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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="common/css/sf.css" rel="stylesheet" type="text/css" />
<title>RecordSub: Subtyping with Records</title>
<link href="common/jquery-ui/jquery-ui.css" rel="stylesheet">
<script src="common/jquery-ui/external/jquery/jquery.js"></script>
<script src="common/jquery-ui/jquery-ui.js"></script>
<script src="common/toggleproofs.js"></script>
<link href="common/css/plf.css" rel="stylesheet" type="text/css"/>
</head>
<body>
<div id="page">
<div id="header">
<div id='logoinheader'><a href='https://softwarefoundations.cis.upenn.edu'>
<img src='common/media/image/sf_logo_sm.png' alt='Software Foundations Logo'></a></div>
<div class='booktitleinheader'><a href='index.html'>Volume 2: Programming Language Foundations</a></div>
<ul id='menu'>
<li class='section_name'><a href='toc.html'>Table of Contents</a></li>
<li class='section_name'><a href='coqindex.html'>Index</a></li>
<li class='section_name'><a href='deps.html'>Roadmap</a></li>
</ul>
</div>
<div id="main">
<h1 class="libtitle">RecordSub<span class="subtitle">Subtyping with Records</span></h1>
<div class="doc">
<div class="paragraph"> </div>
In this chapter, we combine two significant extensions of the pure
STLC -- records (from chapter <a href="Records.html"><span class="inlineref">Records</span></a>) and subtyping (from
chapter <a href="Sub.html"><span class="inlineref">Sub</span></a>) -- and explore their interactions. Most of the
concepts have already been discussed in those chapters, so the
presentation here is somewhat terse. We just comment where things
are nonstandard.
</div>
<div class="code">
<span class="id" title="keyword">Set</span> <span class="id" title="var">Warnings</span> "-notation-overridden,-parsing,-deprecated-hint-without-locality".<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#"><span class="id" title="library">Strings.String</span></a>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">PLF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">Maps</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">PLF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Smallstep.html#"><span class="id" title="library">Smallstep</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Module</span> <a id="RecordSub" class="idref" href="#RecordSub"><span class="id" title="module">RecordSub</span></a>.<br/>
</div>
<div class="doc">
<a id="lab421"></a><h1 class="section">Core Definitions</h1>
</div>
<div class="doc">
<a id="lab422"></a><h3 class="section">Syntax</h3>
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.ty" class="idref" href="#RecordSub.ty"><span class="id" title="inductive">ty</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
<span class="comment">(* proper types *)</span><br/>
| <a id="RecordSub.Ty_Top" class="idref" href="#RecordSub.Ty_Top"><span class="id" title="constructor">Ty_Top</span></a> : <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a><br/>
| <a id="RecordSub.Ty_Base" class="idref" href="#RecordSub.Ty_Base"><span class="id" title="constructor">Ty_Base</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a><br/>
| <a id="RecordSub.Ty_Arrow" class="idref" href="#RecordSub.Ty_Arrow"><span class="id" title="constructor">Ty_Arrow</span></a> : <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a><br/>
<span class="comment">(* record types *)</span><br/>
| <a id="RecordSub.Ty_RNil" class="idref" href="#RecordSub.Ty_RNil"><span class="id" title="constructor">Ty_RNil</span></a> : <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a><br/>
| <a id="RecordSub.Ty_RCons" class="idref" href="#RecordSub.Ty_RCons"><span class="id" title="constructor">Ty_RCons</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#ty:1"><span class="id" title="inductive">ty</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.tm" class="idref" href="#RecordSub.tm"><span class="id" title="inductive">tm</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
<span class="comment">(* proper terms *)</span><br/>
| <a id="RecordSub.tm_var" class="idref" href="#RecordSub.tm_var"><span class="id" title="constructor">tm_var</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a><br/>
| <a id="RecordSub.tm_app" class="idref" href="#RecordSub.tm_app"><span class="id" title="constructor">tm_app</span></a> : <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a><br/>
| <a id="RecordSub.tm_abs" class="idref" href="#RecordSub.tm_abs"><span class="id" title="constructor">tm_abs</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a><br/>
| <a id="RecordSub.tm_rproj" class="idref" href="#RecordSub.tm_rproj"><span class="id" title="constructor">tm_rproj</span></a> : <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a><br/>
<span class="comment">(* record terms *)</span><br/>
| <a id="RecordSub.tm_rnil" class="idref" href="#RecordSub.tm_rnil"><span class="id" title="constructor">tm_rnil</span></a> : <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a><br/>
| <a id="RecordSub.tm_rcons" class="idref" href="#RecordSub.tm_rcons"><span class="id" title="constructor">tm_rcons</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#tm:3"><span class="id" title="inductive">tm</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="var">Declare</span> <span class="id" title="var">Custom</span> <span class="id" title="var">Entry</span> <span class="id" title="var">stlc</span>.<br/>
<span class="id" title="var">Declare</span> <span class="id" title="var">Custom</span> <span class="id" title="var">Entry</span> <span class="id" title="var">stlc_ty</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="d60fde6c3996807cd13393b39ca380c<sub>9</sub>" class="idref" href="#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">"</span></a><{ e }>" := <span class="id" title="var">e</span> (<span class="id" title="var">e</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="6713cbc242333a153601674b9e8101e<sub>9</sub>" class="idref" href="#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation">"</span></a><<span style='letter-spacing:-.4em;'>{</span>{ e <span style='letter-spacing:-.4em;'>}</span>}>" := <span class="id" title="var">e</span> (<span class="id" title="var">e</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="53937f3ba9e51dc428955dd7e0fbd7fe" class="idref" href="#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">"</span></a>( x )" := <span class="id" title="var">x</span> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span>, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="731ae131b4917695ec9910860e6c9fc<sub>4</sub>" class="idref" href="#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">"</span></a>( x )" := <span class="id" title="var">x</span> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span>, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc::x" class="idref" href="#RecordSub.:stlc::x"><span class="id" title="notation">"</span></a>x" := <span class="id" title="var">x</span> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::x" class="idref" href="#RecordSub.:stlc_ty::x"><span class="id" title="notation">"</span></a>x" := <span class="id" title="var">x</span> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::x_'->'_x" class="idref" href="#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">"</span></a>S <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> T" := (<a class="idref" href="RecordSub.html#RecordSub.Ty_Arrow"><span class="id" title="constructor">Ty_Arrow</span></a> <span class="id" title="var">S</span> <span class="id" title="var">T</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 50, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc::x_x" class="idref" href="#RecordSub.:stlc::x_x"><span class="id" title="notation">"</span></a>x y" := (<a class="idref" href="RecordSub.html#RecordSub.tm_app"><span class="id" title="constructor">tm_app</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 1, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="90b3ca4a4628f04e832c34ef6a4befd8" class="idref" href="#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">"</span></a>\ x : t , y" :=<br/>
(<a class="idref" href="RecordSub.html#RecordSub.tm_abs"><span class="id" title="constructor">tm_abs</span></a> <span class="id" title="var">x</span> <span class="id" title="var">t</span> <span class="id" title="var">y</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 90, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/>
<span class="id" title="var">t</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/>
<span class="id" title="var">y</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/>
<span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">tm_var</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">string</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">tm</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="cd5c17a8d7e43fbc5d738c36bb786ce<sub>6</sub>" class="idref" href="#cd5c17a8d7e43fbc5d738c36bb786ce<sub>6</sub>"><span class="id" title="notation">"</span></a>{ x }" := <span class="id" title="var">x</span> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 1, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::'Base'_x" class="idref" href="#RecordSub.:stlc_ty::'Base'_x"><span class="id" title="notation">"</span></a>'Base' x" := (<a class="idref" href="RecordSub.html#RecordSub.Ty_Base"><span class="id" title="constructor">Ty_Base</span></a> <span class="id" title="var">x</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::x_':'_x_'::'_x" class="idref" href="#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">"</span></a> l ':' t<sub>1</sub> '::' t<sub>2</sub>" := (<a class="idref" href="RecordSub.html#RecordSub.Ty_RCons"><span class="id" title="constructor">Ty_RCons</span></a> <span class="id" title="var">l</span> <span class="id" title="var">t<sub>1</sub></span> <span class="id" title="var">t<sub>2</sub></span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="24532377138054657f673a020656af9f" class="idref" href="#24532377138054657f673a020656af9f"><span class="id" title="notation">"</span></a> l := e<sub>1</sub> '::' e<sub>2</sub>" := (<a class="idref" href="RecordSub.html#RecordSub.tm_rcons"><span class="id" title="constructor">tm_rcons</span></a> <span class="id" title="var">l</span> <span class="id" title="var">e<sub>1</sub></span> <span class="id" title="var">e<sub>2</sub></span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::'nil'" class="idref" href="#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">"</span></a>'nil'" := (<a class="idref" href="RecordSub.html#RecordSub.Ty_RNil"><span class="id" title="constructor">Ty_RNil</span></a>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc::'nil'" class="idref" href="#RecordSub.:stlc::'nil'"><span class="id" title="notation">"</span></a>'nil'" := (<a class="idref" href="RecordSub.html#RecordSub.tm_rnil"><span class="id" title="constructor">tm_rnil</span></a>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc::x_'-->'_x" class="idref" href="#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation">"</span></a>o <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span> l" := (<a class="idref" href="RecordSub.html#RecordSub.tm_rproj"><span class="id" title="constructor">tm_rproj</span></a> <span class="id" title="var">o</span> <span class="id" title="var">l</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.:stlc_ty::'Top'" class="idref" href="#RecordSub.:stlc_ty::'Top'"><span class="id" title="notation">"</span></a>'Top'" := (<a class="idref" href="RecordSub.html#RecordSub.Ty_Top"><span class="id" title="constructor">Ty_Top</span></a>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
</div>
<div class="doc">
<a id="lab423"></a><h3 class="section">Well-Formedness</h3>
<div class="paragraph"> </div>
The syntax of terms and types is a bit too loose, in the sense
that it admits things like a record type whose final "tail" is
<span class="inlinecode"><span class="id" title="var">Top</span></span> or some arrow type rather than <span class="inlinecode"><span class="id" title="var">Nil</span></span>. To avoid such cases,
it is useful to assume that all the record types and terms that we
see will obey some simple well-formedness conditions.
