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RecordSub.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>RecordSub: Subtyping with Records</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">RecordSub<span class="subtitle">Subtyping with Records</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">MoreStlc</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab853"></a><h1 class="section">Core Definitions</h1>
</div>
<div class="code code-space">
<br/>
</div>
<div class="doc">
<a name="lab854"></a><h3 class="section">Syntax</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ty</span> : <span class="id" type="keyword">Type</span> :=<br/>
<span class="comment">(* proper types *)</span><br/>
| <span class="id" type="var">TTop</span> : <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TBase</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TArrow</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
<span class="comment">(* record types *)</span><br/>
| <span class="id" type="var">TRNil</span> : <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TRCons</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "T_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TTop" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TBase" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TArrow"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TRNil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TRCons" ].<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">tm</span> : <span class="id" type="keyword">Type</span> :=<br/>
<span class="comment">(* proper terms *)</span><br/>
| <span class="id" type="var">tvar</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tapp</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tabs</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tproj</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
<span class="comment">(* record terms *)</span><br/>
| <span class="id" type="var">trnil</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">trcons</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="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "t_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tvar" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tapp" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tabs"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tproj" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "trnil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "trcons" ].<br/>
<br/>
</div>
<div class="doc">
<a name="lab855"></a><h3 class="section">Well-Formedness</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">record_ty</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">RTnil</span> : <br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">RTcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</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">record_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</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="id" type="keyword">Inductive</span> <span class="id" type="var">record_tm</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">rtnil</span> :<br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">rtcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</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">record_tm</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</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="id" type="keyword">Inductive</span> <span class="id" type="var">well_formed_ty</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">wfTTop</span> : <br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">TTop</span><br/>
| <span class="id" type="var">wfTBase</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span>,<br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TBase</span> <span class="id" type="var">i</span>)<br/>
| <span class="id" type="var">wfTArrow</span> : <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/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</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>)<br/>
| <span class="id" type="var">wfTRNil</span> :<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">wfTRCons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</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">well_formed_ty</span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</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="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">record_ty</span> <span class="id" type="var">record_tm</span> <span class="id" type="var">well_formed_ty</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab856"></a><h3 class="section">Substitution</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="tactic">subst</span> (<span class="id" type="var">x</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">tm</span>) (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) : <span class="id" type="var">tm</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">t</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">tvar</span> <span class="id" type="var">y</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="keyword">then</span> <span class="id" type="var">s</span> <span class="id" type="keyword">else</span> <span class="id" type="var">t</span><br/>
| <span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T</span> <span class="id" type="var">t<sub>1</sub></span> ⇒ <span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T</span> (<span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="keyword">else</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>))<br/>
| <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> ⇒ <span class="id" type="var">tapp</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">tproj</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">i</span> ⇒ <span class="id" type="var">tproj</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) <span class="id" type="var">i</span><br/>
| <span class="id" type="var">trnil</span> ⇒ <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">tr2</span> ⇒ <span class="id" type="var">trcons</span> <span class="id" type="var">i</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">tr2</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "'[' x ':=' s ']' t" := (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 20).<br/>
<br/>
</div>
<div class="doc">
<a name="lab857"></a><h3 class="section">Reduction</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">value</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">v_abs</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>,<br/>
<span class="id" type="var">value</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t</span>)<br/>
| <span class="id" type="var">v_rnil</span> : <span class="id" type="var">value</span> <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">v_rcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">v</span> <span class="id" type="var">vr</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">vr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v</span> <span class="id" type="var">vr</span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">value</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">Tlookup</span> (<span class="id" type="var">i</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">Tr</span>:<span class="id" type="var">ty</span>) : <span class="id" type="var">option</span> <span class="id" type="var">ty</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">Tr</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">TRCons</span> <span class="id" type="var">i'</span> <span class="id" type="var">T</span> <span class="id" type="var">Tr'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">T</span> <span class="id" type="keyword">else</span> <span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">Tr'</span><br/>
