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Typechecking.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Typechecking: A Typechecker for 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">Typechecking<span class="subtitle">A Typechecker for STLC</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="var">has_type</span></span> relation of the STLC defines what it means for a
term to belong to a type (in some context). But it doesn't, by
itself, tell us how to <i>check</i> whether or not a term is well
typed.
<div class="paragraph"> </div>
Fortunately, the rules defining <span class="inlinecode"><span class="id" type="var">has_type</span></span> are <i>syntax directed</i>
— they exactly follow the shape of the term. This makes it
straightforward to translate the typing rules into clauses of a
typechecking <i>function</i> that takes a term and a context and either
returns the term's type or else signals that the term is not
typable. This short chapter constructs such a function and proves
it correct.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Bool.Bool.html#"><span class="id" type="library">Coq.Bool.Bool</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Maps.html#"><span class="id" type="library">Maps</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Smallstep.html#"><span class="id" type="library">Smallstep</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="Stlc.html#"><span class="id" type="library">Stlc</span></a>.<br/>
<br/>
<span class="id" type="keyword">Module</span> <a name="STLCChecker"><span class="id" type="module">STLCChecker</span></a>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab688"></a><h2 class="section">Comparing Types</h2>
<div class="paragraph"> </div>
First, we need a function to compare two types for equality...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> (<span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>:<a class="idref" href="Stlc.html#STLC.ty"><span class="id" type="inductive">ty</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#bool"><span class="id" type="inductive">bool</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#T<sub>1</sub>"><span class="id" type="variable">T<sub>1</sub></span></a>,<a class="idref" href="Typechecking.html#T<sub>2</sub>"><span class="id" type="variable">T<sub>2</sub></span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a>, <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span>, <a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <span class="id" type="var">T<sub>21</sub></span> <span class="id" type="var">T<sub>22</sub></span> ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#andb"><span class="id" type="definition">andb</span></a> (<a class="idref" href="Typechecking.html#beq_ty"><span class="id" type="definition">beq_ty</span></a> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>21</sub></span>) (<a class="idref" href="Typechecking.html#beq_ty"><span class="id" type="definition">beq_ty</span></a> <span class="id" type="var">T<sub>12</sub></span> <span class="id" type="var">T<sub>22</sub></span>)<br/>
| <span class="id" type="var">_</span>,_ ⇒<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
... and we need to establish the usual two-way connection between
the boolean result returned by <span class="inlinecode"><span class="id" type="var">beq_ty</span></span> and the logical
proposition that its inputs are equal.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <a name="STLCChecker.beq_ty_refl"><span class="id" type="lemma">beq_ty_refl</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">T<sub>1</sub></span>,<br/>
<a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <a class="idref" href="Typechecking.html#T<sub>1</sub>"><span class="id" type="variable">T<sub>1</sub></span></a> <a class="idref" href="Typechecking.html#T<sub>1</sub>"><span class="id" type="variable">T<sub>1</sub></span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">T<sub>1</sub></span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">T<sub>1</sub></span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHT1_1</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHT1_2</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="STLCChecker.beq_ty__eq"><span class="id" type="lemma">beq_ty__eq</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>,<br/>
<a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <a class="idref" href="Typechecking.html#T<sub>1</sub>"><span class="id" type="variable">T<sub>1</sub></span></a> <a class="idref" href="Typechecking.html#T<sub>2</sub>"><span class="id" type="variable">T<sub>2</sub></span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Typechecking.html#T<sub>1</sub>"><span class="id" type="variable">T<sub>1</sub></span></a> = <a class="idref" href="Typechecking.html#T<sub>2</sub>"><span class="id" type="variable">T<sub>2</sub></span></a>.<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">T<sub>1</sub></span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">T<sub>1</sub></span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="var">Hbeq</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">T<sub>2</sub></span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hbeq</span>.<br/>
- <span class="comment">(* T<sub>1</sub>=TBool *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* T<sub>1</sub>=TArrow T1_1 T1_2 *)</span><br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Bool.Bool.html#andb_true_iff"><span class="id" type="lemma">andb_true_iff</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H<sub>0</sub></span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>0</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Hbeq1</span> <span class="id" type="var">Hbeq2</span>].<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHT1_1</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hbeq1</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHT1_2</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hbeq2</span>. <span class="id" type="tactic">subst</span>... <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab689"></a><h2 class="section">The Typechecker</h2>
<div class="paragraph"> </div>
