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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Records: Adding Records to STLC</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Records<span class="subtitle">Adding Records to STLC</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)</span><br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Stlc</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab798"></a><h1 class="section">Adding Records</h1>
<div class="paragraph"> </div>
We saw in chapter <span class="inlinecode"><span class="id" type="var">MoreStlc</span></span> how records can be treated as syntactic
sugar for nested uses of products. This is fine for simple
examples, but the encoding is informal (in reality, if we really
treated records this way, it would be carried out in the parser,
which we are eliding here), and anyway it is not very efficient.
So it is also interesting to see how records can be treated as
first-class citizens of the language.
<div class="paragraph"> </div>
Recall the informal definitions we gave before:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
Syntax:
<pre>
t ::= Terms:
| ...
| {i1=t<sub>1</sub>, ..., in=tn} record
| t.i projection
v ::= Values:
| ...
| {i1=v<sub>1</sub>, ..., in=vn} record value
T ::= Types:
| ...
| {i1:T<sub>1</sub>, ..., in:Tn} record type
</pre>
Reduction:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">ti <span style="font-family: arial;">⇒</span> ti' (ST_Rcd)</td>
<td class="infrulenamecol" rowspan="3">
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">{i1=v<sub>1</sub>, ..., im=vm, in=tn, ...} <span style="font-family: arial;">⇒</span> {i1=v<sub>1</sub>, ..., im=vm, in=tn', ...}</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Proj1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">t<sub>1</sub>.i <span style="font-family: arial;">⇒</span> t<sub>1</sub>'.i</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_ProjRcd)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">{..., i=vi, ...}.i <span style="font-family: arial;">⇒</span> vi</td>
<td></td>
</td>
</table></center> Typing:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t<sub>1</sub> : T<sub>1</sub> ... <span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> tn : Tn</td>
<td class="infrulenamecol" rowspan="3">
(T_Rcd)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> {i1=t<sub>1</sub>, ..., in=tn} : {i1:T<sub>1</sub>, ..., in:Tn}</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t : {..., i:Ti, ...}</td>
<td class="infrulenamecol" rowspan="3">
(T_Proj)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> t.i : Ti</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab799"></a><h1 class="section">Formalizing Records</h1>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">STLCExtendedRecords</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab800"></a><h3 class="section">Syntax and Operational Semantics</h3>
<div class="paragraph"> </div>
The most obvious way to formalize the syntax of record types would
be this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">FirstTry</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">alist</span> (<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) := <span class="id" type="var">list</span> (<span class="id" type="var">id</span> × <span class="id" type="var">X</span>).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ty</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">TBase</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TArrow</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TRcd</span> : (<span class="id" type="var">alist</span> <span class="id" type="var">ty</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span>.<br/>
<br/>
</div>
<div class="doc">
Unfortunately, we encounter here a limitation in Coq: this type
does not automatically give us the induction principle we expect
the induction hypothesis in the <span class="inlinecode"><span class="id" type="var">TRcd</span></span> case doesn't give us
any information about the <span class="inlinecode"><span class="id" type="var">ty</span></span> elements of the list, making it
useless for the proofs we want to do.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Check ty_ind. <br/>
====><br/>
ty_ind : <br/>
forall P : ty -> Prop,<br/>
(forall i : id, P (TBase i)) -><br/>
(forall t : ty, P t -> forall t0 : ty, P t0 -> P (TArrow t t0)) -><br/>
(forall a : alist ty, P (TRcd a)) -> <span class="comment">(* ??? *)</span><br/>
forall t : ty, P t<br/>
*)</span><br/>
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">FirstTry</span>.<br/>
<br/>
</div>
<div class="doc">
It is possible to get a better induction principle out of Coq, but
the details of how this is done are not very pretty, and it is not
as intuitive to use as the ones Coq generates automatically for
simple <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definitions.