| <span class="id" type="var">_</span> ⇒ <span class="id" type="var">None</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">tlookup</span> (<span class="id" type="var">i</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">tr</span>:<span class="id" type="var">tm</span>) : <span class="id" type="var">option</span> <span class="id" type="var">tm</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">tr</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">trcons</span> <span class="id" type="var">i'</span> <span class="id" type="var">t</span> <span class="id" type="var">tr'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">t</span> <span class="id" type="keyword">else</span> <span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">tr'</span><br/>
| <span class="id" type="var">_</span> ⇒ <span class="id" type="var">None</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Reserved Notation</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">step</span> : <span class="id" type="var">tm</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">ST_AppAbs</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<sub>12</sub></span> <span class="id" type="var">v<sub>2</sub></span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t<sub>12</sub></span>) <span class="id" type="var">v<sub>2</sub></span>) <span style="font-family: arial;">⇒</span> [<span class="id" type="var">x</span>:=<span class="id" type="var">v<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></span><br/>
| <span class="id" type="var">ST_App1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></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">t<sub>1</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>1</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<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>) <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/>
| <span class="id" type="var">ST_App2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>2</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>)<br/>
| <span class="id" type="var">ST_Proj1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">tr</span> <span class="id" type="var">tr'</span> <span class="id" type="var">i</span>,<br/>
<span class="id" type="var">tr</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tr'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tproj</span> <span class="id" type="var">tr</span> <span class="id" type="var">i</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tproj</span> <span class="id" type="var">tr'</span> <span class="id" type="var">i</span>)<br/>
| <span class="id" type="var">ST_ProjRcd</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">tr</span> <span class="id" type="var">i</span> <span class="id" type="var">vi</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">tr</span> = <span class="id" type="var">Some</span> <span class="id" type="var">vi</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tproj</span> <span class="id" type="var">tr</span> <span class="id" type="var">i</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">vi</span><br/>
| <span class="id" type="var">ST_Rcd_Head</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">tr2</span>,<br/>
<span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>1</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">tr2</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">tr2</span>)<br/>
| <span class="id" type="var">ST_Rcd_Tail</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2</span> <span class="id" type="var">tr2'</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tr2</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tr2'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2'</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" := (<span class="id" type="var">step</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="id" type="keyword">Tactic Notation</span> "step_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_AppAbs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App2"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Proj1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_ProjRcd" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Rcd"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Rcd_Head" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Rcd_Tail" ].<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">step</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab858"></a><h1 class="section">Subtyping</h1>
<div class="paragraph"> </div>
Now we come to the interesting part. We begin by defining
the subtyping relation and developing some of its important
technical properties.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab859"></a><h2 class="section">Definition</h2>
<div class="paragraph"> </div>
The definition of subtyping is essentially just what we
sketched in the motivating discussion, but we need to add
well-formedness side conditions to some of the rules.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">subtype</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="comment">(* Subtyping between proper types *)</span><br/>
| <span class="id" type="var">S_Refl</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">T</span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">T</span> <span class="id" type="var">T</span><br/>
| <span class="id" type="var">S_Trans</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">S</span> <span class="id" type="var">U</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">U</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">U</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span><br/>
| <span class="id" type="var">S_Top</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">S</span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">S</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">TTop</span><br/>
| <span class="id" type="var">S_Arrow</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">S<sub>2</sub></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">subtype</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">S<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S<sub>2</sub></span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">S<sub>2</sub></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>)<br/>
<span class="comment">(* Subtyping between record types *)</span><br/>
| <span class="id" type="var">S_RcdWidth</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</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">well_formed_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</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">subtype</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</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">TRNil</span><br/>
