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Types.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Types: Type Systems</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">Types<span class="subtitle">Type Systems</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Smallstep</span>.<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">multi</span>.<br/>
<br/>
</div>
<div class="doc">
Our next major topic is <i>type systems</i> — static program
analyses that classify expressions according to the "shapes" of
their results. We'll begin with a typed version of a very simple
language with just booleans and numbers, to introduce the basic
ideas of types, typing rules, and the fundamental theorems about
type systems: <i>type preservation</i> and <i>progress</i>. Then we'll move
on to the <i>simply typed lambda-calculus</i>, which lives at the core
of every modern functional programming language (including
Coq).
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab626"></a><h1 class="section">Typed Arithmetic Expressions</h1>
<div class="paragraph"> </div>
To motivate the discussion of type systems, let's begin as
usual with an extremely simple toy language. We want it to have
the potential for programs "going wrong" because of runtime type
errors, so we need something a tiny bit more complex than the
language of constants and addition that we used in chapter
<span class="inlinecode"><span class="id" type="var">Smallstep</span></span>: a single kind of data (just numbers) is too simple,
but just two kinds (numbers and booleans) already gives us enough
material to tell an interesting story.
<div class="paragraph"> </div>
The language definition is completely routine.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab627"></a><h2 class="section">Syntax</h2>
<div class="paragraph"> </div>
Informally:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">t</span> ::= <span class="id" type="var">true</span><br/>
| <span class="id" type="var">false</span><br/>
| <span class="id" type="keyword">if</span> <span class="id" type="var">t</span> <span class="id" type="keyword">then</span> <span class="id" type="var">t</span> <span class="id" type="keyword">else</span> <span class="id" type="var">t</span><br/>
| 0<br/>
| <span class="id" type="var">succ</span> <span class="id" type="var">t</span><br/>
| <span class="id" type="var">pred</span> <span class="id" type="var">t</span><br/>
| <span class="id" type="var">iszero</span> <span class="id" type="var">t</span>
<div class="paragraph"> </div>
</div>
Formally:
</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">ttrue</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tfalse</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tif</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> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tzero</span> : <span class="id" type="var">tm</span><br/>
| <span class="id" type="var">tsucc</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">tpred</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">tiszero</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span>.<br/>
<br/>
</div>
<div class="doc">
<i>Values</i> are <span class="inlinecode"><span class="id" type="var">true</span></span>, <span class="inlinecode"><span class="id" type="var">false</span></span>, and numeric values...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">bvalue</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">bv_true</span> : <span class="id" type="var">bvalue</span> <span class="id" type="var">ttrue</span><br/>
| <span class="id" type="var">bv_false</span> : <span class="id" type="var">bvalue</span> <span class="id" type="var">tfalse</span>.<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">nvalue</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">nv_zero</span> : <span class="id" type="var">nvalue</span> <span class="id" type="var">tzero</span><br/>
| <span class="id" type="var">nv_succ</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>, <span class="id" type="var">nvalue</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nvalue</span> (<span class="id" type="var">tsucc</span> <span class="id" type="var">t</span>).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">value</span> (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) := <span class="id" type="var">bvalue</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">nvalue</span> <span class="id" type="var">t</span>.<br/>
<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">bvalue</span> <span class="id" type="var">nvalue</span>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">value</span>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">extend</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab628"></a><h2 class="section">Operational Semantics</h2>
<div class="paragraph"> </div>
Informally:
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IfTrue)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">if true then t<sub>1</sub> else t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>1</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IfFalse)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">if false then t<sub>1</sub> else t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub></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_If)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">if t<sub>1</sub> then t<sub>2</sub> else t<sub>3</sub> <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">if t<sub>1</sub>' then t<sub>2</sub> else t<sub>3</sub></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_Succ)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">succ t<sub>1</sub> <span style="font-family: arial;">⇒</span> succ t<sub>1</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_PredZero)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">pred 0 <span style="font-family: arial;">⇒</span> 0</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">numeric value v<sub>1</sub></td>
<td class="infrulenamecol" rowspan="3">
(ST_PredSucc)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">pred (succ v<sub>1</sub>) <span style="font-family: arial;">⇒</span> v<sub>1</sub></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_Pred)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">pred t<sub>1</sub> <span style="font-family: arial;">⇒</span> pred t<sub>1</sub>'</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_IszeroZero)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">iszero 0 <span style="font-family: arial;">⇒</span> true</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">numeric value v<sub>1</sub></td>