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" title="var">An</span></span> <span class="inlinecode"><span class="id" title="var">interesting</span></span> <span class="inlinecode"><span class="id" title="var">technical</span></span> <span class="inlinecode"><span class="id" title="var">question</span></span> <span class="inlinecode"><span class="id" title="keyword">is</span></span> <span class="inlinecode"><span class="id" title="var">whether</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">basic</span></span> <span class="inlinecode"><span class="id" title="var">properties</span></span>
<span class="inlinecode"><span class="id" title="keyword">of</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">system</span></span> <span class="inlinecode">--</span> <span class="inlinecode"><span class="id" title="tactic">progress</span></span> <span class="inlinecode"><span class="id" title="var">and</span></span> <span class="inlinecode"><span class="id" title="var">preservation</span></span> <span class="inlinecode">--</span> <span class="inlinecode"><span class="id" title="var">remain</span></span> <span class="inlinecode"><span class="id" title="var">true</span></span> <span class="inlinecode"><span class="id" title="keyword">if</span></span> <span class="inlinecode"><span class="id" title="var">we</span></span>
<span class="inlinecode"><span class="id" title="var">drop</span></span> <span class="inlinecode"><span class="id" title="var">these</span></span> <span class="inlinecode"><span class="id" title="var">conditions</span>.</span> <span class="inlinecode"><span class="id" title="var">I</span></span> <span class="inlinecode"><span class="id" title="var">believe</span></span> <span class="inlinecode"><span class="id" title="var">they</span></span> <span class="inlinecode"><span class="id" title="tactic">do</span>,</span> <span class="inlinecode"><span class="id" title="var">and</span></span> <span class="inlinecode"><span class="id" title="var">I</span></span> <span class="inlinecode"><span class="id" title="var">would</span></span> <span class="inlinecode"><span class="id" title="var">encourage</span></span>
<span class="inlinecode"><span class="id" title="var">motivated</span></span> <span class="inlinecode"><span class="id" title="var">readers</span></span> <span class="inlinecode"><span class="id" title="var">to</span></span> <span class="inlinecode"><span class="id" title="tactic">try</span></span> <span class="inlinecode"><span class="id" title="var">to</span></span> <span class="inlinecode"><span class="id" title="var">check</span></span> <span class="inlinecode"><span class="id" title="var">this</span></span> <span class="inlinecode"><span class="id" title="tactic">by</span></span> <span class="inlinecode"><span class="id" title="var">dropping</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">conditions</span></span>
<span class="inlinecode"><span class="id" title="var">from</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">definitions</span></span> <span class="inlinecode"><span class="id" title="keyword">of</span></span> <span class="inlinecode"><span class="id" title="var">typing</span></span> <span class="inlinecode"><span class="id" title="var">and</span></span> <span class="inlinecode"><span class="id" title="var">subtyping</span></span> <span class="inlinecode"><span class="id" title="var">and</span></span> <span class="inlinecode"><span class="id" title="var">adjusting</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span>
<span class="inlinecode"><span class="id" title="var">proofs</span></span> <span class="inlinecode"><span class="id" title="keyword">in</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">rest</span></span> <span class="inlinecode"><span class="id" title="keyword">of</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">chapter</span></span> <span class="inlinecode"><span class="id" title="var">accordingly</span>.</span> <span class="inlinecode"><span class="id" title="var">This</span></span> <span class="inlinecode"><span class="id" title="keyword">is</span></span> <span class="inlinecode"><span class="id" title="var">not</span></span> <span class="inlinecode"><span class="id" title="var">a</span></span>
<span class="inlinecode"><span class="id" title="tactic">trivial</span></span> <span class="inlinecode"><span class="id" title="var">exercise</span></span> <span class="inlinecode">(<span class="id" title="var">or</span></span> <span class="inlinecode"><span class="id" title="var">I'd</span></span> <span class="inlinecode"><span class="id" title="var">have</span></span> <span class="inlinecode"><span class="id" title="var">done</span></span> <span class="inlinecode"><span class="id" title="var">it</span>!),</span> <span class="inlinecode"><span class="id" title="var">but</span></span> <span class="inlinecode"><span class="id" title="var">it</span></span> <span class="inlinecode"><span class="id" title="var">should</span></span> <span class="inlinecode"><span class="id" title="var">not</span></span> <span class="inlinecode"><span class="id" title="var">involve</span></span>
<span class="inlinecode"><span class="id" title="var">changing</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">basic</span></span> <span class="inlinecode"><span class="id" title="var">structure</span></span> <span class="inlinecode"><span class="id" title="keyword">of</span></span> <span class="inlinecode"><span class="id" title="var">the</span></span> <span class="inlinecode"><span class="id" title="var">proofs</span>.</span> <span class="inlinecode"><span class="id" title="var">If</span></span> <span class="inlinecode"><span class="id" title="var">someone</span></span> <span class="inlinecode"><span class="id" title="var">does</span></span> <span class="inlinecode"><span class="id" title="tactic">do</span></span>
<span class="inlinecode"><span class="id" title="var">it</span>,</span> <span class="inlinecode"><span class="id" title="var">please</span></span> <span class="inlinecode"><span class="id" title="keyword">let</span></span> <span class="inlinecode"><span class="id" title="var">me</span></span> <span class="inlinecode"><span class="id" title="var">know</span>.</span> <span class="inlinecode">--<span class="id" title="var">BCP</span></span> <span class="inlinecode">5/16.</span>
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.record_ty" class="idref" href="#RecordSub.record_ty"><span class="id" title="inductive">record_ty</span></a> : <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.RTnil" class="idref" href="#RecordSub.RTnil"><span class="id" title="constructor">RTnil</span></a> :<br/>
<a class="idref" href="RecordSub.html#record_ty:5"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.RTcons" class="idref" href="#RecordSub.RTcons"><span class="id" title="constructor">RTcons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> <a id="T<sub>1</sub>:8" class="idref" href="#T<sub>1</sub>:8"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:9" class="idref" href="#T<sub>2</sub>:9"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#record_ty:5"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:8"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:9"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.record_tm" class="idref" href="#RecordSub.record_tm"><span class="id" title="inductive">record_tm</span></a> : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.rtnil" class="idref" href="#RecordSub.rtnil"><span class="id" title="constructor">rtnil</span></a> :<br/>
<a class="idref" href="RecordSub.html#record_tm:10"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.rtcons" class="idref" href="#RecordSub.rtcons"><span class="id" title="constructor">rtcons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a> <a id="t<sub>1</sub>:13" class="idref" href="#t<sub>1</sub>:13"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>2</sub>:14" class="idref" href="#t<sub>2</sub>:14"><span class="id" title="binder">t<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#record_tm:10"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:12"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:13"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:14"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.well_formed_ty" class="idref" href="#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> : <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.wfTop" class="idref" href="#RecordSub.wfTop"><span class="id" title="constructor">wfTop</span></a> :<br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Top'"><span class="id" title="notation">Top</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.wfBase" class="idref" href="#RecordSub.wfBase"><span class="id" title="constructor">wfBase</span></a> : <span class="id" title="keyword">∀</span> (<a id="i:17" class="idref" href="#i:17"><span class="id" title="binder">i</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>),<br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Base'_x"><span class="id" title="notation">Base</span></a> <a class="idref" href="RecordSub.html#i:17"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.wfArrow" class="idref" href="#RecordSub.wfArrow"><span class="id" title="constructor">wfArrow</span></a> : <span class="id" title="keyword">∀</span> <a id="T<sub>1</sub>:18" class="idref" href="#T<sub>1</sub>:18"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:19" class="idref" href="#T<sub>2</sub>:19"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:18"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:19"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:18"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:19"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.wfRNil" class="idref" href="#RecordSub.wfRNil"><span class="id" title="constructor">wfRNil</span></a> :<br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.wfRCons" class="idref" href="#RecordSub.wfRCons"><span class="id" title="constructor">wfRCons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:20" class="idref" href="#i:20"><span class="id" title="binder">i</span></a> <a id="T<sub>1</sub>:21" class="idref" href="#T<sub>1</sub>:21"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:22" class="idref" href="#T<sub>2</sub>:22"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:21"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:22"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_ty"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:22"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#well_formed_ty:15"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:20"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:21"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:22"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="RecordSub.html#record_ty"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#record_tm"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
<a id="lab424"></a><h3 class="section">Substitution</h3>
<div class="paragraph"> </div>
Substitution and reduction are as before.
</div>
<div class="code">
<span class="id" title="keyword">Reserved Notation</span> "'[' x ':=' s ']' t" (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 20, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Fixpoint</span> <a id="RecordSub.subst" class="idref" href="#RecordSub.subst"><span class="id" title="definition">subst</span></a> (<a id="x:23" class="idref" href="#x:23"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) (<a id="s:24" class="idref" href="#s:24"><span class="id" title="binder">s</span></a> : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a>) (<a id="t:25" class="idref" href="#t:25"><span class="id" title="binder">t</span></a> : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a>) : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="RecordSub.html#t:25"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="RecordSub.html#RecordSub.tm_var"><span class="id" title="constructor">tm_var</span></a> <span class="id" title="var">y</span> ⇒<br/>
<span class="id" title="keyword">if</span> <span class="id" title="definition">eqb_string</span> <a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="keyword">then</span> <a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="RecordSub.html#t:25"><span class="id" title="variable">t</span></a><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><span class="id" title="var">y</span><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a><span class="id" title="var">T</span><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <span class="id" title="var">t<sub>1</sub></span><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<span class="id" title="keyword">if</span> <span class="id" title="definition">eqb_string</span> <a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="keyword">then</span> <a class="idref" href="RecordSub.html#t:25"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><span class="id" title="var">y</span><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a><span class="id" title="var">T</span><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a><a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>1</sub></span><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><span class="id" title="var">t<sub>1</sub></span> <span class="id" title="var">t<sub>2</sub></span><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a><a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>1</sub></span><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a><a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>2</sub></span><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">t<sub>1</sub></span> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>1</sub></span><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <span class="id" title="var">t<sub>1</sub></span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <span class="id" title="var">tr</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>1</sub></span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:23"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#s:24"><span class="id" title="variable">s</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">tr</span><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
<span class="id" title="keyword">end</span><br/>
<br/>
<span class="id" title="keyword">where</span> <a id="51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>" class="idref" href="#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">"</span></a>'[' x ':=' s ']' t" := (<a class="idref" href="RecordSub.html#subst:26"><span class="id" title="definition">subst</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span>).<br/>
</div>
<div class="doc">
<a id="lab425"></a><h3 class="section">Reduction</h3>