<div class="paragraph"> </div>
Fortunately, there is a different way of formalizing records that
is, in some ways, even simpler and more natural: instead of using
the existing <span class="inlinecode"><span class="id" type="var">list</span></span> type, we can essentially include its
constructors ("nil" and "cons") in the syntax of types.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ty</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">TBase</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TArrow</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TRNil</span> : <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TRCons</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "T_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TBase" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TArrow"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TRNil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "TRCons" ].<br/>
<br/>
</div>
<div class="doc">
Similarly, at the level of terms, we have constructors <span class="inlinecode"><span class="id" type="var">trnil</span></span>
the empty record — and <span class="inlinecode"><span class="id" type="var">trcons</span></span>, which adds a single field to
the front of a list of fields.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">tm</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">tvar</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tapp</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tabs</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
<span class="comment">(* records *)</span><br/>
| <span class="id" type="var">tproj</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">trnil</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">trcons</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "t_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tvar" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tapp" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tabs"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "tproj" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "trnil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "trcons" ].<br/>
<br/>
</div>
<div class="doc">
Some variables, for examples...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">a</span> := (<span class="id" type="var">Id</span> 0).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">f</span> := (<span class="id" type="var">Id</span> 1).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">g</span> := (<span class="id" type="var">Id</span> 2).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">l</span> := (<span class="id" type="var">Id</span> 3).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">A</span> := (<span class="id" type="var">TBase</span> (<span class="id" type="var">Id</span> 4)).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">B</span> := (<span class="id" type="var">TBase</span> (<span class="id" type="var">Id</span> 5)).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">k</span> := (<span class="id" type="var">Id</span> 6).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">i1</span> := (<span class="id" type="var">Id</span> 7).<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">i2</span> := (<span class="id" type="var">Id</span> 8).<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode">{</span> <span class="inlinecode"><span class="id" type="var">i1</span>:<span class="id" type="var">A</span></span> <span class="inlinecode">}</span>
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Check (TRCons i1 A TRNil). *)</span><br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode">{</span> <span class="inlinecode"><span class="id" type="var">i1</span>:<span class="id" type="var">A</span><span style="font-family: arial;">→</span><span class="id" type="var">B</span>,</span> <span class="inlinecode"><span class="id" type="var">i2</span>:<span class="id" type="var">A</span></span> <span class="inlinecode">}</span>
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Check (TRCons i1 (TArrow A B) <br/>
(TRCons i2 A TRNil)). *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab801"></a><h3 class="section">Well-Formedness</h3>
<div class="paragraph"> </div>
Generalizing our abstract syntax for records (from lists to the
nil/cons presentation) introduces the possibility of writing
strange types like this
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">weird_type</span> := <span class="id" type="var">TRCons</span> <span class="id" type="var">X</span> <span class="id" type="var">A</span> <span class="id" type="var">B</span>.<br/>
<br/>
</div>
<div class="doc">
where the "tail" of a record type is not actually a record type!
<div class="paragraph"> </div>
We'll structure our typing judgement so that no ill-formed types
like <span class="inlinecode"><span class="id" type="var">weird_type</span></span> are assigned to terms. To support this, we
define <span class="inlinecode"><span class="id" type="var">record_ty</span></span> and <span class="inlinecode"><span class="id" type="var">record_tm</span></span>, which identify record types
and terms, and <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span> which rules out the ill-formed
types.
<div class="paragraph"> </div>
First, a type is a record type if it is built with just <span class="inlinecode"><span class="id" type="var">TRNil</span></span>
and <span class="inlinecode"><span class="id" type="var">TRCons</span></span> at the outermost level.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">record_ty</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">RTnil</span> : <br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">RTcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>,<br/>
<span class="id" type="var">record_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>).<br/>
<br/>
</div>
<div class="doc">
Similarly, a term is a record term if it is built with <span class="inlinecode"><span class="id" type="var">trnil</span></span>
and <span class="inlinecode"><span class="id" type="var">trcons</span></span>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">record_tm</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">rtnil</span> :<br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">rtcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">record_tm</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>).<br/>
<br/>
</div>
<div class="doc">
Note that <span class="inlinecode"><span class="id" type="var">record_ty</span></span> and <span class="inlinecode"><span class="id" type="var">record_tm</span></span> are not recursive — they
just check the outermost constructor. The <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span>
property, on the other hand, verifies that the whole type is well
formed in the sense that the tail of every record (the second
argument to <span class="inlinecode"><span class="id" type="var">TRCons</span></span>) is a record.