| <span class="id" type="var">S_RcdDepth</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">Sr2</span> <span class="id" type="var">Tr2</span>,<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">Sr2</span> <span class="id" type="var">Tr2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">Sr2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">Tr2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">Sr2</span>) (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">Tr2</span>)<br/>
| <span class="id" type="var">S_RcdPerm</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i1</span> <span class="id" type="var">i2</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">Tr3</span>,<br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i1</span> <span class="id" type="var">T<sub>1</sub></span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i2</span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="var">Tr3</span>)) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">i1</span> ≠ <span class="id" type="var">i2</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i1</span> <span class="id" type="var">T<sub>1</sub></span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i2</span> <span class="id" type="var">T<sub>2</sub></span> <span class="id" type="var">Tr3</span>))<br/>
(<span class="id" type="var">TRCons</span> <span class="id" type="var">i2</span> <span class="id" type="var">T<sub>2</sub></span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i1</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">Tr3</span>)).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">subtype</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "subtype_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_Refl" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_Trans" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_Top"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_Arrow" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_RcdWidth"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_RcdDepth" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "S_RcdPerm" ].<br/>
<br/>
</div>
<div class="doc">
<a name="lab860"></a><h2 class="section">Subtyping Examples and Exercises</h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Examples</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">x</span> := (<span class="id" type="var">Id</span> 0).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">y</span> := (<span class="id" type="var">Id</span> 1).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">z</span> := (<span class="id" type="var">Id</span> 2).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">j</span> := (<span class="id" type="var">Id</span> 3).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">k</span> := (<span class="id" type="var">Id</span> 4).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">i</span> := (<span class="id" type="var">Id</span> 5).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">A</span> := (<span class="id" type="var">TBase</span> (<span class="id" type="var">Id</span> 6)).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">B</span> := (<span class="id" type="var">TBase</span> (<span class="id" type="var">Id</span> 7)).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">C</span> := (<span class="id" type="var">TBase</span> (<span class="id" type="var">Id</span> 8)).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">TRcd_j</span> := <br/>
(<span class="id" type="var">TRCons</span> <span class="id" type="var">j</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">B</span> <span class="id" type="var">B</span>) <span class="id" type="var">TRNil</span>). <span class="comment">(* {j:B->B} *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">TRcd_kj</span> := <br/>
<span class="id" type="var">TRCons</span> <span class="id" type="var">k</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">A</span> <span class="id" type="var">A</span>) <span class="id" type="var">TRcd_j</span>. <span class="comment">(* {k:C->C,j:B->B} *)</span><br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">subtyping_example_0</span> :<br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">C</span> <span class="id" type="var">TRcd_kj</span>) <br/>
(<span class="id" type="var">TArrow</span> <span class="id" type="var">C</span> <span class="id" type="var">TRNil</span>).<br/>
<span class="comment">(* C->{k:A->A,j:B->B} <: C->{} *)</span><br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">S_Arrow</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">S_Refl</span>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">TRcd_kj</span>, <span class="id" type="var">TRcd_j</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">S_RcdWidth</span>; <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The following facts are mostly easy to prove in Coq. To get
full benefit from the exercises, make sure you also
understand how to prove them on paper!
<div class="paragraph"> </div>
<a name="lab861"></a><h4 class="section">Exercise: 2 stars</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">subtyping_example_1</span> :<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">TRcd_kj</span> <span class="id" type="var">TRcd_j</span>.<br/>
<span class="comment">(* {k:A->A,j:B->B} <: {j:B->B} *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab862"></a><h4 class="section">Exercise: 1 star</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">subtyping_example_2</span> :<br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">TTop</span> <span class="id" type="var">TRcd_kj</span>) <br/>
(<span class="id" type="var">TArrow</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">C</span> <span class="id" type="var">C</span>) <span class="id" type="var">TRcd_j</span>).<br/>
<span class="comment">(* Top->{k:A->A,j:B->B} <: (C->C)->{j:B->B} *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab863"></a><h4 class="section">Exercise: 1 star</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">subtyping_example_3</span> :<br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">TRNil</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">j</span> <span class="id" type="var">A</span> <span class="id" type="var">TRNil</span>)) <br/>
(<span class="id" type="var">TArrow</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">k</span> <span class="id" type="var">B</span> <span class="id" type="var">TRNil</span>) <span class="id" type="var">TRNil</span>).<br/>
<span class="comment">(* {}->{j:A} <: {k:B}->{} *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab864"></a><h4 class="section">Exercise: 2 stars</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">subtyping_example_4</span> :<br/>
<span class="id" type="var">subtype</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">x</span> <span class="id" type="var">A</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">y</span> <span class="id" type="var">B</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">z</span> <span class="id" type="var">C</span> <span class="id" type="var">TRNil</span>)))<br/>
(<span class="id" type="var">TRCons</span> <span class="id" type="var">z</span> <span class="id" type="var">C</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">y</span> <span class="id" type="var">B</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">x</span> <span class="id" type="var">A</span> <span class="id" type="var">TRNil</span>))).<br/>
