">extend</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">y</span> <span class="id" type="var">x</span>)...<br/>
+ <span class="comment">(* x<>y *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHt</span>. <span class="id" type="tactic">eapply</span> <span class="id" type="var">context_invariance</span>...<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">z</span> <span class="id" type="var">Hafi</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">extend</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>)...<br/>
<span class="id" type="tactic">subst</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">neq_id</span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
The substitution lemma can be viewed as a kind of "commutation"
property. Intuitively, it says that substitution and typing can
be done in either order: we can either assign types to the terms
<span class="inlinecode"><span class="id" type="var">t</span></span> and <span class="inlinecode"><span class="id" type="var">v</span></span> separately (under suitable contexts) and then combine
them using substitution, or we can substitute first and then
assign a type to <span class="inlinecode"></span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]</span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"></span> — the result is the same either
way.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab687"></a><h2 class="section">Main Theorem</h2>
<div class="paragraph"> </div>
We now have the tools we need to prove preservation: if a closed
term <span class="inlinecode"><span class="id" type="var">t</span></span> has type <span class="inlinecode"><span class="id" type="var">T</span></span>, and takes an evaluation step to <span class="inlinecode"><span class="id" type="var">t'</span></span>, then <span class="inlinecode"><span class="id" type="var">t'</span></span>
is also a closed term with type <span class="inlinecode"><span class="id" type="var">T</span></span>. In other words, the small-step
evaluation relation preserves types.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">preservation</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t'</span> ∈ <span class="id" type="var">T</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Proof</i>: by induction on the derivation of <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> We can immediately rule out <span class="inlinecode"><span class="id" type="var">T_Var</span></span>, <span class="inlinecode"><span class="id" type="var">T_Abs</span></span>, <span class="inlinecode"><span class="id" type="var">T_True</span></span>, and
<span class="inlinecode"><span class="id" type="var">T_False</span></span> as the final rules in the derivation, since in each of
these cases <span class="inlinecode"><span class="id" type="var">t</span></span> cannot take a step.
<div class="paragraph"> </div>
</li>
<li> If the last rule in the derivation was <span class="inlinecode"><span class="id" type="var">T_App</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>. There are three cases to consider, one for each rule that
could have been used to show that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> takes a step to <span class="inlinecode"><span class="id" type="var">t'</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> takes a step by <span class="inlinecode"><span class="id" type="var">ST_App1</span></span>, with <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> stepping to
<span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>, then by the IH <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span> has the same type as <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>, and
hence <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> has the same type as <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>.
<div class="paragraph"> </div>
</li>
<li> The <span class="inlinecode"><span class="id" type="var">ST_App2</span></span> case is similar.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> takes a step by <span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span>, then <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">=</span>
<span class="inlinecode">\<span class="id" type="var">x</span>:<span class="id" type="var">T<sub>11</sub>.t<sub>12</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> steps to <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">t<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></span></span>; the
desired result now follows from the fact that substitution
preserves types.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If the last rule in the derivation was <span class="inlinecode"><span class="id" type="var">T_If</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, and there are again three cases depending on
how <span class="inlinecode"><span class="id" type="var">t</span></span> steps.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t</span></span> steps to <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> or <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, the result is immediate, since
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span> have the same type as <span class="inlinecode"><span class="id" type="var">t</span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" type="var">t</span></span> steps by <span class="inlinecode"><span class="id" type="var">ST_If</span></span>, and the desired conclusion
follows directly from the induction hypothesis.
</li>
</ul>
</li>
</ul>
</div>
<div class="code code-tight">
<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> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="var">remember</span> (@<span class="id" type="var">empty</span> <span class="id" type="var">ty</span>) <span class="id" type="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span> <span class="id" type="var">HT</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">t'</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">HT</span>;<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t'</span> <span class="id" type="var">HE</span>; <span class="id" type="tactic">subst</span> <span style="font-family: serif; font-size:85%;">Γ</span>; <span class="id" type="tactic">subst</span>; <br/>
<span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> [<span class="id" type="tactic">inversion</span> <span class="id" type="var">HE</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">auto</span>].<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HE</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="comment">(* Most of the cases are immediate by induction, <br/>
and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> takes care of them *)</span><br/>
+ <span class="comment">(* ST_AppAbs *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">substitution_preserves_typing</span> <span class="id" type="keyword">with</span> <span class="id" type="var">T<sub>11</sub></span>...<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HT1</span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab688"></a><h4 class="section">Exercise: 2 stars (subject_expansion_stlc)</h4>
An exercise in the <span class="inlinecode"><span class="id" type="keyword">Types</span></span> chapter asked about the subject
expansion property for the simple language of arithmetic and
boolean expressions. Does this property hold for STLC? That is,
is it always the case that, if <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> and <span class="inlinecode"><span class="id" type="var">has_type</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span>,
then <span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>? If so, prove it. If not, give a
counter-example not involving conditionals.
<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="lab689"></a><h1 class="section">Type Soundness</h1>
<div class="paragraph"> </div>
<a name="lab690"></a><h4 class="section">Exercise: 2 stars, optional (type_soundness)</h4>
<div class="paragraph"> </div>
Put progress and preservation together and show that a well-typed
term can <i>never</i> reach a stuck state.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">stuck</span> (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
(<span class="id" type="var">normal_form</span> <span class="id" type="var">step</span>) <span class="id" type="var">t</span> <span style="font-family: arial;">∧</span> ¬ <span class="id" type="var">value</span> <span class="id" type="var">t</span>.<br/>
<br/>
<span class="id" type="keyword">Corollary</span> <span class="id" type="var">soundness</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">t</span> <span style="font-family: arial;">⇒*</span> <span class="id" type="var">t'</span> <span style="font-family: arial;">→</span><br/>
~(<span class="id" type="var">stuck</span> <span class="id" type="var">t'</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span> <span class="id" type="var">Hhas_type</span> <span class="id" type="var">Hmulti</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">stuck</span>.<br/>
<span class="id" type="tactic">intros</span> [<span class="id" type="var">Hnf</span> <span class="id" type="var">Hnot_val</span>]. <span class="id" type="tactic">unfold</span> <span class="id" type="var">normal_form</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hnf</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Hmulti</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab691"></a><h1 class="section">Uniqueness of Types</h1>
<div class="paragraph"> </div>
<a name="lab692"></a><h4 class="section">Exercise: 3 stars (types_unique)</h4>
Another pleasant property of the STLC is that types are
unique: a given term (in a given context) has at most one
type. Formalize this statement and prove it.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab693"></a><h1 class="section">Additional Exercises</h1>