<div class="paragraph"> </div>
Of course, we should also be concerned about ill-formed terms, not
just types; but typechecking can rules those out without the help
of an extra <span class="inlinecode"><span class="id" type="var">well_formed_tm</span></span> definition because it already
examines the structure of terms. LATER : should they fill in part of this as an exercise? We
didn't give rules for it above
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">well_formed_ty</span> : <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">wfTBase</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span>,<br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TBase</span> <span class="id" type="var">i</span>)<br/>
| <span class="id" type="var">wfTArrow</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>)<br/>
| <span class="id" type="var">wfTRNil</span> :<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">wfTRCons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">record_ty</span> <span class="id" type="var">record_tm</span> <span class="id" type="var">well_formed_ty</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab802"></a><h3 class="section">Substitution</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="tactic">subst</span> (<span class="id" type="var">x</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">tm</span>) (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) : <span class="id" type="var">tm</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">t</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">tvar</span> <span class="id" type="var">y</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="keyword">then</span> <span class="id" type="var">s</span> <span class="id" type="keyword">else</span> <span class="id" type="var">t</span><br/>
| <span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T</span> <span class="id" type="var">t<sub>1</sub></span> ⇒ <span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T</span> (<span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="keyword">else</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>))<br/>
| <span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> ⇒ <span class="id" type="var">tapp</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">tproj</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">i</span> ⇒ <span class="id" type="var">tproj</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) <span class="id" type="var">i</span><br/>
| <span class="id" type="var">trnil</span> ⇒ <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">tr1</span> ⇒ <span class="id" type="var">trcons</span> <span class="id" type="var">i</span> (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t<sub>1</sub></span>) (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">tr1</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "'[' x ':=' s ']' t" := (<span class="id" type="tactic">subst</span> <span class="id" type="var">x</span> <span class="id" type="var">s</span> <span class="id" type="var">t</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 20).<br/>
<br/>
</div>
<div class="doc">
<a name="lab803"></a><h3 class="section">Reduction</h3>
<div class="paragraph"> </div>
Next we define the values of our language. A record is a value if
all of its fields are.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">value</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">v_abs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>,<br/>
<span class="id" type="var">value</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>)<br/>
| <span class="id" type="var">v_rnil</span> : <span class="id" type="var">value</span> <span class="id" type="var">trnil</span><br/>
| <span class="id" type="var">v_rcons</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">vr</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">vr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">vr</span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">value</span>.<br/>
<br/>
</div>
<div class="doc">
Utility functions for extracting one field from record type or
term:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">Tlookup</span> (<span class="id" type="var">i</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">Tr</span>:<span class="id" type="var">ty</span>) : <span class="id" type="var">option</span> <span class="id" type="var">ty</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">Tr</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">TRCons</span> <span class="id" type="var">i'</span> <span class="id" type="var">T</span> <span class="id" type="var">Tr'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">T</span> <span class="id" type="keyword">else</span> <span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">Tr'</span><br/>
| <span class="id" type="var">_</span> ⇒ <span class="id" type="var">None</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">tlookup</span> (<span class="id" type="var">i</span>:<span class="id" type="var">id</span>) (<span class="id" type="var">tr</span>:<span class="id" type="var">tm</span>) : <span class="id" type="var">option</span> <span class="id" type="var">tm</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">tr</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">trcons</span> <span class="id" type="var">i'</span> <span class="id" type="var">t</span> <span class="id" type="var">tr'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">t</span> <span class="id" type="keyword">else</span> <span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">tr'</span><br/>
| <span class="id" type="var">_</span> ⇒ <span class="id" type="var">None</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="var">step</span></span> function uses the term-level lookup function (for the
projection rule), while the type-level lookup is needed for
<span class="inlinecode"><span class="id" type="var">has_type</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">step</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ST_AppAbs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span> <span class="id" type="var">v<sub>2</sub></span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>2</sub></span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>) <span class="id" type="var">v<sub>2</sub></span>) <span style="font-family: arial;">⇒</span> ([<span class="id" type="var">x</span>:=<span class="id" type="var">v<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></span>)<br/>
| <span class="id" type="var">ST_App1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>1</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">ST_App2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>2</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tapp</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>)<br/>
| <span class="id" type="var">ST_Proj1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">i</span>,<br/>
<span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>1</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tproj</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">i</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tproj</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">i</span>)<br/>
| <span class="id" type="var">ST_ProjRcd</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">tr</span> <span class="id" type="var">i</span> <span class="id" type="var">vi</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">tr</span> = <span class="id" type="var">Some</span> <span class="id" type="var">vi</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tproj</span> <span class="id" type="var">tr</span> <span class="id" type="var">i</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">vi</span><br/>
| <span class="id" type="var">ST_Rcd_Head</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">tr2</span>,<br/>
<span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t<sub>1</sub>'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">tr2</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">tr2</span>)<br/>
| <span class="id" type="var">ST_Rcd_Tail</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2</span> <span class="id" type="var">tr2'</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tr2</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tr2'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">tr2'</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒</span>' t<sub>2</sub>" := (<span class="id" type="var">step</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>).<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "step_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_AppAbs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_App2"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Proj1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_ProjRcd"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Rcd_Head" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Rcd_Tail" ].<br/>
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">multistep</span> := (<span class="id" type="var">multi</span> <span class="id" type="var">step</span>).<br/>
<span class="id" type="keyword">Notation</span> "t<sub>1</sub> '<span style="font-family: arial;">⇒*</span>' t<sub>2</sub>" := (<span class="id" type="var">multistep</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">step</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab804"></a><h3 class="section">Typing</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">context</span> := <span class="id" type="var">partial_map</span> <span class="id" type="var">ty</span>.<br/>
<br/>
</div>
<div class="doc">
Next we define the typing rules. These are nearly direct
transcriptions of the inference rules shown above. The only major
difference is the use of <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span>. In the informal
presentation we used a grammar that only allowed well formed
record types, so we didn't have to add a separate check.