<span class="comment">(* {x:A,y:B,z:C} <: {z:C,y:B,x:A} *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">trcd_kj</span> := <br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">k</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> <span class="id" type="var">A</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">z</span>)) <br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">j</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> <span class="id" type="var">B</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">z</span>)) <br/>
<span class="id" type="var">trnil</span>)).<br/>
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">Examples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab865"></a><h2 class="section">Properties of Subtyping</h2>
<div class="paragraph"> </div>
<a name="lab866"></a><h3 class="section">Well-Formedness</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">subtype__wf</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">S</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">well_formed_ty</span> <span class="id" type="var">S</span>.<br/>
<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">S</span> <span class="id" type="var">T</span> <span class="id" type="var">Hsub</span>.<br/>
<span class="id" type="var">subtype_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Hsub</span>) <span class="id" type="var">Case</span>; <br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">try</span> (<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHsub1</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHsub2</span>)...<br/>
<span class="id" type="var">Case</span> "S_RcdPerm".<br/>
<span class="id" type="tactic">split</span>... <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">wf_rcd_lookup</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">T</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">Ti</span>.<br/>
<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">i</span> <span class="id" type="var">T</span>.<br/>
<span class="id" type="var">T_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">T</span>) <span class="id" type="var">Case</span>; <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">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="var">Case</span> "TRCons".<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">unfold</span> <span class="id" type="var">Tlookup</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i0</span>)... <span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>; <span class="id" type="tactic">subst</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab867"></a><h3 class="section">Field Lookup</h3>
<div class="paragraph"> </div>
Our record matching lemmas get a little more complicated in
the presence of subtyping for two reasons: First, record
types no longer necessarily describe the exact structure of
corresponding terms. Second, reasoning by induction on
<span class="inlinecode"><span class="id" type="var">has_type</span></span> derivations becomes harder in general, because
<span class="inlinecode"><span class="id" type="var">has_type</span></span> is no longer syntax directed.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">rcd_types_match</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">S</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">Si</span>, <span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">S</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Si</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">subtype</span> <span class="id" type="var">Si</span> <span class="id" type="var">Ti</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> (<span class="id" type="tactic">eauto</span> <span class="id" type="keyword">using</span> <span class="id" type="var">wf_rcd_lookup</span>).<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span> <span class="id" type="var">Hsub</span> <span class="id" type="var">Hget</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">Ti</span>.<br/>
<span class="id" type="var">subtype_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Hsub</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">Ti</span> <span class="id" type="var">Hget</span>;<br/>
<span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="var">Case</span> "S_Refl".<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">Ti</span>...<br/>
<span class="id" type="var">Case</span> "S_Trans".<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">IHHsub2</span> <span class="id" type="var">Ti</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">Ui</span> <span class="id" type="var">Hui</span>]... <span class="id" type="tactic">destruct</span> <span class="id" type="var">Hui</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">IHHsub1</span> <span class="id" type="var">Ui</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">Si</span> <span class="id" type="var">Hsi</span>]... <span class="id" type="tactic">destruct</span> <span class="id" type="var">Hsi</span>.<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">Si</span>...<br/>
<span class="id" type="var">Case</span> "S_RcdDepth".<br/>
<span class="id" type="tactic">rename</span> <span class="id" type="var">i0</span> <span class="id" type="var">into</span> <span class="id" type="var">k</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">Tlookup</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">Tlookup</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hget</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">k</span>)...<br/>
<span class="id" type="var">SCase</span> "i = k -- we're looking up the first field".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hget</span>. <span class="id" type="tactic">subst</span>. <span style="font-family: arial;">∃</span><span class="id" type="var">S<sub>1</sub></span>...<br/>
<span class="id" type="var">Case</span> "S_RcdPerm".<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">Ti</span>. <span class="id" type="tactic">split</span>.<br/>
<span class="id" type="var">SCase</span> "lookup".<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">Tlookup</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">Tlookup</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hget</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i1</span>)...<br/>
<span class="id" type="var">SSCase</span> "i = i1 -- we're looking up the first field".<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i2</span>)...<br/>
<span class="id" type="var">SSSCase</span> "i = i2 - -contradictory".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">SCase</span> "subtype".<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">inversion</span> <span class="id" type="var">H5</span>. <span class="id" type="tactic">subst</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab868"></a><h4 class="section">Exercise: 3 stars (rcd_types_match_informal)</h4>
Write a careful informal proof of the <span class="inlinecode"><span class="id" type="var">rcd_types_match</span></span>
lemma.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab869"></a><h3 class="section">Inversion Lemmas</h3>
<div class="paragraph"> </div>