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.value" class="idref" href="#RecordSub.value"><span class="id" title="inductive">value</span></a> : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.v_abs" class="idref" href="#RecordSub.v_abs"><span class="id" title="constructor">v_abs</span></a> : <span class="id" title="keyword">∀</span> <a id="x:30" class="idref" href="#x:30"><span class="id" title="binder">x</span></a> <a id="T<sub>2</sub>:31" class="idref" href="#T<sub>2</sub>:31"><span class="id" title="binder">T<sub>2</sub></span></a> <a id="t<sub>1</sub>:32" class="idref" href="#t<sub>1</sub>:32"><span class="id" title="binder">t<sub>1</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#value:28"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#x:30"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:31"><span class="id" title="variable">T<sub>2</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:32"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.v_rnil" class="idref" href="#RecordSub.v_rnil"><span class="id" title="constructor">v_rnil</span></a> : <a class="idref" href="RecordSub.html#value:28"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.v_rcons" class="idref" href="#RecordSub.v_rcons"><span class="id" title="constructor">v_rcons</span></a> : <span class="id" title="keyword">∀</span> <a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> <a id="v<sub>1</sub>:34" class="idref" href="#v<sub>1</sub>:34"><span class="id" title="binder">v<sub>1</sub></span></a> <a id="vr:35" class="idref" href="#vr:35"><span class="id" title="binder">vr</span></a>,<br/>
<a class="idref" href="RecordSub.html#value:28"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:34"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#value:28"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#vr:35"><span class="id" title="variable">vr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#value:28"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:34"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#vr:35"><span class="id" title="variable">vr</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="RecordSub.html#value"><span class="id" title="inductive">value</span></a> : <span class="id" title="var">core</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Fixpoint</span> <a id="RecordSub.Tlookup" class="idref" href="#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> (<a id="i:36" class="idref" href="#i:36"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) (<a id="Tr:37" class="idref" href="#Tr:37"><span class="id" title="binder">Tr</span></a>:<a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a>) : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="RecordSub.html#Tr:37"><span class="id" title="variable">Tr</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <span class="id" title="var">i'</span> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">Tr'</span> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> ⇒<br/>
<span class="id" title="keyword">if</span> <span class="id" title="definition">eqb_string</span> <a class="idref" href="RecordSub.html#i:36"><span class="id" title="variable">i</span></a> <span class="id" title="var">i'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">T</span> <span class="id" title="keyword">else</span> <a class="idref" href="RecordSub.html#Tlookup:38"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:36"><span class="id" title="variable">i</span></a> <span class="id" title="var">Tr'</span><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Fixpoint</span> <a id="RecordSub.tlookup" class="idref" href="#RecordSub.tlookup"><span class="id" title="definition">tlookup</span></a> (<a id="i:40" class="idref" href="#i:40"><span class="id" title="binder">i</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) (<a id="tr:41" class="idref" href="#tr:41"><span class="id" title="binder">tr</span></a>:<a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a>) : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="RecordSub.html#tr:41"><span class="id" title="variable">tr</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">i'</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <span class="id" title="var">t</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <span class="id" title="var">tr'</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ⇒<br/>
<span class="id" title="keyword">if</span> <span class="id" title="definition">eqb_string</span> <a class="idref" href="RecordSub.html#i:40"><span class="id" title="variable">i</span></a> <span class="id" title="var">i'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">t</span> <span class="id" title="keyword">else</span> <a class="idref" href="RecordSub.html#tlookup:42"><span class="id" title="definition">tlookup</span></a> <a class="idref" href="RecordSub.html#i:40"><span class="id" title="variable">i</span></a> <span class="id" title="var">tr'</span><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Reserved Notation</span> "t '<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span>' t'" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.step" class="idref" href="#RecordSub.step"><span class="id" title="inductive">step</span></a> : <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.ST_AppAbs" class="idref" href="#RecordSub.ST_AppAbs"><span class="id" title="constructor">ST_AppAbs</span></a> : <span class="id" title="keyword">∀</span> <a id="x:46" class="idref" href="#x:46"><span class="id" title="binder">x</span></a> <a id="T<sub>2</sub>:47" class="idref" href="#T<sub>2</sub>:47"><span class="id" title="binder">T<sub>2</sub></span></a> <a id="t<sub>1</sub>:48" class="idref" href="#t<sub>1</sub>:48"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="v<sub>2</sub>:49" class="idref" href="#v<sub>2</sub>:49"><span class="id" title="binder">v<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#v<sub>2</sub>:49"><span class="id" title="variable">v<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#x:46"><span class="id" title="variable">x</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a><a class="idref" href="RecordSub.html#T<sub>2</sub>:47"><span class="id" title="variable">T<sub>2</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:48"><span class="id" title="variable">t<sub>1</sub></span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#v<sub>2</sub>:49"><span class="id" title="variable">v<sub>2</sub></span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="RecordSub.html#x:46"><span class="id" title="variable">x</span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a><a class="idref" href="RecordSub.html#v<sub>2</sub>:49"><span class="id" title="variable">v<sub>2</sub></span></a><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a><a class="idref" href="RecordSub.html#t<sub>1</sub>:48"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.ST_App1" class="idref" href="#RecordSub.ST_App1"><span class="id" title="constructor">ST_App1</span></a> : <span class="id" title="keyword">∀</span> <a id="t<sub>1</sub>:50" class="idref" href="#t<sub>1</sub>:50"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>1</sub>':51" class="idref" href="#t<sub>1</sub>':51"><span class="id" title="binder">t<sub>1</sub>'</span></a> <a id="t<sub>2</sub>:52" class="idref" href="#t<sub>2</sub>:52"><span class="id" title="binder">t<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#t<sub>1</sub>:50"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>':51"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#t<sub>1</sub>:50"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:52"><span class="id" title="variable">t<sub>2</sub></span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#t<sub>1</sub>':51"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:52"><span class="id" title="variable">t<sub>2</sub></span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.ST_App2" class="idref" href="#RecordSub.ST_App2"><span class="id" title="constructor">ST_App2</span></a> : <span class="id" title="keyword">∀</span> <a id="v<sub>1</sub>:53" class="idref" href="#v<sub>1</sub>:53"><span class="id" title="binder">v<sub>1</sub></span></a> <a id="t<sub>2</sub>:54" class="idref" href="#t<sub>2</sub>:54"><span class="id" title="binder">t<sub>2</sub></span></a> <a id="t<sub>2</sub>':55" class="idref" href="#t<sub>2</sub>':55"><span class="id" title="binder">t<sub>2</sub>'</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:53"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#t<sub>2</sub>:54"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>':55"><span class="id" title="variable">t<sub>2</sub>'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#v<sub>1</sub>:53"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:54"><span class="id" title="variable">t<sub>2</sub></span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a><a class="idref" href="RecordSub.html#v<sub>1</sub>:53"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>':55"><span class="id" title="variable">t<sub>2</sub>'</span></a><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.ST_Proj1" class="idref" href="#RecordSub.ST_Proj1"><span class="id" title="constructor">ST_Proj1</span></a> : <span class="id" title="keyword">∀</span> <a id="t<sub>1</sub>:56" class="idref" href="#t<sub>1</sub>:56"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>1</sub>':57" class="idref" href="#t<sub>1</sub>':57"><span class="id" title="binder">t<sub>1</sub>'</span></a> <a id="i:58" class="idref" href="#i:58"><span class="id" title="binder">i</span></a>,<br/>
<a class="idref" href="RecordSub.html#t<sub>1</sub>:56"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>':57"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:56"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#i:58"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>':57"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#i:58"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.ST_ProjRcd" class="idref" href="#RecordSub.ST_ProjRcd"><span class="id" title="constructor">ST_ProjRcd</span></a> : <span class="id" title="keyword">∀</span> <a id="tr:59" class="idref" href="#tr:59"><span class="id" title="binder">tr</span></a> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a> <a id="vi:61" class="idref" href="#vi:61"><span class="id" title="binder">vi</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#tr:59"><span class="id" title="variable">tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.tlookup"><span class="id" title="definition">tlookup</span></a> <a class="idref" href="RecordSub.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#tr:59"><span class="id" title="variable">tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#vi:61"><span class="id" title="variable">vi</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#tr:59"><span class="id" title="variable">tr</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#i:60"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#vi:61"><span class="id" title="variable">vi</span></a><br/>
| <a id="RecordSub.ST_Rcd_Head" class="idref" href="#RecordSub.ST_Rcd_Head"><span class="id" title="constructor">ST_Rcd_Head</span></a> : <span class="id" title="keyword">∀</span> <a id="i:62" class="idref" href="#i:62"><span class="id" title="binder">i</span></a> <a id="t<sub>1</sub>:63" class="idref" href="#t<sub>1</sub>:63"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>1</sub>':64" class="idref" href="#t<sub>1</sub>':64"><span class="id" title="binder">t<sub>1</sub>'</span></a> <a id="tr<sub>2</sub>:65" class="idref" href="#tr<sub>2</sub>:65"><span class="id" title="binder">tr<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#t<sub>1</sub>:63"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>':64"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:63"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#tr<sub>2</sub>:65"><span class="id" title="variable">tr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>':64"><span class="id" title="variable">t<sub>1</sub>'</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#tr<sub>2</sub>:65"><span class="id" title="variable">tr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
| <a id="RecordSub.ST_Rcd_Tail" class="idref" href="#RecordSub.ST_Rcd_Tail"><span class="id" title="constructor">ST_Rcd_Tail</span></a> : <span class="id" title="keyword">∀</span> <a id="i:66" class="idref" href="#i:66"><span class="id" title="binder">i</span></a> <a id="v<sub>1</sub>:67" class="idref" href="#v<sub>1</sub>:67"><span class="id" title="binder">v<sub>1</sub></span></a> <a id="tr<sub>2</sub>:68" class="idref" href="#tr<sub>2</sub>:68"><span class="id" title="binder">tr<sub>2</sub></span></a> <a id="tr<sub>2</sub>':69" class="idref" href="#tr<sub>2</sub>':69"><span class="id" title="binder">tr<sub>2</sub>'</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:67"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#tr<sub>2</sub>:68"><span class="id" title="variable">tr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#tr<sub>2</sub>':69"><span class="id" title="variable">tr<sub>2</sub>'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:66"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:67"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#tr<sub>2</sub>:68"><span class="id" title="variable">tr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#i:66"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#v<sub>1</sub>:67"><span class="id" title="variable">v<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#tr<sub>2</sub>':69"><span class="id" title="variable">tr<sub>2</sub>'</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a><br/>
<br/>
<span class="id" title="keyword">where</span> <a id="RecordSub.:::x_'-->'_x" class="idref" href="#RecordSub.:::x_'-->'_x"><span class="id" title="notation">"</span></a>t '<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span>' t'" := (<a class="idref" href="RecordSub.html#step:45"><span class="id" title="inductive">step</span></a> <span class="id" title="var">t</span> <span class="id" title="var">t'</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="RecordSub.html#step"><span class="id" title="inductive">step</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
<a id="lab426"></a><h1 class="section">Subtyping</h1>
<div class="paragraph"> </div>
Now we come to the interesting part, where the features we've
added start to interact. We begin by defining the subtyping
relation and developing some of its important technical
properties.
</div>
<div class="doc">
<a id="lab427"></a><h2 class="section">Definition</h2>
<div class="paragraph"> </div>
The definition of subtyping is essentially just what we sketched
in the discussion of record subtyping in chapter <a href="Sub.html"><span class="inlineref">Sub</span></a>, but we
need to add well-formedness side conditions to some of the rules.
Also, we replace the "n-ary" width, depth, and permutation
subtyping rules by binary rules that deal with just the first
field.