<div class="paragraph"> </div>
We'd like to set things up so that that whenever <span class="inlinecode"><span class="id" type="var">has_type</span></span> <span class="inlinecode"><span style="font-family: serif; font-size:85%;">Γ</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span>
<span class="inlinecode"><span class="id" type="var">T</span></span> holds, we also have <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span>. That is, <span class="inlinecode"><span class="id" type="var">has_type</span></span>
never assigns ill-formed types to terms. In fact, we prove this
theorem below.
<div class="paragraph"> </div>
However, we don't want to clutter the definition of <span class="inlinecode"><span class="id" type="var">has_type</span></span>
with unnecessary uses of <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span>. Instead, we place
<span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span> checks only where needed - where an inductive
call to <span class="inlinecode"><span class="id" type="var">has_type</span></span> won't already be checking the well-formedness
of a type.
<div class="paragraph"> </div>
For example, we check <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> in the <span class="inlinecode"><span class="id" type="var">T_Var</span></span> case,
because there is no inductive <span class="inlinecode"><span class="id" type="var">has_type</span></span> call that would
enforce this. Similarly, in the <span class="inlinecode"><span class="id" type="var">T_Abs</span></span> case, we require a
proof of <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> because the inductive call to
<span class="inlinecode"><span class="id" type="var">has_type</span></span> only guarantees that <span class="inlinecode"><span class="id" type="var">T<sub>12</sub></span></span> is well-formed.
<div class="paragraph"> </div>
In the rules you must write, the only necessary <span class="inlinecode"><span class="id" type="var">well_formed_ty</span></span>
check comes in the <span class="inlinecode"><span class="id" type="var">tnil</span></span> case.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "Gamma '<span style="font-family: arial;">⊢</span>' t '∈' T" (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">has_type</span> : <span class="id" type="var">context</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ty</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">T_Var</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> = <span class="id" type="var">Some</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>) ∈ <span class="id" type="var">T</span><br/>
| <span class="id" type="var">T_Abs</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span> <span class="id" type="var">t<sub>12</sub></span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T<sub>11</sub></span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">extend</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span>) <span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>12</sub></span> ∈ <span class="id" type="var">T<sub>12</sub></span> <span style="font-family: arial;">→</span> <br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>) ∈ (<span class="id" type="var">TArrow</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">T<sub>12</sub></span>)<br/>
| <span class="id" type="var">T_App</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>1</sub></span> ∈ (<span class="id" type="var">TArrow</span> <span class="id" type="var">T<sub>1</sub></span> <span class="id" type="var">T<sub>2</sub></span>) <span style="font-family: arial;">→</span> <br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>2</sub></span> ∈ <span class="id" type="var">T<sub>1</sub></span> <span style="font-family: arial;">→</span> <br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>) ∈ <span class="id" type="var">T<sub>2</sub></span><br/>
<span class="comment">(* records: *)</span><br/>
| <span class="id" type="var">T_Proj</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">Ti</span> <span class="id" type="var">Tr</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">Tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">Tr</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (<span class="id" type="var">tproj</span> <span class="id" type="var">t</span> <span class="id" type="var">i</span>) ∈ <span class="id" type="var">Ti</span><br/>
| <span class="id" type="var">T_RNil</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">trnil</span> ∈ <span class="id" type="var">TRNil</span><br/>
| <span class="id" type="var">T_RCons</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">tr</span> <span class="id" type="var">Tr</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">tr</span> ∈ <span class="id" type="var">Tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_ty</span> <span class="id" type="var">Tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">tr</span>) ∈ (<span class="id" type="var">TRCons</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> <span class="id" type="var">Tr</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "Gamma '<span style="font-family: arial;">⊢</span>' t '∈' T" := (<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">has_type</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "has_type_cases" <span class="id" type="var">tactic</span>(<span class="id" type="var">first</span>) <span class="id" type="var">ident</span>(<span class="id" type="var">c</span>) :=<br/>
<span class="id" type="var">first</span>;<br/>
[ <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Var" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Abs" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_App"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_Proj" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_RNil" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "T_RCons" ].<br/>
<br/>
</div>
<div class="doc">
<a name="lab805"></a><h2 class="section">Examples</h2>
<div class="paragraph"> </div>
<a name="lab806"></a><h4 class="section">Exercise: 2 stars (examples)</h4>
Finish the proofs.