<a name="lab870"></a><h4 class="section">Exercise: 3 stars, optional (sub_inversion_arrow)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">sub_inversion_arrow</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">U</span> <span class="id" type="var">V1</span> <span class="id" type="var">V2</span>,<br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">U</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">V1</span> <span class="id" type="var">V2</span>) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">U1</span>, <span style="font-family: arial;">∃</span><span class="id" type="var">U2</span>, <br/>
(<span class="id" type="var">U</span>=(<span class="id" type="var">TArrow</span> <span class="id" type="var">U1</span> <span class="id" type="var">U2</span>)) <span style="font-family: arial;">∧</span> (<span class="id" type="var">subtype</span> <span class="id" type="var">V1</span> <span class="id" type="var">U1</span>) <span style="font-family: arial;">∧</span> (<span class="id" type="var">subtype</span> <span class="id" type="var">U2</span> <span class="id" type="var">V2</span>).<br/>
<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">U</span> <span class="id" type="var">V1</span> <span class="id" type="var">V2</span> <span class="id" type="var">Hs</span>.<br/>
<span class="id" type="var">remember</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">V1</span> <span class="id" type="var">V2</span>) <span class="id" type="keyword">as</span> <span class="id" type="var">V</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">V2</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">V1</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab871"></a><h1 class="section">Typing</h1>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">context</span> := <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> (<span class="id" type="var">option</span> <span class="id" type="var">ty</span>).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">empty</span> : <span class="id" type="var">context</span> := (<span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="var">None</span>).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">extend</span> (<span style="font-family: serif; font-size:85%;">Γ</span> : <span class="id" type="var">context</span>) (<span class="id" type="var">x</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">T</span> : <span class="id" type="var">ty</span>) :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">x'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">x'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">T</span> <span class="id" type="keyword">else</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x'</span>.<br/>
<br/>
<span class="id" type="keyword">Reserved Notation</span> "Gamma '<span style="font-family: arial;">⊢</span>' t '∈' T" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">has_type</span> : <span class="id" type="var">context</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">T_Var</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">T</span>,<br/>
<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> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>) <span class="id" type="var">T</span><br/>
| <span class="id" type="var">T_Abs</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">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span> <span class="id" type="var">t<sub>12</sub></span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>11</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> (<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">T<sub>11</sub></span>) <span class="id" type="var">t<sub>12</sub></span> <span class="id" type="var">T<sub>12</sub></span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> (<span class="id" type="var">tabs</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="var">TArrow</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">T_App</span> : <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> <span style="font-family: serif; font-size:85%;">Γ</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">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>1</sub></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">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</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>) <span class="id" type="var">T<sub>2</sub></span><br/>
| <span class="id" type="var">T_Proj</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">has_type</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 style="font-family: arial;">→</span><br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> (<span class="id" type="var">tproj</span> <span class="id" type="var">t</span> <span class="id" type="var">i</span>) <span class="id" type="var">Ti</span><br/>
<span class="comment">(* Subsumption *)</span><br/>
| <span class="id" type="var">T_Sub</span> : <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">S</span> <span class="id" type="var">T</span>,<br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t</span> <span class="id" type="var">S</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</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 class="comment">(* Rules for record terms *)</span><br/>
| <span class="id" type="var">T_RNil</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span>,<br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">trnil</span> <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">T_RCons</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">tr</span> <span class="id" type="var">Tr</span>,<br/>
<span class="id" type="var">has_type</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 style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">tr</span> <span class="id" type="var">Tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">Tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">tr</span>) (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> <span class="id" type="var">Tr</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "Gamma '<span style="font-family: arial;">⊢</span>' t '∈' T" := (<span class="id" type="var">has_type</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/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">has_type</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "has_type_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Var" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Abs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_App"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Proj" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Sub"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_RNil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_RCons" ].<br/>
<br/>
</div>
<div class="doc">
<a name="lab872"></a><h2 class="section">Typing Examples</h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Examples2</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">Examples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab873"></a><h4 class="section">Exercise: 1 star</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_example_0</span> : <br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">k</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> <span class="id" type="var">A</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">z</span>)) <br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">j</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> <span class="id" type="var">B</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">z</span>)) <br/>
<span class="id" type="var">trnil</span>))<br/>