The typechecker works by walking over the structure of the given
term, returning either <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> or <span class="inlinecode"><span class="id" type="var">None</span></span>. Each time we make a
recursive call to find out the types of the subterms, we need to
pattern-match on the results to make sure that they are not
<span class="inlinecode"><span class="id" type="var">None</span></span>. Also, in the <span class="inlinecode"><span class="id" type="var">tapp</span></span> case, we use pattern matching to
extract the left- and right-hand sides of the function's arrow
type (and fail if the type of the function is not <span class="inlinecode"><span class="id" type="var">TArrow</span></span> <span class="inlinecode"><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>
for some <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>).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> (<span style="font-family: serif; font-size:85%;">Γ</span>:<a class="idref" href="Stlc.html#STLC.context"><span class="id" type="definition">context</span></a>) (<span class="id" type="var">t</span>:<a class="idref" href="Stlc.html#STLC.tm"><span class="id" type="inductive">tm</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#option"><span class="id" type="inductive">option</span></a> <a class="idref" href="Stlc.html#STLC.ty"><span class="id" type="inductive">ty</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Stlc.html#STLC.tvar"><span class="id" type="constructor">tvar</span></a> <span class="id" type="var">x</span> ⇒ <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span><br/>
| <a class="idref" href="Stlc.html#STLC.tabs"><span class="id" type="constructor">tabs</span></a> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span> ⇒ <span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> (<a class="idref" href="Maps.html#update"><span class="id" type="definition">update</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span>) <span class="id" type="var">t<sub>12</sub></span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>12</sub></span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> (<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <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">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
| <a class="idref" href="Stlc.html#STLC.tapp"><span class="id" type="constructor">tapp</span></a> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> ⇒ <span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>1</sub></span>, <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> (<a class="idref" href="Stlc.html#STLC.TArrow"><span class="id" type="constructor">TArrow</span></a> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span>),<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>2</sub></span> ⇒<br/>
<span class="id" type="keyword">if</span> <a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="keyword">then</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>12</sub></span> <span class="id" type="keyword">else</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <span class="id" type="var">_</span>,_ ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
| <a class="idref" href="Stlc.html#STLC.ttrue"><span class="id" type="constructor">ttrue</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.tfalse"><span class="id" type="constructor">tfalse</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a><br/>
| <a class="idref" href="Stlc.html#STLC.tif"><span class="id" type="constructor">tif</span></a> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">f</span> ⇒ <span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="Stlc.html#STLC.TBool"><span class="id" type="constructor">TBool</span></a> ⇒<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a>, <a class="idref" href="Typechecking.html#type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">f</span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>1</sub></span>, <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>2</sub></span> ⇒<br/>
<span class="id" type="keyword">if</span> <a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="keyword">then</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="keyword">else</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <span class="id" type="var">_</span>,_ ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
| <span class="id" type="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab690"></a><h2 class="section">Properties</h2>
<div class="paragraph"> </div>
To verify that this typechecking algorithm is correct, we show
that it is <i>sound</i> and <i>complete</i> for the original <span class="inlinecode"><span class="id" type="var">has_type</span></span>
relation — that is, <span class="inlinecode"><span class="id" type="var">type_check</span></span> and <span class="inlinecode"><span class="id" type="var">has_type</span></span> define the same
partial function.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="STLCChecker.type_checking_sound"><span class="id" type="lemma">type_checking_sound</span></a> : <span style="font-family: arial;">∀</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/>
<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="Typechecking.html#T"><span class="id" type="variable">T</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Stlc.html#STLC.has_type"><span class="id" type="inductive">has_type</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="Typechecking.html#T"><span class="id" type="variable">T</span></a>.<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> <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">t</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">induction</span> <span class="id" type="var">t</span>; <span class="id" type="tactic">intros</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">T</span> <span class="id" type="var">Htc</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">Htc</span>.<br/>