<div class="paragraph"> </div>
Feel free to use Coq's automation features in this proof.
However, if you are not confident about how the type system works,
you may want to carry out the proof first using the basic
features (<span class="inlinecode"><span class="id" type="tactic">apply</span></span> instead of <span class="inlinecode"><span class="id" type="tactic">eapply</span></span>, in particular) and then
perhaps compress it using automation.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">typing_example_2</span> : <br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">a</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i1</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">A</span> <span class="id" type="var">A</span>)<br/>
(<span class="id" type="var">TRCons</span> <span class="id" type="var">i2</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">B</span> <span class="id" type="var">B</span>)<br/>
<span class="id" type="var">TRNil</span>))<br/>
(<span class="id" type="var">tproj</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>) <span class="id" type="var">i2</span>))<br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i1</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">a</span> <span class="id" type="var">A</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>)) <br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i2</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">a</span> <span class="id" type="var">B</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>))<br/>
<span class="id" type="var">trnil</span>))) ∈<br/>
(<span class="id" type="var">TArrow</span> <span class="id" type="var">B</span> <span class="id" type="var">B</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
Before starting to prove this fact (or the one above!), make sure
you understand what it is saying.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_nonexample</span> : <br/>
¬ <span style="font-family: arial;">∃</span><span class="id" type="var">T</span>,<br/>
(<span class="id" type="var">extend</span> <span class="id" type="var">empty</span> <span class="id" type="var">a</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i2</span> (<span class="id" type="var">TArrow</span> <span class="id" type="var">A</span> <span class="id" type="var">A</span>) <br/>
<span class="id" type="var">TRNil</span>)) <span style="font-family: arial;">⊢</span><br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i1</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">a</span> <span class="id" type="var">B</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>)) (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>)) ∈<br/>
<span class="id" type="var">T</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_nonexample_2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">y</span>,<br/>
¬ <span style="font-family: arial;">∃</span><span class="id" type="var">T</span>,<br/>
(<span class="id" type="var">extend</span> <span class="id" type="var">empty</span> <span class="id" type="var">y</span> <span class="id" type="var">A</span>) <span style="font-family: arial;">⊢</span><br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">a</span> (<span class="id" type="var">TRCons</span> <span class="id" type="var">i1</span> <span class="id" type="var">A</span> <span class="id" type="var">TRNil</span>)<br/>
(<span class="id" type="var">tproj</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">a</span>) <span class="id" type="var">i1</span>))<br/>
(<span class="id" type="var">trcons</span> <span class="id" type="var">i1</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">y</span>) (<span class="id" type="var">trcons</span> <span class="id" type="var">i2</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">y</span>) <span class="id" type="var">trnil</span>))) ∈<br/>
<span class="id" type="var">T</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab807"></a><h2 class="section">Properties of Typing</h2>
<div class="paragraph"> </div>
The proofs of progress and preservation for this system are
essentially the same as for the pure simply typed lambda-calculus,
but we need to add some technical lemmas involving records.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab808"></a><h3 class="section">Well-Formedness</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">wf_rcd_lookup</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">i</span> <span class="id" type="var">T</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">well_formed_ty</span> <span class="id" type="var">Ti</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span>.<br/>
<span class="id" type="var">T_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">T</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="var">Case</span> "TRCons".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">Tlookup</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i0</span>)...<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>. <span class="id" type="tactic">subst</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">step_preserves_record_tm</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">tr</span> <span class="id" type="var">tr'</span>,<br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">tr</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">tr'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">record_tm</span> <span class="id" type="var">tr'</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">tr</span> <span class="id" type="var">tr'</span> <span class="id" type="var">Hrt</span> <span class="id" type="var">Hstp</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hrt</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hstp</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">has_type__wf</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <span class="id" type="var">well_formed_ty</span> <span class="id" type="var">T</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">Htyp</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Htyp</span>) <span class="id" type="var">Case</span>...<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">IHHtyp1</span>...<br/>
<span class="id" type="var">Case</span> "T_Proj".<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">wf_rcd_lookup</span>...<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab809"></a><h3 class="section">Field Lookup</h3>
<div class="paragraph"> </div>
Lemma: If <span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span class="id" type="var">Tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> returns <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">Ti</span></span>,
then <span class="inlinecode"><span class="id" type="var">tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">v</span></span> returns <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span> for some term <span class="inlinecode"><span class="id" type="var">ti</span></span> such
that <span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">Ti</span></span>.