<span class="id" type="var">TRcd_kj</span>.<br/>
<span class="comment">(* empty |- {k=(\z:A.z), j=(\z:B.z)} : {k:A->A,j:B->B} *)</span><br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab874"></a><h4 class="section">Exercise: 2 stars</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_example_1</span> : <br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">TRcd_j</span> (<span class="id" type="var">tproj</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>) <span class="id" type="var">j</span>))<br/>
(<span class="id" type="var">trcd_kj</span>))<br/>
(<span class="id" type="var">TArrow</span> <span class="id" type="var">B</span> <span class="id" type="var">B</span>).<br/>
<span class="comment">(* empty |- (\x:{k:A->A,j:B->B}. x.j) {k=(\z:A.z), j=(\z:B.z)} : B->B *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab875"></a><h4 class="section">Exercise: 2 stars, optional</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_example_2</span> : <br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> (<span class="id" type="var">TArrow</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">C</span> <span class="id" type="var">C</span>) <span class="id" type="var">TRcd_j</span>) <br/>
(<span class="id" type="var">tproj</span> (<span class="id" type="var">tapp</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">z</span>) <br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">C</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)))<br/>
<span class="id" type="var">j</span>))<br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">z</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">C</span> <span class="id" type="var">C</span>) <span class="id" type="var">trcd_kj</span>))<br/>
(<span class="id" type="var">TArrow</span> <span class="id" type="var">B</span> <span class="id" type="var">B</span>).<br/>
<span class="comment">(* empty |- (\z:(C->C)->{j:B->B}. (z (\x:C.x)).j)<br/>
(\z:C->C. {k=(\z:A.z), j=(\z:B.z)})<br/>
: B->B *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">Examples2</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab876"></a><h2 class="section">Properties of Typing</h2>
<div class="paragraph"> </div>
<a name="lab877"></a><h3 class="section">Well-Formedness</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">has_type__wf</span> : <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/>
<span class="id" type="var">has_type</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 style="font-family: arial;">→</span> <span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span>.<br/>
<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="var">T</span> <span class="id" type="var">Htyp</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Htyp</span>) <span class="id" type="var">Case</span>...<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">IHHtyp1</span>...<br/>
<span class="id" type="var">Case</span> "T_Proj".<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">wf_rcd_lookup</span>...<br/>
<span class="id" type="var">Case</span> "T_Sub".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">subtype__wf</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H</span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_preserves_record_tm</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">tr</span> <span class="id" type="var">tr'</span>,<br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tr</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tr'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr'</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">tr</span> <span class="id" type="var">tr'</span> <span class="id" type="var">Hrt</span> <span class="id" type="var">Hstp</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hrt</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hstp</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab878"></a><h3 class="section">Field Lookup</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">lookup_field_in_value</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">v</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <span class="id" type="var">v</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span> <br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">vi</span>, <span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">v</span> = <span class="id" type="var">Some</span> <span class="id" type="var">vi</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <span class="id" type="var">vi</span> <span class="id" type="var">Ti</span>.<br/>
<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="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">i</span> <span class="id" type="var">Ti</span> <span class="id" type="var">Hval</span> <span class="id" type="var">Htyp</span>. <span class="id" type="var">revert</span> <span class="id" type="var">Ti</span> <span class="id" type="var">HeqGamma</span> <span class="id" type="var">Hval</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Htyp</span>) <span class="id" type="var">Case</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="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="var">Case</span> "T_Sub".<br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">rcd_types_match</span> <span class="id" type="var">S</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span>... <span class="id" type="tactic">destruct</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Si</span> [<span class="id" type="var">HgetSi</span> <span class="id" type="var">Hsub</span>]].<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">IHHtyp</span> <span class="id" type="var">Si</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">vi</span> [<span class="id" type="var">Hget</span> <span class="id" type="var">Htyvi</span>]]...<br/>
<span class="id" type="var">Case</span> "T_RCons".<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i0</span>).<br/>
<span class="id" type="var">SCase</span> "i is first".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H1</span>. <span class="id" type="tactic">subst</span>. <span style="font-family: arial;">∃</span><span class="id" type="var">t</span>...<br/>
<span class="id" type="var">SCase</span> "i in tail".<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">IHHtyp2</span> <span class="id" type="var">Ti</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">vi</span> [<span class="id" type="var">get</span> <span class="id" type="var">Htyvi</span>]]...<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hval</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab879"></a><h3 class="section">Progress</h3>
<div class="paragraph"> </div>
<a name="lab880"></a><h4 class="section">Exercise: 3 stars (canonical_forms_of_arrow_types)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">canonical_forms_of_arrow_types</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">s</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">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">s</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">s</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span style="font-family: arial;">∃</span><span class="id" type="var">S<sub>1</sub></span>, <span style="font-family: arial;">∃</span><span class="id" type="var">s<sub>2</sub></span>,<br/>