</div>
<div class="code">
<span class="id" title="keyword">Reserved Notation</span> "T '<:' U" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.subtype" class="idref" href="#RecordSub.subtype"><span class="id" title="inductive">subtype</span></a> : <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
<span class="comment">(* Subtyping between proper types *)</span><br/>
| <a id="RecordSub.S_Refl" class="idref" href="#RecordSub.S_Refl"><span class="id" title="constructor">S_Refl</span></a> : <span class="id" title="keyword">∀</span> <a id="T:72" class="idref" href="#T:72"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T:72"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#T:72"><span class="id" title="variable">T</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T:72"><span class="id" title="variable">T</span></a><br/>
| <a id="RecordSub.S_Trans" class="idref" href="#RecordSub.S_Trans"><span class="id" title="constructor">S_Trans</span></a> : <span class="id" title="keyword">∀</span> <a id="S:73" class="idref" href="#S:73"><span class="id" title="binder">S</span></a> <a id="U:74" class="idref" href="#U:74"><span class="id" title="binder">U</span></a> <a id="T:75" class="idref" href="#T:75"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#S:73"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#U:74"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#U:74"><span class="id" title="variable">U</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T:75"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#S:73"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T:75"><span class="id" title="variable">T</span></a><br/>
| <a id="RecordSub.S_Top" class="idref" href="#RecordSub.S_Top"><span class="id" title="constructor">S_Top</span></a> : <span class="id" title="keyword">∀</span> <a id="S:76" class="idref" href="#S:76"><span class="id" title="binder">S</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#S:76"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#S:76"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Top'"><span class="id" title="notation">Top</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.S_Arrow" class="idref" href="#RecordSub.S_Arrow"><span class="id" title="constructor">S_Arrow</span></a> : <span class="id" title="keyword">∀</span> <a id="S<sub>1</sub>:77" class="idref" href="#S<sub>1</sub>:77"><span class="id" title="binder">S<sub>1</sub></span></a> <a id="S<sub>2</sub>:78" class="idref" href="#S<sub>2</sub>:78"><span class="id" title="binder">S<sub>2</sub></span></a> <a id="T<sub>1</sub>:79" class="idref" href="#T<sub>1</sub>:79"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:80" class="idref" href="#T<sub>2</sub>:80"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#T<sub>1</sub>:79"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:77"><span class="id" title="variable">S<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#S<sub>2</sub>:78"><span class="id" title="variable">S<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:80"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:77"><span class="id" title="variable">S<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#S<sub>2</sub>:78"><span class="id" title="variable">S<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:79"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:80"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
<span class="comment">(* Subtyping between record types *)</span><br/>
| <a id="RecordSub.S_RcdWidth" class="idref" href="#RecordSub.S_RcdWidth"><span class="id" title="constructor">S_RcdWidth</span></a> : <span class="id" title="keyword">∀</span> <a id="i:81" class="idref" href="#i:81"><span class="id" title="binder">i</span></a> <a id="T<sub>1</sub>:82" class="idref" href="#T<sub>1</sub>:82"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:83" class="idref" href="#T<sub>2</sub>:83"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:82"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:83"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:81"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:82"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:83"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.S_RcdDepth" class="idref" href="#RecordSub.S_RcdDepth"><span class="id" title="constructor">S_RcdDepth</span></a> : <span class="id" title="keyword">∀</span> <a id="i:84" class="idref" href="#i:84"><span class="id" title="binder">i</span></a> <a id="S<sub>1</sub>:85" class="idref" href="#S<sub>1</sub>:85"><span class="id" title="binder">S<sub>1</sub></span></a> <a id="T<sub>1</sub>:86" class="idref" href="#T<sub>1</sub>:86"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="Sr<sub>2</sub>:87" class="idref" href="#Sr<sub>2</sub>:87"><span class="id" title="binder">Sr<sub>2</sub></span></a> <a id="Tr<sub>2</sub>:88" class="idref" href="#Tr<sub>2</sub>:88"><span class="id" title="binder">Tr<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#S<sub>1</sub>:85"><span class="id" title="variable">S<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:86"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Sr<sub>2</sub>:87"><span class="id" title="variable">Sr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#Tr<sub>2</sub>:88"><span class="id" title="variable">Tr<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_ty"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#Sr<sub>2</sub>:87"><span class="id" title="variable">Sr<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_ty"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#Tr<sub>2</sub>:88"><span class="id" title="variable">Tr<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:84"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:85"><span class="id" title="variable">S<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Sr<sub>2</sub>:87"><span class="id" title="variable">Sr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i:84"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:86"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Tr<sub>2</sub>:88"><span class="id" title="variable">Tr<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
| <a id="RecordSub.S_RcdPerm" class="idref" href="#RecordSub.S_RcdPerm"><span class="id" title="constructor">S_RcdPerm</span></a> : <span class="id" title="keyword">∀</span> <a id="i<sub>1</sub>:89" class="idref" href="#i<sub>1</sub>:89"><span class="id" title="binder">i<sub>1</sub></span></a> <a id="i<sub>2</sub>:90" class="idref" href="#i<sub>2</sub>:90"><span class="id" title="binder">i<sub>2</sub></span></a> <a id="T<sub>1</sub>:91" class="idref" href="#T<sub>1</sub>:91"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:92" class="idref" href="#T<sub>2</sub>:92"><span class="id" title="binder">T<sub>2</sub></span></a> <a id="Tr<sub>3</sub>:93" class="idref" href="#Tr<sub>3</sub>:93"><span class="id" title="binder">Tr<sub>3</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i<sub>1</sub>:89"><span class="id" title="variable">i<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:91"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#i<sub>2</sub>:90"><span class="id" title="variable">i<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:92"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Tr<sub>3</sub>:93"><span class="id" title="variable">Tr<sub>3</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#i<sub>1</sub>:89"><span class="id" title="variable">i<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'<>'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="RecordSub.html#i<sub>2</sub>:90"><span class="id" title="variable">i<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i<sub>1</sub>:89"><span class="id" title="variable">i<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:91"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#i<sub>2</sub>:90"><span class="id" title="variable">i<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:92"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Tr<sub>3</sub>:93"><span class="id" title="variable">Tr<sub>3</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#i<sub>2</sub>:90"><span class="id" title="variable">i<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:92"><span class="id" title="variable">T<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#i<sub>1</sub>:89"><span class="id" title="variable">i<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:91"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Tr<sub>3</sub>:93"><span class="id" title="variable">Tr<sub>3</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a><br/>
<br/>
<span class="id" title="keyword">where</span> <a id="RecordSub.:::x_'<:'_x" class="idref" href="#RecordSub.:::x_'<:'_x"><span class="id" title="notation">"</span></a>T '<:' U" := (<a class="idref" href="RecordSub.html#subtype:71"><span class="id" title="inductive">subtype</span></a> <span class="id" title="var">T</span> <span class="id" title="var">U</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="RecordSub.html#subtype"><span class="id" title="inductive">subtype</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
<a id="lab428"></a><h2 class="section">Examples</h2>
</div>
<div class="code">
<span class="id" title="keyword">Module</span> <a id="RecordSub.Examples" class="idref" href="#RecordSub.Examples"><span class="id" title="module">Examples</span></a>.<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">string_scope</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.x" class="idref" href="#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> := "x".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.y" class="idref" href="#RecordSub.Examples.y"><span class="id" title="abbreviation">y</span></a> := "y".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.z" class="idref" href="#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> := "z".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.j" class="idref" href="#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a> := "j".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.k" class="idref" href="#RecordSub.Examples.k"><span class="id" title="abbreviation">k</span></a> := "k".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.i" class="idref" href="#RecordSub.Examples.i"><span class="id" title="abbreviation">i</span></a> := "i".<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.A" class="idref" href="#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a> := <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Base'_x"><span class="id" title="notation">Base</span></a> "A" <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.B" class="idref" href="#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> := <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Base'_x"><span class="id" title="notation">Base</span></a> "B" <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a id="RecordSub.Examples.C" class="idref" href="#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> := <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Base'_x"><span class="id" title="notation">Base</span></a> "C" <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="RecordSub.Examples.TRcd_j" class="idref" href="#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a> :=<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>. <span class="comment">(* {j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B} *)</span><br/>
<span class="id" title="keyword">Definition</span> <a id="RecordSub.Examples.TRcd_kj" class="idref" href="#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a> :=<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.k"><span class="id" title="abbreviation">k</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>. <span class="comment">(* {k:C<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>C,j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B} *)</span><br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples.subtyping_example_0" class="idref" href="#RecordSub.Examples.subtyping_example_0"><span class="id" title="definition">subtyping_example_0</span></a> :<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.S_Arrow"><span class="id" title="constructor">S_Arrow</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.S_Refl"><span class="id" title="constructor">S_Refl</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a>, <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.S_RcdWidth"><span class="id" title="constructor">S_RcdWidth</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The following facts are mostly easy to prove in Coq. To get full
benefit, make sure you also understand how to prove them on
paper!
<div class="paragraph"> </div>
<a id="lab429"></a><h4 class="section">Exercise: 2 stars, standard (subtyping_example_1)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples.subtyping_example_1" class="idref" href="#RecordSub.Examples.subtyping_example_1"><span class="id" title="definition">subtyping_example_1</span></a> :<br/>
<a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a>.<br/>
<span class="comment">(* {k:A<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>A,j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B} <: {j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B} *)</span><br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab430"></a><h4 class="section">Exercise: 1 star, standard (subtyping_example_2)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples.subtyping_example_2" class="idref" href="#RecordSub.Examples.subtyping_example_2"><span class="id" title="definition">subtyping_example_2</span></a> :<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'Top'"><span class="id" title="notation">Top</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab431"></a><h4 class="section">Exercise: 1 star, standard (subtyping_example_3)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples.subtyping_example_3" class="idref" href="#RecordSub.Examples.subtyping_example_3"><span class="id" title="definition">subtyping_example_3</span></a> :<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.k"><span class="id" title="abbreviation">k</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="comment">(* {}<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>{j:A} <: {k:B}<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>{} *)</span><br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab432"></a><h4 class="section">Exercise: 2 stars, standard (subtyping_example_4)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples.subtyping_example_4" class="idref" href="#RecordSub.Examples.subtyping_example_4"><span class="id" title="definition">subtyping_example_4</span></a> :<br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.y"><span class="id" title="abbreviation">y</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a><br/>
<a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.y"><span class="id" title="abbreviation">y</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a>.<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="code">
<span class="id" title="keyword">End</span> <a class="idref" href="RecordSub.html#RecordSub.Examples"><span class="id" title="module">Examples</span></a>.<br/>
</div>
<div class="doc">
<a id="lab433"></a><h2 class="section">Properties of Subtyping</h2>
<div class="paragraph"> </div>
<a id="lab434"></a><h3 class="section">Well-Formedness</h3>
<div class="paragraph"> </div>
To get started proving things about subtyping, we need a couple of
technical lemmas that intuitively (1) allow us to extract the
well-formedness assumptions embedded in subtyping derivations
and (2) record the fact that fields of well-formed record types
are themselves well-formed types.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.subtype__wf" class="idref" href="#RecordSub.subtype__wf"><span class="id" title="lemma">subtype__wf</span></a> : <span class="id" title="keyword">∀</span> <a id="S:94" class="idref" href="#S:94"><span class="id" title="binder">S</span></a> <a id="T:95" class="idref" href="#T:95"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.subtype"><span class="id" title="inductive">subtype</span></a> <a class="idref" href="RecordSub.html#S:94"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#T:95"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T:95"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#S:94"><span class="id" title="variable">S</span></a>.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">S</span> <span class="id" title="var">T</span> <span class="id" title="var">Hsub</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">Hsub</span>;<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">try</span> (<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHsub1</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHsub2</span>)...<br/>