<div class="paragraph"> </div>
Proof: By induction on the typing derivation <span class="inlinecode"><span class="id" type="var">Htyp</span></span>. Since
<span class="inlinecode"><span class="id" type="var">Tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">Ti</span></span>, <span class="inlinecode"><span class="id" type="var">T</span></span> must be a record type, this and
the fact that <span class="inlinecode"><span class="id" type="var">v</span></span> is a value eliminate most cases by inspection,
leaving only the <span class="inlinecode"><span class="id" type="var">T_RCons</span></span> case.
<div class="paragraph"> </div>
If the last step in the typing derivation is by <span class="inlinecode"><span class="id" type="var">T_RCons</span></span>, then
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i0</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> and <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">TRCons</span></span> <span class="inlinecode"><span class="id" type="var">i0</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode"><span class="id" type="var">Tr</span></span> for some <span class="inlinecode"><span class="id" type="var">i0</span></span>,
<span class="inlinecode"><span class="id" type="var">t</span></span>, <span class="inlinecode"><span class="id" type="var">tr</span></span>, <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span class="id" type="var">Tr</span></span>.
<div class="paragraph"> </div>
This leaves two possiblities to consider - either <span class="inlinecode"><span class="id" type="var">i0</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">i</span></span> or
not.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">i0</span></span>, then since <span class="inlinecode"><span class="id" type="var">Tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode">(<span class="id" type="var">TRCons</span></span> <span class="inlinecode"><span class="id" type="var">i0</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode"><span class="id" type="var">Tr</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Some</span></span>
<span class="inlinecode"><span class="id" type="var">Ti</span></span> we have <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Ti</span></span>. It follows that <span class="inlinecode"><span class="id" type="var">t</span></span> itself satisfies
the theorem.
<div class="paragraph"> </div>
</li>
<li> On the other hand, suppose <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode">≠</span> <span class="inlinecode"><span class="id" type="var">i0</span></span>. Then
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">Tr</span>
<div class="paragraph"> </div>
</div>
and
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> = <span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">tr</span>,
<div class="paragraph"> </div>
</div>
so the result follows from the induction hypothesis. <font size=-2>☐</font>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">lookup_field_in_value</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">v</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">v</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">T</span> = <span class="id" type="var">Some</span> <span class="id" type="var">Ti</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">ti</span>, <span class="id" type="var">tlookup</span> <span class="id" type="var">i</span> <span class="id" type="var">v</span> = <span class="id" type="var">Some</span> <span class="id" type="var">ti</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">ti</span> ∈ <span class="id" type="var">Ti</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">v</span> <span class="id" type="var">T</span> <span class="id" type="var">i</span> <span class="id" type="var">Ti</span> <span class="id" type="var">Hval</span> <span class="id" type="var">Htyp</span> <span class="id" type="var">Hget</span>.<br/>
<span class="id" type="var">remember</span> (@<span class="id" type="var">empty</span> <span class="id" type="var">ty</span>) <span class="id" type="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Htyp</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>...<br/>
<span class="id" type="var">Case</span> "T_RCons".<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hget</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">i</span> <span class="id" type="var">i0</span>).<br/>
<span class="id" type="var">SCase</span> "i is first".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hget</span>. <span class="id" type="tactic">subst</span>.<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">t</span>...<br/>
<span class="id" type="var">SCase</span> "get tail".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHtyp2</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">vi</span> [<span class="id" type="var">Hgeti</span> <span class="id" type="var">Htypi</span>]]...<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hval</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab810"></a><h3 class="section">Progress</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="tactic">progress</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T</span>, <br/>
<span class="id" type="var">empty</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span>, <span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* Theorem: Suppose empty |- t : T. Then either<br/>
1. t is a value, or<br/>
2. t ==> t' for some t'.<br/>
Proof: By induction on the given typing derivation. *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">Ht</span>.<br/>
<span class="id" type="var">remember</span> (@<span class="id" type="var">empty</span> <span class="id" type="var">ty</span>) <span class="id" type="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">HeqGamma</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Ht</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">HeqGamma</span>; <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">Case</span> "T_Var".<br/>
<span class="comment">(* The final rule in the given typing derivation cannot be <span class="inlinecode"><span class="id" type="var">T_Var</span></span>,<br/>
since it can never be the case that <span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T</span></span> (since the<br/>
context is empty). *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Case</span> "T_Abs".<br/>
<span class="comment">(* If the <span class="inlinecode"><span class="id" type="var">T_Abs</span></span> rule was the last used, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">tabs</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>,<br/>