<td class="infrulenamecol" rowspan="3">
(ST_IszeroSucc)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">iszero (succ v<sub>1</sub>) <span style="font-family: arial;">⇒</span> false</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_Iszero)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">iszero t<sub>1</sub> <span style="font-family: arial;">⇒</span> iszero t<sub>1</sub>'</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
Formally:
</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_IfTrue</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">tif</span> <span class="id" type="var">ttrue</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">t<sub>1</sub></span><br/>
| <span class="id" type="var">ST_IfFalse</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">tif</span> <span class="id" type="var">tfalse</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">t<sub>2</sub></span><br/>
| <span class="id" type="var">ST_If</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> <span class="id" type="var">t<sub>3</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">tif</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>)<br/>
| <span class="id" type="var">ST_Succ</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>,<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">tsucc</span> <span class="id" type="var">t<sub>1</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tsucc</span> <span class="id" type="var">t<sub>1</sub>'</span>)<br/>
| <span class="id" type="var">ST_PredZero</span> :<br/>
(<span class="id" type="var">tpred</span> <span class="id" type="var">tzero</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">tzero</span><br/>
| <span class="id" type="var">ST_PredSucc</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span>,<br/>
<span class="id" type="var">nvalue</span> <span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tpred</span> (<span class="id" type="var">tsucc</span> <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><br/>
| <span class="id" type="var">ST_Pred</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>,<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">tpred</span> <span class="id" type="var">t<sub>1</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tpred</span> <span class="id" type="var">t<sub>1</sub>'</span>)<br/>
| <span class="id" type="var">ST_IszeroZero</span> :<br/>
(<span class="id" type="var">tiszero</span> <span class="id" type="var">tzero</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">ttrue</span><br/>
| <span class="id" type="var">ST_IszeroSucc</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span>,<br/>
<span class="id" type="var">nvalue</span> <span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tiszero</span> (<span class="id" type="var">tsucc</span> <span class="id" type="var">t<sub>1</sub></span>)) <span style="font-family: arial;">⇒</span> <span class="id" type="var">tfalse</span><br/>
| <span class="id" type="var">ST_Iszero</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>,<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">tiszero</span> <span class="id" type="var">t<sub>1</sub></span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tiszero</span> <span class="id" type="var">t<sub>1</sub>'</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/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">step</span>.<br/>
</div>
</div>
<div class="doc">
Notice that the <span class="inlinecode"><span class="id" type="var">step</span></span> relation doesn't care about whether
expressions make global sense — it just checks that the operation
in the <i>next</i> reduction step is being applied to the right kinds
of operands.
<div class="paragraph"> </div>
For example, the term <span class="inlinecode"><span class="id" type="var">succ</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span> (i.e., <span class="inlinecode"><span class="id" type="var">tsucc</span></span> <span class="inlinecode"><span class="id" type="var">ttrue</span></span> in the
formal syntax) cannot take a step, but the almost as obviously
nonsensical term
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">succ</span> (<span class="id" type="keyword">if</span> <span class="id" type="var">true</span> <span class="id" type="keyword">then</span> <span class="id" type="var">true</span> <span class="id" type="keyword">else</span> <span class="id" type="var">true</span>)
<div class="paragraph"> </div>
</div>
can take a step (once, before becoming stuck).
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab629"></a><h2 class="section">Normal Forms and Values</h2>
<div class="paragraph"> </div>
The first interesting thing about the <span class="inlinecode"><span class="id" type="var">step</span></span> relation in this
language is that the strong progress theorem from the Smallstep
chapter fails! That is, there are terms that are normal
forms (they can't take a step) but not values (because we have not
included them in our definition of possible "results of
evaluation"). Such terms are <i>stuck</i>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> <span class="id" type="var">step_normal_form</span> := (<span class="id" type="var">normal_form</span> <span class="id" type="var">step</span>).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">stuck</span> (<span class="id" type="var">t</span>:<span class="id" type="var">tm</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">step_normal_form</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∧</span> ¬ <span class="id" type="var">value</span> <span class="id" type="var">t</span>.<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">stuck</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab630"></a><h4 class="section">Exercise: 2 stars (some_term_is_stuck)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">some_term_is_stuck</span> :<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">t</span>, <span class="id" type="var">stuck</span> <span class="id" type="var">t</span>.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
However, although values and normal forms are not the same in this
language, the former set is included in the latter. This is
important because it shows we did not accidentally define things
so that some value could still take a step.