<span class="id" type="var">s</span> = <span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">S<sub>1</sub></span> <span class="id" type="var">s<sub>2</sub></span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="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">has_type</span> <span class="id" type="var">empty</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> <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="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="var">revert</span> <span class="id" type="var">HeqGamma</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Ht</span>) <span class="id" type="var">Case</span>; <br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">HeqGamma</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">Case</span> "T_Var".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="id" type="var">right</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">SCase</span> "t<sub>1</sub> is a value".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">SSCase</span> "t<sub>2</sub> is a value".<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">canonical_forms_of_arrow_types</span> <span class="id" type="var">empty</span> <span class="id" type="var">t<sub>1</sub></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="keyword">as</span> [<span class="id" type="var">x</span> [<span class="id" type="var">S<sub>1</sub></span> [<span class="id" type="var">t<sub>12</sub></span> <span class="id" type="var">Heqt1</span>]]]...<br/>
<span class="id" type="tactic">subst</span>. <span style="font-family: arial;">∃</span>([<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>)...<br/>
<span class="id" type="var">SSCase</span> "t<sub>2</sub> steps".<br/>
<span class="id" type="tactic">destruct</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/>
<span class="id" type="var">SCase</span> "t<sub>1</sub> steps".<br/>
<span class="id" type="tactic">destruct</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/>
<span class="id" type="var">Case</span> "T_Proj".<br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt</span>...<br/>
<span class="id" type="var">SCase</span> "rcd is value".<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">lookup_field_in_value</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">t'</span> [<span class="id" type="var">Hget</span> <span class="id" type="var">Ht'</span>]]...<br/>
<span class="id" type="var">SCase</span> "rcd_steps".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tproj</span> <span class="id" type="var">t'</span> <span class="id" type="var">i</span>)...<br/>
<span class="id" type="var">Case</span> "T_RCons".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>...<br/>
<span class="id" type="var">SCase</span> "head is a value".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span>...<br/>
<span class="id" type="var">SSCase</span> "tail steps".<br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">H2</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">tr'</span> <span class="id" type="var">Hstp</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">tr'</span>)...<br/>
<span class="id" type="var">SCase</span> "head steps".<br/>
<span class="id" type="var">right</span>. <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">t'</span> <span class="id" type="var">Hstp</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t'</span> <span class="id" type="var">tr</span>)... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Informal proof of progress:
<div class="paragraph"> </div>
Theorem : For any term <span class="inlinecode"><span class="id" type="var">t</span></span> and type <span class="inlinecode"><span class="id" type="var">T</span></span>, if <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>
then <span class="inlinecode"><span class="id" type="var">t</span></span> is a value or <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> for some term <span class="inlinecode"><span class="id" type="var">t'</span></span>.
<div class="paragraph"> </div>
Proof : Let <span class="inlinecode"><span class="id" type="var">t</span></span> and <span class="inlinecode"><span class="id" type="var">T</span></span> be given such that <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>. We go
by induction on the typing derivation. Cases <span class="inlinecode"><span class="id" type="var">T_Abs</span></span> and
<span class="inlinecode"><span class="id" type="var">T_RNil</span></span> are immediate because abstractions and <span class="inlinecode">{}</span> are always
values. Case <span class="inlinecode"><span class="id" type="var">T_Var</span></span> is vacuous because variables cannot be
typed in the empty context.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last step in the typing derivation is by <span class="inlinecode"><span class="id" type="var">T_App</span></span>, then
there are terms <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 types <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> such that
<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>, <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>2</sub></span></span>, <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<sub>1</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></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> and
<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<sub>2</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span>.
<div class="paragraph"> </div>
The induction hypotheses for these typing derivations yield
that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value or steps, and that <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> is a value or
steps. We consider each case:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></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> for some term <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> <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<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>
<li> Otherwise <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <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<sub>2</sub>'</span></span> for some term <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span>. Then
<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> <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 class="id" type="var">t<sub>2</sub>'</span></span> by rule <span class="inlinecode"><span class="id" type="var">ST_App2</span></span> because <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> is a value. By lemma
<span class="inlinecode"><span class="id" type="var">canonical_forms_for_arrow_types</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">x</span>:<span class="id" type="var">S<sub>1</sub>.s<sub>2</sub></span></span> for some
<span class="inlinecode"><span class="id" type="var">x</span></span>, <span class="inlinecode"><span class="id" type="var">S<sub>1</sub></span></span>, and <span class="inlinecode"><span class="id" type="var">s<sub>2</sub></span></span>. And <span class="inlinecode">(\<span class="id" type="var">x</span>:<span class="id" type="var">S<sub>1</sub>.s<sub>2</sub></span>)</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">x</span>:=<span class="id" type="var">t<sub>2</sub></span>]<span class="id" type="var">s<sub>2</sub></span></span> by
<span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span>, since <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> is a value.