which is a value. *)</span><br/>
<span class="id" type="var">left</span>...<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="comment">(* If the last rule applied was T_App, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>, and we know <br/>
from the form of the rule that<br/>
<span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>2</sub></span></span><br/>
<span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T<sub>1</sub></span></span><br/>
By the induction hypothesis, each of t<sub>1</sub> and t<sub>2</sub> either is a value <br/>
or can take a step. *)</span><br/>
<span class="id" type="var">right</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">SCase</span> "t<sub>1</sub> is a value".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">SSCase</span> "t<sub>2</sub> is a value".<br/>
<span class="comment">(* If both <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> are values, then we know that <br/>
<span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">tabs</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span></span>, since abstractions are the only values<br/>
that can have an arrow type. But <br/>
<span class="inlinecode">(<span class="id" type="var">tabs</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T<sub>11</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>12</sub></span>)</span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode">[<span class="id" type="var">x</span>:=<span class="id" type="var">t<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></span></span> by <span class="inlinecode"><span class="id" type="var">ST_AppAbs</span></span>. *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> (<span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>).<br/>
<span style="font-family: arial;">∃</span>([<span class="id" type="var">x</span>:=<span class="id" type="var">t<sub>2</sub></span>]<span class="id" type="var">t<sub>12</sub></span>)...<br/>
<span class="id" type="var">SSCase</span> "t<sub>2</sub> steps".<br/>
<span class="comment">(* If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span>, then <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span> <br/>
by <span class="inlinecode"><span class="id" type="var">ST_App2</span></span>. *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t<sub>2</sub>'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>)...<br/>
<span class="id" type="var">SCase</span> "t<sub>1</sub> steps".<br/>
<span class="comment">(* Finally, If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>, then <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" type="var">ST_App1</span></span>. *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span>)...<br/>
<span class="id" type="var">Case</span> "T_Proj".<br/>
<span class="comment">(* If the last rule in the given derivation is <span class="inlinecode"><span class="id" type="var">T_Proj</span></span>, then <br/>
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">tproj</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> and<br/>
<span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode">(<span class="id" type="var">TRcd</span></span> <span class="inlinecode"><span class="id" type="var">Tr</span>)</span><br/>
By the IH, <span class="inlinecode"><span class="id" type="var">t</span></span> either is a value or takes a step. *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt</span>...<br/>
<span class="id" type="var">SCase</span> "rcd is value".<br/>
<span class="comment">(* If <span class="inlinecode"><span class="id" type="var">t</span></span> is a value, then we may use lemma<br/>
<span class="inlinecode"><span class="id" type="var">lookup_field_in_value</span></span> to show <span class="inlinecode"><span class="id" type="var">tlookup</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">Some</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span> for<br/>
some <span class="inlinecode"><span class="id" type="var">ti</span></span> which gives us <span class="inlinecode"><span class="id" type="var">tproj</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">ti</span></span> by <span class="inlinecode"><span class="id" type="var">ST_ProjRcd</span></span><br/>
*)</span><br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">lookup_field_in_value</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H0</span> <span class="id" type="var">Ht</span> <span class="id" type="var">H</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">ti</span> [<span class="id" type="var">Hlkup</span> <span class="id" type="var">_</span>]].<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">ti</span>...<br/>
<span class="id" type="var">SCase</span> "rcd_steps".<br/>
<span class="comment">(* On the other hand, if <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span>, then <span class="inlinecode"><span class="id" type="var">tproj</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">tproj</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span><br/>
by <span class="inlinecode"><span class="id" type="var">ST_Proj1</span></span>. *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t'</span> <span class="id" type="var">Hstp</span>]. <span style="font-family: arial;">∃</span>(<span class="id" type="var">tproj</span> <span class="id" type="var">t'</span> <span class="id" type="var">i</span>)...<br/>
<span class="id" type="var">Case</span> "T_RNil".<br/>
<span class="comment">(* If the last rule in the given derivation is <span class="inlinecode"><span class="id" type="var">T_RNil</span></span>, then <br/>
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">trnil</span></span>, which is a value. *)</span><br/>
<span class="id" type="var">left</span>...<br/>
<span class="id" type="var">Case</span> "T_RCons".<br/>
<span class="comment">(* If the last rule is <span class="inlinecode"><span class="id" type="var">T_RCons</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> and<br/>
<span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">T</span></span><br/>
<span class="inlinecode"><span class="id" type="var">empty</span></span> <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">Tr</span></span><br/>
By the IH, each of <span class="inlinecode"><span class="id" type="var">t</span></span> and <span class="inlinecode"><span class="id" type="var">tr</span></span> either is a value or can take<br/>
a step. *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span>...<br/>
<span class="id" type="var">SCase</span> "head is a value".<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">SSCase</span> "tail is a value".<br/>