- <span class="comment">(* S_RcdPerm *)</span><br/>
<span class="id" title="tactic">split</span>... <span class="id" title="tactic">inversion</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">inversion</span> <span class="id" title="var">H<sub>5</sub></span>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.wf_rcd_lookup" class="idref" href="#RecordSub.wf_rcd_lookup"><span class="id" title="lemma">wf_rcd_lookup</span></a> : <span class="id" title="keyword">∀</span> <a id="i:96" class="idref" href="#i:96"><span class="id" title="binder">i</span></a> <a id="T:97" class="idref" href="#T:97"><span class="id" title="binder">T</span></a> <a id="Ti:98" class="idref" href="#Ti:98"><span class="id" title="binder">Ti</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T:97"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:96"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#T:97"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#Ti:98"><span class="id" title="variable">Ti</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#Ti:98"><span class="id" title="variable">Ti</span></a>.<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">i</span> <span class="id" title="var">T</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">T</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">solve_by_invert</span>.<br/>
- <span class="comment">(* RCons *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>0</sub></span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="definition">eqb_string</span> <span class="id" title="var">i</span> <span class="id" title="var">s</span>)... <span class="id" title="tactic">inversion</span> <span class="id" title="var">H<sub>0</sub></span>; <span class="id" title="tactic">subst</span>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a id="lab435"></a><h3 class="section">Field Lookup</h3>
<div class="paragraph"> </div>
The 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 the
corresponding terms. And second, reasoning by induction on typing
derivations becomes harder in general, because typing is no longer
syntax directed.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.rcd_types_match" class="idref" href="#RecordSub.rcd_types_match"><span class="id" title="lemma">rcd_types_match</span></a> : <span class="id" title="keyword">∀</span> <a id="S:99" class="idref" href="#S:99"><span class="id" title="binder">S</span></a> <a id="T:100" class="idref" href="#T:100"><span class="id" title="binder">T</span></a> <a id="i:101" class="idref" href="#i:101"><span class="id" title="binder">i</span></a> <a id="Ti:102" class="idref" href="#Ti:102"><span class="id" title="binder">Ti</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.subtype"><span class="id" title="inductive">subtype</span></a> <a class="idref" href="RecordSub.html#S:99"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#T:100"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:101"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#T:100"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#Ti:102"><span class="id" title="variable">Ti</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="Si:103" class="idref" href="#Si:103"><span class="id" title="binder">Si</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:101"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#S:99"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#Si:103"><span class="id" title="variable">Si</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="RecordSub.html#RecordSub.subtype"><span class="id" title="inductive">subtype</span></a> <a class="idref" href="RecordSub.html#Si:103"><span class="id" title="variable">Si</span></a> <a class="idref" href="RecordSub.html#Ti:102"><span class="id" title="variable">Ti</span></a>.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> (<span class="id" title="tactic">eauto</span> <span class="id" title="keyword">using</span> <a class="idref" href="RecordSub.html#RecordSub.wf_rcd_lookup"><span class="id" title="lemma">wf_rcd_lookup</span></a>).<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">S</span> <span class="id" title="var">T</span> <span class="id" title="var">i</span> <span class="id" title="var">Ti</span> <span class="id" title="var">Hsub</span> <span class="id" title="var">Hget</span>. <span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">Ti</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">Hsub</span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">Ti</span> <span class="id" title="var">Hget</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="var">solve_by_invert</span>.<br/>
- <span class="comment">(* S_Refl *)</span><br/>
<span class="id" title="tactic">∃</span> <span class="id" title="var">Ti</span>...<br/>
- <span class="comment">(* S_Trans *)</span><br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="var">IHHsub2</span> <span class="id" title="var">Ti</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Ui</span> <span class="id" title="var">Hui</span>]... <span class="id" title="tactic">destruct</span> <span class="id" title="var">Hui</span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="var">IHHsub1</span> <span class="id" title="var">Ui</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Si</span> <span class="id" title="var">Hsi</span>]... <span class="id" title="tactic">destruct</span> <span class="id" title="var">Hsi</span>.<br/>
<span class="id" title="tactic">∃</span> <span class="id" title="var">Si</span>...<br/>
- <span class="comment">(* S_RcdDepth *)</span><br/>
<span class="id" title="tactic">rename</span> <span class="id" title="var">i<sub>0</sub></span> <span class="id" title="var">into</span> <span class="id" title="var">k</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a>. <span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hget</span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="definition">eqb_string</span> <span class="id" title="var">i</span> <span class="id" title="var">k</span>)...<br/>
+ <span class="comment">(* i = k -- we're looking up the first field *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Hget</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">∃</span> <span class="id" title="var">S<sub>1</sub></span>...<br/>
- <span class="comment">(* S_RcdPerm *)</span><br/>
<span class="id" title="tactic">∃</span> <span class="id" title="var">Ti</span>. <span class="id" title="tactic">split</span>.<br/>
+ <span class="comment">(* lookup *)</span><br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a>. <span class="id" title="tactic">unfold</span> <a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hget</span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="axiom">eqb_stringP</span> <span class="id" title="var">i</span> <span class="id" title="var">i<sub>1</sub></span>)...<br/>
× <span class="comment">(* i = i<sub>1</sub> -- we're looking up the first field *)</span><br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="axiom">eqb_stringP</span> <span class="id" title="var">i</span> <span class="id" title="var">i<sub>2</sub></span>)...<br/>
<span class="comment">(* i = i<sub>2</sub> -- contradictory *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>0</sub></span>.<br/>
<span class="id" title="tactic">subst</span>...<br/>
+ <span class="comment">(* subtype *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">inversion</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">subst</span>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a id="lab436"></a><h4 class="section">Exercise: 3 stars, standard (rcd_types_match_informal)</h4>
Write a careful informal proof of the <span class="inlinecode"><span class="id" title="var">rcd_types_match</span></span>
lemma.
</div>
<div class="code">
<span class="comment">(* FILL IN HERE *)</span><br/><hr class='doublespaceincode'/>
<span class="comment">(* Do not modify the following line: *)</span><br/>
<span class="id" title="keyword">Definition</span> <a id="RecordSub.manual_grade_for_rcd_types_match_informal" class="idref" href="#RecordSub.manual_grade_for_rcd_types_match_informal"><span class="id" title="definition">manual_grade_for_rcd_types_match_informal</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac<sub>4</sub>"><span class="id" title="notation">×</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) := <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab437"></a><h3 class="section">Inversion Lemmas</h3>
<div class="paragraph"> </div>
<a id="lab438"></a><h4 class="section">Exercise: 3 stars, standard, optional (sub_inversion_arrow)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.sub_inversion_arrow" class="idref" href="#RecordSub.sub_inversion_arrow"><span class="id" title="lemma">sub_inversion_arrow</span></a> : <span class="id" title="keyword">∀</span> <a id="U:104" class="idref" href="#U:104"><span class="id" title="binder">U</span></a> <a id="V<sub>1</sub>:105" class="idref" href="#V<sub>1</sub>:105"><span class="id" title="binder">V<sub>1</sub></span></a> <a id="V<sub>2</sub>:106" class="idref" href="#V<sub>2</sub>:106"><span class="id" title="binder">V<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#U:104"><span class="id" title="variable">U</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#V<sub>1</sub>:105"><span class="id" title="variable">V<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#V<sub>2</sub>:106"><span class="id" title="variable">V<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="U<sub>1</sub>:107" class="idref" href="#U<sub>1</sub>:107"><span class="id" title="binder">U<sub>1</sub></span></a> <a id="U<sub>2</sub>:108" class="idref" href="#U<sub>2</sub>:108"><span class="id" title="binder">U<sub>2</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#U:104"><span class="id" title="variable">U</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#U<sub>1</sub>:107"><span class="id" title="variable">U<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#U<sub>2</sub>:108"><span class="id" title="variable">U<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#V<sub>1</sub>:105"><span class="id" title="variable">V<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#U<sub>1</sub>:107"><span class="id" title="variable">U<sub>1</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#U<sub>2</sub>:108"><span class="id" title="variable">U<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#V<sub>2</sub>:106"><span class="id" title="variable">V<sub>2</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
<div class="togglescript" id="proofcontrol4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')"><span class="show"></span></div>
<div class="proofscript" id="proof4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">U</span> <span class="id" title="var">V<sub>1</sub></span> <span class="id" title="var">V<sub>2</sub></span> <span class="id" title="var">Hs</span>.<br/>
<span class="id" title="var">remember</span> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <span class="id" title="var">V<sub>1</sub></span> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">V<sub>2</sub></span> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">V</span>.<br/>
<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">V<sub>2</sub></span>. <span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">V<sub>1</sub></span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<br/>
</div>
<div class="doc">
<a id="lab439"></a><h1 class="section">Typing</h1>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="RecordSub.context" class="idref" href="#RecordSub.context"><span class="id" title="definition">context</span></a> := <span class="id" title="definition">partial_map</span> <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Reserved Notation</span> "Gamma '⊢' t '∈' T" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40,<br/>
<span class="id" title="var">t</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">T</span> <span class="id" title="var">custom</span> <span class="id" title="var">stlc_ty</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="RecordSub.has_type" class="idref" href="#RecordSub.has_type"><span class="id" title="inductive">has_type</span></a> : <a class="idref" href="RecordSub.html#RecordSub.context"><span class="id" title="definition">context</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.tm"><span class="id" title="inductive">tm</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.ty"><span class="id" title="inductive">ty</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="RecordSub.T_Var" class="idref" href="#RecordSub.T_Var"><span class="id" title="constructor">T_Var</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:111" class="idref" href="#Gamma:111"><span class="id" title="binder">Gamma</span></a> (<a id="x:112" class="idref" href="#x:112"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) <a id="T:113" class="idref" href="#T:113"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:111"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#x:112"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#T:113"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T:113"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:111"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#x:112"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:113"><span class="id" title="variable">T</span></a><br/>
| <a id="RecordSub.T_Abs" class="idref" href="#RecordSub.T_Abs"><span class="id" title="constructor">T_Abs</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:114" class="idref" href="#Gamma:114"><span class="id" title="binder">Gamma</span></a> <a id="x:115" class="idref" href="#x:115"><span class="id" title="binder">x</span></a> <a id="T<sub>11</sub>:116" class="idref" href="#T<sub>11</sub>:116"><span class="id" title="binder">T<sub>11</sub></span></a> <a id="T<sub>12</sub>:117" class="idref" href="#T<sub>12</sub>:117"><span class="id" title="binder">T<sub>12</sub></span></a> <a id="t<sub>12</sub>:118" class="idref" href="#t<sub>12</sub>:118"><span class="id" title="binder">t<sub>12</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T<sub>11</sub>:116"><span class="id" title="variable">T<sub>11</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#x:115"><span class="id" title="variable">x</span></a> <span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span> <a class="idref" href="RecordSub.html#T<sub>11</sub>:116"><span class="id" title="variable">T<sub>11</sub></span></a><span class="id" title="notation">;</span> <a class="idref" href="RecordSub.html#Gamma:114"><span class="id" title="variable">Gamma</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t<sub>12</sub>:118"><span class="id" title="variable">t<sub>12</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T<sub>12</sub>:117"><span class="id" title="variable">T<sub>12</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:114"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#x:115"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T<sub>11</sub>:116"><span class="id" title="variable">T<sub>11</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#t<sub>12</sub>:118"><span class="id" title="variable">t<sub>12</sub></span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#T<sub>11</sub>:116"><span class="id" title="variable">T<sub>11</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>12</sub>:117"><span class="id" title="variable">T<sub>12</sub></span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a><br/>
| <a id="RecordSub.T_App" class="idref" href="#RecordSub.T_App"><span class="id" title="constructor">T_App</span></a> : <span class="id" title="keyword">∀</span> <a id="T<sub>1</sub>:119" class="idref" href="#T<sub>1</sub>:119"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:120" class="idref" href="#T<sub>2</sub>:120"><span class="id" title="binder">T<sub>2</sub></span></a> <a id="Gamma:121" class="idref" href="#Gamma:121"><span class="id" title="binder">Gamma</span></a> <a id="t<sub>1</sub>:122" class="idref" href="#t<sub>1</sub>:122"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>2</sub>:123" class="idref" href="#t<sub>2</sub>:123"><span class="id" title="binder">t<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:121"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:122"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#T<sub>1</sub>:119"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:120"><span class="id" title="variable">T<sub>2</sub></span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:121"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:123"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T<sub>1</sub>:119"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:121"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t<sub>1</sub>:122"><span class="id" title="variable">t<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:123"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:120"><span class="id" title="variable">T<sub>2</sub></span></a><br/>