- <span class="comment">(* tvar *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* tapp *)</span><br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>1</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TO<sub>1</sub></span>.<br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>2</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TO<sub>2</sub></span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TO<sub>1</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">T<sub>1</sub></span>|]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>;<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span>]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TO<sub>2</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">T<sub>2</sub></span>|]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>2</sub></span>) <span class="id" type="var">eqn</span>: <span class="id" type="var">Heqb</span>;<br/>
<span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Typechecking.html#STLCChecker.beq_ty__eq"><span class="id" type="lemma">beq_ty__eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Heqb</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>0</sub></span>; <span class="id" type="tactic">subst</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">rename</span> <span class="id" type="var">t</span> <span class="id" type="var">into</span> <span class="id" type="var">T<sub>1</sub></span>.<br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Maps.html#update"><span class="id" type="definition">update</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">y</span> <span class="id" type="var">T<sub>1</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">G'</span>.<br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span class="id" type="var">G'</span> <span class="id" type="var">t<sub>0</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TO<sub>2</sub></span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TO<sub>2</sub></span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>0</sub></span>; <span class="id" type="tactic">subst</span>...<br/>
- <span class="comment">(* ttrue *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* tfalse *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* tif *)</span><br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>1</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TOc</span>.<br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>2</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TO<sub>1</sub></span>.<br/>
<span class="id" type="var">remember</span> (<a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>3</sub></span>) <span class="id" type="keyword">as</span> <span class="id" type="var">TO<sub>2</sub></span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TOc</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Tc</span>|]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">Tc</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TO<sub>1</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">T<sub>1</sub></span>|]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">TO<sub>2</sub></span> <span class="id" type="keyword">as</span> [<span class="id" type="var">T<sub>2</sub></span>|]; <span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="Typechecking.html#STLCChecker.beq_ty"><span class="id" type="definition">beq_ty</span></a> <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">eqn</span>:<span class="id" type="var">Heqb</span>;<br/>
<span class="id" type="tactic">try</span> <span class="id" type="var">solve_by_invert</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="Typechecking.html#STLCChecker.beq_ty__eq"><span class="id" type="lemma">beq_ty__eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Heqb</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>0</sub></span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="STLCChecker.type_checking_complete"><span class="id" type="lemma">type_checking_complete</span></a> : <span style="font-family: arial;">∀</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/>
<a class="idref" href="Stlc.html#STLC.has_type"><span class="id" type="inductive">has_type</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a> <a class="idref" href="Typechecking.html#T"><span class="id" type="variable">T</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Typechecking.html#STLCChecker.type_check"><span class="id" type="definition">type_check</span></a> <span style="font-family: serif; font-size:85%;">Γ</span> <a class="idref" href="Typechecking.html#t"><span class="id" type="variable">t</span></a> = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="Typechecking.html#T"><span class="id" type="variable">T</span></a>.<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">auto</span>.<br/>
<span class="id" type="tactic">intros</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> <span class="id" type="var">Hty</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Hty</span>; <span class="id" type="tactic">simpl</span>.<br/>
- <span class="comment">(* T_Var *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* T_Abs *)</span> <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty</span>...<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty1</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty2</span>.<br/>
<span class="id" type="tactic">rewrite</span> (<a class="idref" href="Typechecking.html#STLCChecker.beq_ty_refl"><span class="id" type="lemma">beq_ty_refl</span></a> <span class="id" type="var">T<sub>11</sub></span>)...<br/>
- <span class="comment">(* T_True *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* T_False *)</span> <span class="id" type="tactic">eauto</span>.<br/>
- <span class="comment">(* T_If *)</span> <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty1</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty2</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHHty3</span>. <span class="id" type="tactic">rewrite</span> (<a class="idref" href="Typechecking.html#STLCChecker.beq_ty_refl"><span class="id" type="lemma">beq_ty_refl</span></a> <span class="id" type="var">T</span>)...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">End</span> <a class="idref" href="Typechecking.html#STLCChecker"><span class="id" type="module">STLCChecker</span></a>.<br/>
<br/>
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