<div class="paragraph"> </div>
</li>
</ul>
</li>
</ul>
</li>
<li> If the last step of the derivation is by <span class="inlinecode"><span class="id" type="var">T_Proj</span></span>, then there
is a term <span class="inlinecode"><span class="id" type="var">tr</span></span>, type <span class="inlinecode"><span class="id" type="var">Tr</span></span> and label <span class="inlinecode"><span class="id" type="var">i</span></span> such that <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">tr.i</span></span>,
<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">tr</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">Tr</span></span>, and <span class="inlinecode"><span class="id" type="var">Tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">Tr</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
The IH for the typing subderivation gives us that either <span class="inlinecode"><span class="id" type="var">tr</span></span>
is a value or it steps. If <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">tr'</span></span> for some term <span class="inlinecode"><span class="id" type="var">tr'</span></span>,
then <span class="inlinecode"><span class="id" type="var">tr.i</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">tr'.i</span></span> by rule <span class="inlinecode"><span class="id" type="var">ST_Proj1</span></span>.
<div class="paragraph"> </div>
Otherwise, <span class="inlinecode"><span class="id" type="var">tr</span></span> is a value. In this case, lemma
<span class="inlinecode"><span class="id" type="var">lookup_field_in_value</span></span> yields that there is a term <span class="inlinecode"><span class="id" type="var">ti</span></span> such
that <span class="inlinecode"><span class="id" type="var">tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span>. It follows that <span class="inlinecode"><span class="id" type="var">tr.i</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span>
by rule <span class="inlinecode"><span class="id" type="var">ST_ProjRcd</span></span>.
<div class="paragraph"> </div>
</li>
<li> If the final step of the derivation is by <span class="inlinecode"><span class="id" type="var">T_Sub</span></span>, then there
is a type <span class="inlinecode"><span class="id" type="var">S</span></span> such that <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><:</span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <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">S</span></span>. The
desired result is exactly the induction hypothesis for the
typing subderivation.
<div class="paragraph"> </div>
</li>
<li> If the final step of the derivation is by <span class="inlinecode"><span class="id" type="var">T_RCons</span></span>, then there
exist some terms <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span>, types <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">Tr</span></span> and a label <span class="inlinecode"><span class="id" type="var">t</span></span> such
that <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">tr</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">i</span>:<span class="id" type="var">T<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">Tr</span>}</span>, <span class="inlinecode"><span class="id" type="var">record_tm</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span>,
<span class="inlinecode"><span class="id" type="var">record_tm</span></span> <span class="inlinecode"><span class="id" type="var">Tr</span></span>, <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<sub>1</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> and <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">tr</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">Tr</span></span>.
<div class="paragraph"> </div>
The induction hypotheses for these typing derivations yield
that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value or steps, and that <span class="inlinecode"><span class="id" type="var">tr</span></span> is a value or
steps. We consider each case:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></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> for some term <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>. Then
<span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">tr</span>}</span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub>'</span>,</span> <span class="inlinecode"><span class="id" type="var">tr</span>}</span> by rule <span class="inlinecode"><span class="id" type="var">ST_Rcd_Head</span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">tr'</span></span> for some term <span class="inlinecode"><span class="id" type="var">tr'</span></span>. Then
<span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">tr</span>}</span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">tr'</span>}</span> by rule <span class="inlinecode"><span class="id" type="var">ST_Rcd_Tail</span></span>,
since <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" type="var">tr</span></span> is also a value. So, <span class="inlinecode">{<span class="id" type="var">i</span>=<span class="id" type="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" type="var">tr</span>}</span> is a
value by <span class="inlinecode"><span class="id" type="var">v_rcons</span></span>.
</li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab881"></a><h3 class="section">Inversion Lemmas</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">typing_inversion_var</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">T</span>,<br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> (<span class="id" type="var">tvar</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: arial;">∃</span><span class="id" type="var">S</span>,<br/>
<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">S</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">subtype</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span>.<br/>
<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">T</span> <span class="id" type="var">Hty</span>.<br/>
<span class="id" type="var">remember</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>) <span class="id" type="keyword">as</span> <span class="id" type="var">t</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Hty</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">intros</span>; <br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Heqt</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="var">Case</span> "T_Var".<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">T</span>...<br/>
<span class="id" type="var">Case</span> "T_Sub".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHty</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">U</span> [<span class="id" type="var">Hctx</span> <span class="id" type="var">HsubU</span>]]... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">typing_inversion_app</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</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>2</sub></span>,<br/>