| <a id="RecordSub.T_Proj" class="idref" href="#RecordSub.T_Proj"><span class="id" title="constructor">T_Proj</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:124" class="idref" href="#Gamma:124"><span class="id" title="binder">Gamma</span></a> <a id="i:125" class="idref" href="#i:125"><span class="id" title="binder">i</span></a> <a id="t:126" class="idref" href="#t:126"><span class="id" title="binder">t</span></a> <a id="T:127" class="idref" href="#T:127"><span class="id" title="binder">T</span></a> <a id="Ti:128" class="idref" href="#Ti:128"><span class="id" title="binder">Ti</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:124"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:126"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:127"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:125"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#T:127"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#Ti:128"><span class="id" title="variable">Ti</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:124"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:126"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#i:125"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#Ti:128"><span class="id" title="variable">Ti</span></a><br/>
<span class="comment">(* Subsumption *)</span><br/>
| <a id="RecordSub.T_Sub" class="idref" href="#RecordSub.T_Sub"><span class="id" title="constructor">T_Sub</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:129" class="idref" href="#Gamma:129"><span class="id" title="binder">Gamma</span></a> <a id="t:130" class="idref" href="#t:130"><span class="id" title="binder">t</span></a> <a id="S:131" class="idref" href="#S:131"><span class="id" title="binder">S</span></a> <a id="T:132" class="idref" href="#T:132"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:129"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:130"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#S:131"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.subtype"><span class="id" title="inductive">subtype</span></a> <a class="idref" href="RecordSub.html#S:131"><span class="id" title="variable">S</span></a> <a class="idref" href="RecordSub.html#T:132"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:129"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:130"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:132"><span class="id" title="variable">T</span></a><br/>
<span class="comment">(* Rules for record terms *)</span><br/>
| <a id="RecordSub.T_RNil" class="idref" href="#RecordSub.T_RNil"><span class="id" title="constructor">T_RNil</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:133" class="idref" href="#Gamma:133"><span class="id" title="binder">Gamma</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:133"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::'nil'"><span class="id" title="notation">nil</span></a><br/>
| <a id="RecordSub.T_RCons" class="idref" href="#RecordSub.T_RCons"><span class="id" title="constructor">T_RCons</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:134" class="idref" href="#Gamma:134"><span class="id" title="binder">Gamma</span></a> <a id="i:135" class="idref" href="#i:135"><span class="id" title="binder">i</span></a> <a id="t:136" class="idref" href="#t:136"><span class="id" title="binder">t</span></a> <a id="T:137" class="idref" href="#T:137"><span class="id" title="binder">T</span></a> <a id="tr:138" class="idref" href="#tr:138"><span class="id" title="binder">tr</span></a> <a id="Tr:139" class="idref" href="#Tr:139"><span class="id" title="binder">Tr</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:134"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:136"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:137"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:134"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#tr:138"><span class="id" title="variable">tr</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#Tr:139"><span class="id" title="variable">Tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_ty"><span class="id" title="inductive">record_ty</span></a> <a class="idref" href="RecordSub.html#Tr:139"><span class="id" title="variable">Tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_tm"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#tr:138"><span class="id" title="variable">tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#Gamma:134"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#i:135"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#t:136"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#tr:138"><span class="id" title="variable">tr</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#i:135"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#T:137"><span class="id" title="variable">T</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_':'_x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#Tr:139"><span class="id" title="variable">Tr</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a><br/>
<br/>
<span class="id" title="keyword">where</span> <a id="0295dc7f5e0fb00e1819de789c6e3774" class="idref" href="#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">"</span></a>Gamma '⊢' t '∈' T" := (<a class="idref" href="RecordSub.html#has_type:110"><span class="id" title="inductive">has_type</span></a> <span class="id" title="var">Gamma</span> <span class="id" title="var">t</span> <span class="id" title="var">T</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="RecordSub.html#has_type"><span class="id" title="inductive">has_type</span></a> : <span class="id" title="var">core</span>.<br/>
</div>
<div class="doc">
<a id="lab440"></a><h2 class="section">Typing Examples</h2>
</div>
<div class="code">
<span class="id" title="keyword">Module</span> <a id="RecordSub.Examples2" class="idref" href="#RecordSub.Examples2"><span class="id" title="module">Examples2</span></a>.<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">Examples</span>.<br/>
</div>
<div class="doc">
<a id="lab441"></a><h4 class="section">Exercise: 1 star, standard (typing_example_0)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="RecordSub.Examples2.trcd_kj" class="idref" href="#RecordSub.Examples2.trcd_kj"><span class="id" title="definition">trcd_kj</span></a> :=<br/>
<a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.k"><span class="id" title="abbreviation">k</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.A"><span class="id" title="abbreviation">A</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::'nil'"><span class="id" title="notation">nil</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples2.typing_example_0" class="idref" href="#RecordSub.Examples2.typing_example_0"><span class="id" title="definition">typing_example_0</span></a> :<br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples2.trcd_kj"><span class="id" title="definition">trcd_kj</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_kj"><span class="id" title="definition">TRcd_kj</span></a>.<br/>
<span class="comment">(* empty ⊢ {k=(\z:A.z), j=(\z:B.z)} : {k:A<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>A,j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B} *)</span><br/>
<div class="togglescript" id="proofcontrol5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')"><span class="show"></span></div>
<div class="proofscript" id="proof5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')">
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<br/>
</div>
<div class="doc">
<a id="lab442"></a><h4 class="section">Exercise: 2 stars, standard (typing_example_1)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples2.typing_example_1" class="idref" href="#RecordSub.Examples2.typing_example_1"><span class="id" title="definition">typing_example_1</span></a> :<br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples2.trcd_kj"><span class="id" title="definition">trcd_kj</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a>.<br/>
<span class="comment">(* empty ⊢ (\x:{k:A<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>A,j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B}. x.j)<br/>
{k=(\z:A.z), j=(\z:B.z)}<br/>
: B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B *)</span><br/>
<div class="togglescript" id="proofcontrol6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')"><span class="show"></span></div>
<div class="proofscript" id="proof6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<br/>
</div>
<div class="doc">
<a id="lab443"></a><h4 class="section">Exercise: 2 stars, standard, optional (typing_example_2)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Example</span> <a id="RecordSub.Examples2.typing_example_2" class="idref" href="#RecordSub.Examples2.typing_example_2"><span class="id" title="definition">typing_example_2</span></a> :<br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.TRcd_j"><span class="id" title="definition">TRcd_j</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.x"><span class="id" title="abbreviation">x</span></a><a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.j"><span class="id" title="abbreviation">j</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a><br/>
<a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">(</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.z"><span class="id" title="abbreviation">z</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.C"><span class="id" title="abbreviation">C</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples2.trcd_kj"><span class="id" title="definition">trcd_kj</span></a> <a class="idref" href="RecordSub.html#53937f3ba9e51dc428955dd7e0fbd7fe"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.Examples.B"><span class="id" title="abbreviation">B</span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a>.<br/>
<span class="comment">(* empty ⊢ (\z:(C<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>C)<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>{j:B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B}. (z (\x:C.x)).j)<br/>
(\z:C<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>C. {k=(\z:A.z), j=(\z:B.z)})<br/>
: B<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span>B *)</span><br/>
<div class="togglescript" id="proofcontrol7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')"><span class="show"></span></div>
<div class="proofscript" id="proof7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="RecordSub.html#RecordSub.Examples2"><span class="id" title="module">Examples2</span></a>.<br/>
</div>
<div class="doc">
<a id="lab444"></a><h2 class="section">Properties of Typing</h2>
<div class="paragraph"> </div>
<a id="lab445"></a><h3 class="section">Well-Formedness</h3>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.has_type__wf" class="idref" href="#RecordSub.has_type__wf"><span class="id" title="lemma">has_type__wf</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:140" class="idref" href="#Gamma:140"><span class="id" title="binder">Gamma</span></a> <a id="t:141" class="idref" href="#t:141"><span class="id" title="binder">t</span></a> <a id="T:142" class="idref" href="#T:142"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.has_type"><span class="id" title="inductive">has_type</span></a> <a class="idref" href="RecordSub.html#Gamma:140"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#t:141"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#T:142"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#RecordSub.well_formed_ty"><span class="id" title="inductive">well_formed_ty</span></a> <a class="idref" href="RecordSub.html#T:142"><span class="id" title="variable">T</span></a>.<br/>
<div class="togglescript" id="proofcontrol8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')"><span class="show"></span></div>
<div class="proofscript" id="proof8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Gamma</span> <span class="id" title="var">t</span> <span class="id" title="var">T</span> <span class="id" title="var">Htyp</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">Htyp</span>...<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">IHHtyp1</span>...<br/>
- <span class="comment">(* T_Proj *)</span><br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="RecordSub.html#RecordSub.wf_rcd_lookup"><span class="id" title="lemma">wf_rcd_lookup</span></a>...<br/>
- <span class="comment">(* T_Sub *)</span><br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.subtype__wf"><span class="id" title="lemma">subtype__wf</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span>...<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.step_preserves_record_tm" class="idref" href="#RecordSub.step_preserves_record_tm"><span class="id" title="lemma">step_preserves_record_tm</span></a> : <span class="id" title="keyword">∀</span> <a id="tr:143" class="idref" href="#tr:143"><span class="id" title="binder">tr</span></a> <a id="tr':144" class="idref" href="#tr':144"><span class="id" title="binder">tr'</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.record_tm"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#tr:143"><span class="id" title="variable">tr</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#tr:143"><span class="id" title="variable">tr</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#tr':144"><span class="id" title="variable">tr'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.record_tm"><span class="id" title="inductive">record_tm</span></a> <a class="idref" href="RecordSub.html#tr':144"><span class="id" title="variable">tr'</span></a>.<br/>
<div class="togglescript" id="proofcontrol9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')"><span class="show"></span></div>
<div class="proofscript" id="proof9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')">
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">tr</span> <span class="id" title="var">tr'</span> <span class="id" title="var">Hrt</span> <span class="id" title="var">Hstp</span>.<br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Hrt</span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">Hstp</span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a id="lab446"></a><h3 class="section">Field Lookup</h3>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.lookup_field_in_value" class="idref" href="#RecordSub.lookup_field_in_value"><span class="id" title="lemma">lookup_field_in_value</span></a> : <span class="id" title="keyword">∀</span> <a id="v:145" class="idref" href="#v:145"><span class="id" title="binder">v</span></a> <a id="T:146" class="idref" href="#T:146"><span class="id" title="binder">T</span></a> <a id="i:147" class="idref" href="#i:147"><span class="id" title="binder">i</span></a> <a id="Ti:148" class="idref" href="#Ti:148"><span class="id" title="binder">Ti</span></a>,<br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#v:145"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#v:145"><span class="id" title="variable">v</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:146"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.Tlookup"><span class="id" title="definition">Tlookup</span></a> <a class="idref" href="RecordSub.html#i:147"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#T:146"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#Ti:148"><span class="id" title="variable">Ti</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="vi:149" class="idref" href="#vi:149"><span class="id" title="binder">vi</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#RecordSub.tlookup"><span class="id" title="definition">tlookup</span></a> <a class="idref" href="RecordSub.html#i:147"><span class="id" title="variable">i</span></a> <a class="idref" href="RecordSub.html#v:145"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="RecordSub.html#vi:149"><span class="id" title="variable">vi</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#vi:149"><span class="id" title="variable">vi</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#Ti:148"><span class="id" title="variable">Ti</span></a>.<br/>
<div class="togglescript" id="proofcontrol10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')"><span class="show"></span></div>