<span class="comment">(* If <span class="inlinecode"><span class="id" type="var">t</span></span> and <span class="inlinecode"><span class="id" type="var">tr</span></span> are both values, then <span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span><br/>
is a value as well. *)</span><br/>
<span class="id" type="var">left</span>...<br/>
<span class="id" type="var">SSCase</span> "tail steps".<br/>
<span class="comment">(* If <span class="inlinecode"><span class="id" type="var">t</span></span> is a value and <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">tr'</span></span>, then <br/>
<span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr'</span></span> by <br/>
<span class="inlinecode"><span class="id" type="var">ST_Rcd_Tail</span></span>. *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">H2</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">tr'</span> <span class="id" type="var">Hstp</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t</span> <span class="id" type="var">tr'</span>)...<br/>
<span class="id" type="var">SCase</span> "head steps".<br/>
<span class="comment">(* If <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span>, then <br/>
<span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">trcons</span></span> <span class="inlinecode"><span class="id" type="var">i</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode"><span class="id" type="var">tr</span></span> <br/>
by <span class="inlinecode"><span class="id" type="var">ST_Rcd_Head</span></span>. *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">H1</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t'</span> <span class="id" type="var">Hstp</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">t'</span> <span class="id" type="var">tr</span>)... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab811"></a><h3 class="section">Context Invariance</h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">appears_free_in</span> : <span class="id" type="var">id</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">afi_var</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>)<br/>
| <span class="id" type="var">afi_app1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">afi_app2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tapp</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span>)<br/>
| <span class="id" type="var">afi_abs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>,<br/>
<span class="id" type="var">y</span> ≠ <span class="id" type="var">x</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t<sub>12</sub></span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">T<sub>11</sub></span> <span class="id" type="var">t<sub>12</sub></span>)<br/>
| <span class="id" type="var">afi_proj</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">i</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">tproj</span> <span class="id" type="var">t</span> <span class="id" type="var">i</span>)<br/>
| <span class="id" type="var">afi_rhead</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">i</span> <span class="id" type="var">ti</span> <span class="id" type="var">tr</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">ti</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">ti</span> <span class="id" type="var">tr</span>)<br/>
| <span class="id" type="var">afi_rtail</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">i</span> <span class="id" type="var">ti</span> <span class="id" type="var">tr</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">tr</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">trcons</span> <span class="id" type="var">i</span> <span class="id" type="var">ti</span> <span class="id" type="var">tr</span>).<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">appears_free_in</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">context_invariance</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: serif; font-size:85%;">Γ'</span> <span class="id" type="var">t</span> <span class="id" type="var">S</span>,<br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">S</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> = <span style="font-family: serif; font-size:85%;">Γ'</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ'</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">S</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span style="font-family: serif; font-size:85%;">Γ'</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>) <span class="id" type="var">Case</span>; <br/>
<span class="id" type="tactic">intros</span> <span style="font-family: serif; font-size:85%;">Γ'</span> <span class="id" type="var">Heqv</span>...<br/>
<span class="id" type="var">Case</span> "T_Var".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Var</span>... <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">Heqv</span>...<br/>
<span class="id" type="var">Case</span> "T_Abs".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Abs</span>... <span class="id" type="tactic">apply</span> <span class="id" type="var">IHhas_type</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">y</span> <span class="id" type="var">Hafi</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">extend</span>. <span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_id_dec</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>)...<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_App</span> <span class="id" type="keyword">with</span> <span class="id" type="var">T<sub>1</sub></span>...<br/>
<span class="id" type="var">Case</span> "T_RCons".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_RCons</span>... <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">free_in_context</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span style="font-family: serif; font-size:85%;">Γ</span>,<br/>
<span class="id" type="var">appears_free_in</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: serif; font-size:85%;">Γ</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">T'</span>, <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> = <span class="id" type="var">Some</span> <span class="id" type="var">T'</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">x</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">Hafi</span> <span class="id" type="var">Htyp</span>.<br/>
<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Htyp</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hafi</span>; <span class="id" type="tactic">subst</span>...<br/>
<span class="id" type="var">Case</span> "T_Abs".<br/>