<div class="proofscript" id="proof10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="var">remember</span> <span class="id" title="definition">empty</span> <span class="id" title="keyword">as</span> <span class="id" title="var">Gamma</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">t</span> <span class="id" title="var">T</span> <span class="id" title="var">i</span> <span class="id" title="var">Ti</span> <span class="id" title="var">Hval</span> <span class="id" title="var">Htyp</span>. <span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">Ti</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">Htyp</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">solve_by_invert</span>.<br/>
- <span class="comment">(* T_Sub *)</span><br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="RecordSub.html#RecordSub.rcd_types_match"><span class="id" title="lemma">rcd_types_match</span></a> <span class="id" title="var">S</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>0</sub></span>...<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>0</sub></span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Si</span> [<span class="id" title="var">HgetSi</span> <span class="id" title="var">Hsub</span>]].<br/>
<span class="id" title="tactic">eapply</span> <span class="id" title="var">IHHtyp</span> <span class="id" title="keyword">in</span> <span class="id" title="var">HgetSi</span>...<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">HgetSi</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">vi</span> [<span class="id" title="var">Hget</span> <span class="id" title="var">Htyvi</span>]]...<br/>
- <span class="comment">(* T_RCons *)</span><br/>
<span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>0</sub></span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>1</sub></span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="definition">eqb_string</span> <span class="id" title="var">i</span> <span class="id" title="var">i<sub>0</sub></span>).<br/>
+ <span class="comment">(* i is first *)</span><br/>
<span class="id" title="tactic">injection</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="keyword">as</span> <span class="id" title="var">H<sub>1</sub></span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">∃</span> <span class="id" title="var">t</span>...<br/>
+ <span class="comment">(* i in tail *)</span><br/>
<span class="id" title="tactic">eapply</span> <span class="id" title="var">IHHtyp2</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>1</sub></span>...<br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Hval</span>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a id="lab447"></a><h3 class="section">Progress</h3>
<div class="paragraph"> </div>
<a id="lab448"></a><h4 class="section">Exercise: 3 stars, standard (canonical_forms_of_arrow_types)</h4>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.canonical_forms_of_arrow_types" class="idref" href="#RecordSub.canonical_forms_of_arrow_types"><span class="id" title="lemma">canonical_forms_of_arrow_types</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:150" class="idref" href="#Gamma:150"><span class="id" title="binder">Gamma</span></a> <a id="s:151" class="idref" href="#s:151"><span class="id" title="binder">s</span></a> <a id="T<sub>1</sub>:152" class="idref" href="#T<sub>1</sub>:152"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:153" class="idref" href="#T<sub>2</sub>:153"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:150"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#s:151"><span class="id" title="variable">s</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#T<sub>1</sub>:152"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:153"><span class="id" title="variable">T<sub>2</sub></span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#s:151"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:154" class="idref" href="#x:154"><span class="id" title="binder">x</span></a> <a id="S<sub>1</sub>:155" class="idref" href="#S<sub>1</sub>:155"><span class="id" title="binder">S<sub>1</sub></span></a> <a id="s<sub>2</sub>:156" class="idref" href="#s<sub>2</sub>:156"><span class="id" title="binder">s<sub>2</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
<a class="idref" href="RecordSub.html#s:151"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#x:154"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:155"><span class="id" title="variable">S<sub>1</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#s<sub>2</sub>:156"><span class="id" title="variable">s<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>.<br/>
<div class="togglescript" id="proofcontrol11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')"><span class="show"></span></div>
<div class="proofscript" id="proof11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<br/>
<span class="id" title="keyword">Theorem</span> <a id="RecordSub.progress" class="idref" href="#RecordSub.progress"><span class="id" title="lemma">progress</span></a> : <span class="id" title="keyword">∀</span> <a id="t:157" class="idref" href="#t:157"><span class="id" title="binder">t</span></a> <a id="T:158" class="idref" href="#T:158"><span class="id" title="binder">T</span></a>,<br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t:157"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:158"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#RecordSub.value"><span class="id" title="inductive">value</span></a> <a class="idref" href="RecordSub.html#t:157"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e<sub>25</sub>"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="t':159" class="idref" href="#t':159"><span class="id" title="binder">t'</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#t:157"><span class="id" title="variable">t</span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <a class="idref" href="RecordSub.html#t':159"><span class="id" title="variable">t'</span></a>.<br/>
<div class="togglescript" id="proofcontrol12" onclick="toggleDisplay('proof12');toggleDisplay('proofcontrol12')"><span class="show"></span></div>
<div class="proofscript" id="proof12" onclick="toggleDisplay('proof12');toggleDisplay('proofcontrol12')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">t</span> <span class="id" title="var">T</span> <span class="id" title="var">Ht</span>.<br/>
<span class="id" title="var">remember</span> <span class="id" title="definition">empty</span> <span class="id" title="keyword">as</span> <span class="id" title="var">Gamma</span>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">HeqGamma</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">Ht</span>;<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">HeqGamma</span>; <span class="id" title="tactic">subst</span>...<br/>
- <span class="comment">(* T_Var *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">H</span>.<br/>
- <span class="comment">(* T_App *)</span><br/>
<span class="id" title="tactic">right</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHt1</span>; <span class="id" title="tactic">subst</span>...<br/>
+ <span class="comment">(* t<sub>1</sub> is a value *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHt2</span>; <span class="id" title="tactic">subst</span>...<br/>
× <span class="comment">(* t<sub>2</sub> is a value *)</span><br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="RecordSub.html#RecordSub.canonical_forms_of_arrow_types"><span class="id" title="axiom">canonical_forms_of_arrow_types</span></a> <span class="id" title="definition">empty</span> <span class="id" title="var">t<sub>1</sub></span> <span class="id" title="var">T<sub>1</sub></span> <span class="id" title="var">T<sub>2</sub></span>)<br/>
<span class="id" title="keyword">as</span> [<span class="id" title="var">x</span> [<span class="id" title="var">S<sub>1</sub></span> [<span class="id" title="var">t<sub>12</sub></span> <span class="id" title="var">Heqt1</span>]]]...<br/>
<span class="id" title="tactic">subst</span>. <span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">[</span></a><span class="id" title="var">x</span><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">:=</span></a><span class="id" title="var">t<sub>2</sub></span><a class="idref" href="RecordSub.html#51bec6b6e9a9f3b871de8b1a020b69d<sub>7</sub>"><span class="id" title="notation">]</span></a> <span class="id" title="var">t<sub>12</sub></span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>...<br/>
× <span class="comment">(* t<sub>2</sub> steps *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>0</sub></span> <span class="id" title="keyword">as</span> [<span class="id" title="var">t<sub>2</sub>'</span> <span class="id" title="var">Hstp</span>]. <span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">t<sub>1</sub></span> <span class="id" title="var">t<sub>2</sub>'</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> ...<br/>
+ <span class="comment">(* t<sub>1</sub> steps *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">t<sub>1</sub>'</span> <span class="id" title="var">Hstp</span>]. <span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">t<sub>1</sub>'</span> <span class="id" title="var">t<sub>2</sub></span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>...<br/>
- <span class="comment">(* T_Proj *)</span><br/>
<span class="id" title="tactic">right</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHt</span>...<br/>
+ <span class="comment">(* rcd is value *)</span><br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="RecordSub.html#RecordSub.lookup_field_in_value"><span class="id" title="lemma">lookup_field_in_value</span></a> <span class="id" title="var">t</span> <span class="id" title="var">T</span> <span class="id" title="var">i</span> <span class="id" title="var">Ti</span>)<br/>
<span class="id" title="keyword">as</span> [<span class="id" title="var">t'</span> [<span class="id" title="var">Hget</span> <span class="id" title="var">Ht'</span>]]...<br/>
+ <span class="comment">(* rcd_steps *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>0</sub></span> <span class="id" title="keyword">as</span> [<span class="id" title="var">t'</span> <span class="id" title="var">Hstp</span>]. <span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">t'</span> <a class="idref" href="RecordSub.html#RecordSub.:stlc::x_'-->'_x"><span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>...<br/>
- <span class="comment">(* T_RCons *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHt1</span>...<br/>
+ <span class="comment">(* head is a value *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHHt2</span>...<br/>
× <span class="comment">(* tail steps *)</span><br/>
<span class="id" title="tactic">right</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>2</sub></span> <span class="id" title="keyword">as</span> [<span class="id" title="var">tr'</span> <span class="id" title="var">Hstp</span>].<br/>
<span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <span class="id" title="var">t</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <span class="id" title="var">tr'</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>...<br/>
+ <span class="comment">(* head steps *)</span><br/>
<span class="id" title="tactic">right</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="keyword">as</span> [<span class="id" title="var">t'</span> <span class="id" title="var">Hstp</span>].<br/>
<span class="id" title="tactic">∃</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <span class="id" title="var">i</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">:=</span></a> <span class="id" title="var">t'</span> <a class="idref" href="RecordSub.html#24532377138054657f673a020656af9f"><span class="id" title="notation">::</span></a> <span class="id" title="var">tr</span><a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<i>Theorem</i> : For any term <span class="inlinecode"><span class="id" title="var">t</span></span> and type <span class="inlinecode"><span class="id" title="var">T</span></span>, if <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T</span></span>
then <span class="inlinecode"><span class="id" title="var">t</span></span> is a value or <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">t'</span></span> for some term <span class="inlinecode"><span class="id" title="var">t'</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: Let <span class="inlinecode"><span class="id" title="var">t</span></span> and <span class="inlinecode"><span class="id" title="var">T</span></span> be given such that <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T</span></span>. We
proceed by induction on the given typing derivation.
<div class="paragraph"> </div>
<ul class="doclist">
<li> The cases where the last step in the typing derivation is
<span class="inlinecode"><span class="id" title="var">T_Abs</span></span> or <span class="inlinecode"><span class="id" title="var">T_RNil</span></span> are immediate because abstractions and
<span class="inlinecode">{}</span> are always values. The case for <span class="inlinecode"><span class="id" title="var">T_Var</span></span> is vacuous
because variables cannot be typed in the empty context.
<div class="paragraph"> </div>
</li>
<li> If the last step in the typing derivation is by <span class="inlinecode"><span class="id" title="var">T_App</span></span>, then
there are terms <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> and types <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">T<sub>2</sub></span></span> such that <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span>, <span class="inlinecode"><span class="id" title="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">T<sub>2</sub></span></span>, <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">T<sub>2</sub></span></span> and <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> <span class="inlinecode">:</span>
<span class="inlinecode"><span class="id" title="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" title="var">t<sub>1</sub></span></span> is a value or steps, and that <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> is a value or
steps.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>1</sub>'</span></span> for some term <span class="inlinecode"><span class="id" title="var">t<sub>1</sub>'</span></span>. Then <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span>
<span class="inlinecode"><span class="id" title="var">t<sub>1</sub>'</span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" title="var">ST_App1</span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise <span class="inlinecode"><span class="id" title="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" title="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub>'</span></span> for some term <span class="inlinecode"><span class="id" title="var">t<sub>2</sub>'</span></span>. Then <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span>
<span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub>'</span></span> by rule <span class="inlinecode"><span class="id" title="var">ST_App2</span></span> because <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> is a value.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> is a value. By Lemma
<span class="inlinecode"><span class="id" title="var">canonical_forms_for_arrow_types</span></span>, <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode">\<span class="id" title="var">x</span>:<span class="id" title="var">S<sub>1</sub>.s2</span></span> for
some <span class="inlinecode"><span class="id" title="var">x</span></span>, <span class="inlinecode"><span class="id" title="var">S<sub>1</sub></span></span>, and <span class="inlinecode"><span class="id" title="var">s<sub>2</sub></span></span>. But then <span class="inlinecode">(\<span class="id" title="var">x</span>:<span class="id" title="var">S<sub>1</sub>.s2</span>)</span> <span class="inlinecode"><span class="id" title="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span>
<span class="inlinecode">[<span class="id" title="var">x</span>:=<span class="id" title="var">t<sub>2</sub></span>]<span class="id" title="var">s<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" title="var">ST_AppAbs</span></span>, since <span class="inlinecode"><span class="id" title="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" title="var">T_Proj</span></span>, then there
are a term <span class="inlinecode"><span class="id" title="var">tr</span></span>, a type <span class="inlinecode"><span class="id" title="var">Tr</span></span>, and a label <span class="inlinecode"><span class="id" title="var">i</span></span> such that <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" title="var">tr.i</span></span>, <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">tr</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">Tr</span></span>, and <span class="inlinecode"><span class="id" title="var">Tlookup</span></span> <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode"><span class="id" title="var">Tr</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">Some</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span>.
<div class="paragraph"> </div>
By the IH, either <span class="inlinecode"><span class="id" title="var">tr</span></span> is a value or it steps. If <span class="inlinecode"><span class="id" title="var">tr</span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span>
<span class="inlinecode"><span class="id" title="var">tr'</span></span> for some term <span class="inlinecode"><span class="id" title="var">tr'</span></span>, then <span class="inlinecode"><span class="id" title="var">tr.i</span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">tr'.i</span></span> by rule
<span class="inlinecode"><span class="id" title="var">ST_Proj1</span></span>.
<div class="paragraph"> </div>
If <span class="inlinecode"><span class="id" title="var">tr</span></span> is a value, then Lemma <span class="inlinecode"><span class="id" title="var">lookup_field_in_value</span></span> yields
that there is a term <span class="inlinecode"><span class="id" title="var">ti</span></span> such that <span class="inlinecode"><span class="id" title="var">tlookup</span></span> <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode"><span class="id" title="var">tr</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">Some</span></span> <span class="inlinecode"><span class="id" title="var">ti</span></span>.
It follows that <span class="inlinecode"><span class="id" title="var">tr.i</span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">ti</span></span> by rule <span class="inlinecode"><span class="id" title="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" title="var">T_Sub</span></span>, then there
is a type <span class="inlinecode"><span class="id" title="var">S</span></span> such that <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode"><:</span> <span class="inlinecode"><span class="id" title="var">T</span></span> and <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="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" title="var">T_RCons</span></span>, then
there exist some terms <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">tr</span></span>, types <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span> <span class="inlinecode"><span class="id" title="var">Tr</span></span> and a label
<span class="inlinecode"><span class="id" title="var">t</span></span> such that <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" title="var">tr</span>}</span>, <span class="inlinecode"><span class="id" title="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode">{<span class="id" title="var">i</span>:<span class="id" title="var">T<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" title="var">Tr</span>}</span>, <span class="inlinecode"><span class="id" title="var">record_ty</span></span>
<span class="inlinecode"><span class="id" title="var">tr</span></span>, <span class="inlinecode"><span class="id" title="var">record_tm</span></span> <span class="inlinecode"><span class="id" title="var">Tr</span></span>, <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" title="var">empty</span></span> <span class="inlinecode">⊢</span> <span class="inlinecode"><span class="id" title="var">tr</span></span> <span class="inlinecode">:</span>
<span class="inlinecode"><span class="id" title="var">Tr</span></span>.
<div class="paragraph"> </div>
The induction hypotheses for these typing derivations yield
that <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> is a value or steps, and that <span class="inlinecode"><span class="id" title="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" title="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">t<sub>1</sub>'</span></span> for some term <span class="inlinecode"><span class="id" title="var">t<sub>1</sub>'</span></span>. Then <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" title="var">tr</span>}</span>
<span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub>'</span>,</span> <span class="inlinecode"><span class="id" title="var">tr</span>}</span> by rule <span class="inlinecode"><span class="id" title="var">ST_Rcd_Head</span></span>.
<div class="paragraph"> </div>
</li>
<li> Otherwise <span class="inlinecode"><span class="id" title="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" title="var">tr</span></span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode"><span class="id" title="var">tr'</span></span> for some term <span class="inlinecode"><span class="id" title="var">tr'</span></span>. Then <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub></span>,</span>
<span class="inlinecode"><span class="id" title="var">tr</span>}</span> <span class="inlinecode"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:6%;'><span style='letter-spacing:-.2em;'>-</span><span style='letter-spacing:-.2em;'>-</span></span>></span></span></span> <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" title="var">tr'</span>}</span> by rule <span class="inlinecode"><span class="id" title="var">ST_Rcd_Tail</span></span>, since <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span></span> is
a value.
<div class="paragraph"> </div>
</li>
<li> Otherwise, <span class="inlinecode"><span class="id" title="var">tr</span></span> is also a value. So, <span class="inlinecode">{<span class="id" title="var">i</span>=<span class="id" title="var">t<sub>1</sub></span>,</span> <span class="inlinecode"><span class="id" title="var">tr</span>}</span> is a
value by <span class="inlinecode"><span class="id" title="var">v_rcons</span></span>.
</li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
<div class="doc">
<a id="lab449"></a><h3 class="section">Inversion Lemma</h3>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.typing_inversion_abs" class="idref" href="#RecordSub.typing_inversion_abs"><span class="id" title="lemma">typing_inversion_abs</span></a> : <span class="id" title="keyword">∀</span> <a id="Gamma:160" class="idref" href="#Gamma:160"><span class="id" title="binder">Gamma</span></a> <a id="x:161" class="idref" href="#x:161"><span class="id" title="binder">x</span></a> <a id="S<sub>1</sub>:162" class="idref" href="#S<sub>1</sub>:162"><span class="id" title="binder">S<sub>1</sub></span></a> <a id="t<sub>2</sub>:163" class="idref" href="#t<sub>2</sub>:163"><span class="id" title="binder">t<sub>2</sub></span></a> <a id="T:164" class="idref" href="#T:164"><span class="id" title="binder">T</span></a>,<br/>
<a class="idref" href="RecordSub.html#Gamma:160"><span class="id" title="variable">Gamma</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <a class="idref" href="RecordSub.html#x:161"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:162"><span class="id" title="variable">S<sub>1</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:163"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T:164"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="S<sub>2</sub>:165" class="idref" href="#S<sub>2</sub>:165"><span class="id" title="binder">S<sub>2</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><<span style='letter-spacing:-.4em;'>{</span>{</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:162"><span class="id" title="variable">S<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#S<sub>2</sub>:165"><span class="id" title="variable">S<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#6713cbc242333a153601674b9e8101e<sub>9</sub>"><span class="id" title="notation"><span style='letter-spacing:-.4em;'>}</span>}></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#T:164"><span class="id" title="variable">T</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#x:161"><span class="id" title="variable">x</span></a> <span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span> <a class="idref" href="RecordSub.html#S<sub>1</sub>:162"><span class="id" title="variable">S<sub>1</sub></span></a><span class="id" title="notation">;</span> <a class="idref" href="RecordSub.html#Gamma:160"><span class="id" title="variable">Gamma</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#t<sub>2</sub>:163"><span class="id" title="variable">t<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#S<sub>2</sub>:165"><span class="id" title="variable">S<sub>2</sub></span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/>
<div class="togglescript" id="proofcontrol13" onclick="toggleDisplay('proof13');toggleDisplay('proofcontrol13')"><span class="show"></span></div>
<div class="proofscript" id="proof13" onclick="toggleDisplay('proof13');toggleDisplay('proofcontrol13')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Gamma</span> <span class="id" title="var">x</span> <span class="id" title="var">S<sub>1</sub></span> <span class="id" title="var">t<sub>2</sub></span> <span class="id" title="var">T</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="var">remember</span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation"><{</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a> <span class="id" title="var">x</span> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <span class="id" title="var">S<sub>1</sub></span><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <span class="id" title="var">t<sub>2</sub></span> <a class="idref" href="RecordSub.html#d60fde6c3996807cd13393b39ca380c<sub>9</sub>"><span class="id" title="notation">}></span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">t</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">H</span>;<br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Heqt</span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">solve_by_invert</span>.<br/>
- <span class="comment">(* T_Abs *)</span><br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hwf</span> := <a class="idref" href="RecordSub.html#RecordSub.has_type__wf"><span class="id" title="lemma">has_type__wf</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H<sub>0</sub></span>).<br/>
<span class="id" title="tactic">∃</span> <span class="id" title="var">T<sub>12</sub></span>...<br/>
- <span class="comment">(* T_Sub *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">IHhas_type</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">S<sub>2</sub></span> [<span class="id" title="var">Hsub</span> <span class="id" title="var">Hty</span>]]...<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="RecordSub.abs_arrow" class="idref" href="#RecordSub.abs_arrow"><span class="id" title="lemma">abs_arrow</span></a> : <span class="id" title="keyword">∀</span> <a id="x:166" class="idref" href="#x:166"><span class="id" title="binder">x</span></a> <a id="S<sub>1</sub>:167" class="idref" href="#S<sub>1</sub>:167"><span class="id" title="binder">S<sub>1</sub></span></a> <a id="s<sub>2</sub>:168" class="idref" href="#s<sub>2</sub>:168"><span class="id" title="binder">s<sub>2</sub></span></a> <a id="T<sub>1</sub>:169" class="idref" href="#T<sub>1</sub>:169"><span class="id" title="binder">T<sub>1</sub></span></a> <a id="T<sub>2</sub>:170" class="idref" href="#T<sub>2</sub>:170"><span class="id" title="binder">T<sub>2</sub></span></a>,<br/>
<span class="id" title="definition">empty</span> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#x:166"><span class="id" title="variable">x</span></a> <a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:167"><span class="id" title="variable">S<sub>1</sub></span></a><a class="idref" href="RecordSub.html#90b3ca4a4628f04e832c34ef6a4befd8"><span class="id" title="notation">,</span></a> <a class="idref" href="RecordSub.html#s<sub>2</sub>:168"><span class="id" title="variable">s<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#T<sub>1</sub>:169"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:stlc_ty::x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:170"><span class="id" title="variable">T<sub>2</sub></span></a><a class="idref" href="RecordSub.html#731ae131b4917695ec9910860e6c9fc<sub>4</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="RecordSub.html#T<sub>1</sub>:169"><span class="id" title="variable">T<sub>1</sub></span></a> <a class="idref" href="RecordSub.html#RecordSub.:::x_'<:'_x"><span class="id" title="notation"><:</span></a> <a class="idref" href="RecordSub.html#S<sub>1</sub>:167"><span class="id" title="variable">S<sub>1</sub></span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">(</span></a><a class="idref" href="RecordSub.html#x:166"><span class="id" title="variable">x</span></a> <span class="id" title="notation"><span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>⊢</span><span style='font-size:90%;'>></span></span></span></span></span> <a class="idref" href="RecordSub.html#S<sub>1</sub>:167"><span class="id" title="variable">S<sub>1</sub></span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">)</span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">⊢</span></a> <a class="idref" href="RecordSub.html#s<sub>2</sub>:168"><span class="id" title="variable">s<sub>2</sub></span></a> <a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">\</span></a><a class="idref" href="RecordSub.html#0295dc7f5e0fb00e1819de789c6e3774"><span class="id" title="notation">in</span></a> <a class="idref" href="RecordSub.html#T<sub>2</sub>:170"><span class="id" title="variable">T<sub>2</sub></span></a>.<br/>
<div class="togglescript" id="proofcontrol14" onclick="toggleDisplay('proof14');toggleDisplay('proofcontrol14')"><span class="show"></span></div>
<div class="proofscript" id="proof14" onclick="toggleDisplay('proof14');toggleDisplay('proofcontrol14')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">S<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">T<sub>1</sub></span> <span class="id" title="var">T<sub>2</sub></span> <span class="id" title="var">Hty</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.typing_inversion_abs"><span class="id" title="lemma">typing_inversion_abs</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hty</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">Hty</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">S<sub>2</sub></span> [<span class="id" title="var">Hsub</span> <span class="id" title="var">Hty</span>]].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="RecordSub.html#RecordSub.sub_inversion_arrow"><span class="id" title="axiom">sub_inversion_arrow</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hsub</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">Hsub</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">U<sub>1</sub></span> [<span class="id" title="var">U<sub>2</sub></span> [<span class="id" title="var">Heq</span> [<span class="id" title="var">Hsub1</span> <span class="id" title="var">Hsub2</span>]]]].<br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Heq</span>; <span class="id" title="tactic">subst</span>... <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>