<div class="paragraph"> </div>
<a name="lab631"></a><h4 class="section">Exercise: 3 stars, advanced (value_is_nf)</h4>
Hint: You will reach a point in this proof where you need to
use an induction to reason about a term that is known to be a
numeric value. This induction can be performed either over the
term itself or over the evidence that it is a numeric value. The
proof goes through in either case, but you will find that one way
is quite a bit shorter than the other. For the sake of the
exercise, try to complete the proof both ways.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">value_is_nf</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="var">step_normal_form</span> <span class="id" type="var">t</span>.<br/>
<div class="togglescript" id="proofcontrol4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')"><span class="show"></span></div>
<div class="proofscript" id="proof4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab632"></a><h4 class="section">Exercise: 3 stars, optional (step_deterministic)</h4>
Using <span class="inlinecode"><span class="id" type="var">value_is_nf</span></span>, we can show that the <span class="inlinecode"><span class="id" type="var">step</span></span> relation is
also deterministic...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">step_deterministic</span>:<br/>
<span class="id" type="var">deterministic</span> <span class="id" type="var">step</span>.<br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab633"></a><h2 class="section">Typing</h2>
<div class="paragraph"> </div>
The next critical observation about this language is that,
although there are stuck terms, they are all "nonsensical", mixing
booleans and numbers in a way that we don't even <i>want</i> to have a
meaning. We can easily exclude such ill-typed terms by defining a
<i>typing relation</i> that relates terms to the types (either numeric
or boolean) of their final results.
</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">TBool</span> : <span class="id" type="var">ty</span><br/>
| <span class="id" type="var">TNat</span> : <span class="id" type="var">ty</span>.<br/>
<br/>
</div>
<div class="doc">
In informal notation, the typing relation is often written
<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>, pronounced "<span class="inlinecode"><span class="id" type="var">t</span></span> has type <span class="inlinecode"><span class="id" type="var">T</span></span>." The <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> symbol is
called a "turnstile". (Below, we're going to see richer typing
relations where an additional "context" argument is written to the
left of the turnstile. Here, the context is always empty.) <center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(T_True)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> true ∈ Bool</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(T_False)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> false ∈ Bool</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> t<sub>1</sub> ∈ Bool <span style="font-family: arial;">⊢</span> t<sub>2</sub> ∈ T <span style="font-family: arial;">⊢</span> t<sub>3</sub> ∈ T</td>
<td class="infrulenamecol" rowspan="3">
(T_If)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> if t<sub>1</sub> then t<sub>2</sub> else t<sub>3</sub> ∈ T</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(T_Zero)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> 0 ∈ Nat</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> t<sub>1</sub> ∈ Nat</td>
<td class="infrulenamecol" rowspan="3">
(T_Succ)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> succ t<sub>1</sub> ∈ Nat</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> t<sub>1</sub> ∈ Nat</td>
<td class="infrulenamecol" rowspan="3">
(T_Pred)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> pred t<sub>1</sub> ∈ Nat</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> t<sub>1</sub> ∈ Nat</td>
<td class="infrulenamecol" rowspan="3">
(T_IsZero)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="font-family: arial;">⊢</span> iszero t<sub>1</sub> ∈ Bool</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "'<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">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_True</span> : <br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">ttrue</span> ∈ <span class="id" type="var">TBool</span><br/>
| <span class="id" type="var">T_False</span> : <br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tfalse</span> ∈ <span class="id" type="var">TBool</span><br/>
| <span class="id" type="var">T_If</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 class="id" type="var">t<sub>3</sub></span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TBool</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>2</sub></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<sub>3</sub></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">tif</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span> ∈ <span class="id" type="var">T</span><br/>
| <span class="id" type="var">T_Zero</span> : <br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tzero</span> ∈ <span class="id" type="var">TNat</span><br/>
| <span class="id" type="var">T_Succ</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TNat</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tsucc</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TNat</span><br/>
| <span class="id" type="var">T_Pred</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TNat</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tpred</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TNat</span><br/>
| <span class="id" type="var">T_Iszero</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TNat</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tiszero</span> <span class="id" type="var">t<sub>1</sub></span> ∈ <span class="id" type="var">TBool</span><br/>
<br/>
<span class="id" type="keyword">where</span> "'<span style="font-family: arial;">⊢</span>' t '∈' T" := (<span class="id" type="var">has_type</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span>).<br/>
<br/>
<div class="togglescript" id="proofcontrol5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')"><span class="show"></span></div>
<div class="proofscript" id="proof5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')">
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">has_type</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab634"></a><h3 class="section">Examples</h3>
<div class="paragraph"> </div>
It's important to realize that the typing relation is a
<i>conservative</i> (or <i>static</i>) approximation: it does not calculate
the type of the normal form of a term.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">has_type_1</span> : <br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tif</span> <span class="id" type="var">tfalse</span> <span class="id" type="var">tzero</span> (<span class="id" type="var">tsucc</span> <span class="id" type="var">tzero</span>) ∈ <span class="id" type="var">TNat</span>.<br/>
<div class="togglescript" id="proofcontrol6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')"><span class="show"></span></div>
<div class="proofscript" id="proof6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_If</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_False</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Zero</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Succ</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">T_Zero</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
(Since we've included all the constructors of the typing relation
in the hint database, the <span class="inlinecode"><span class="id" type="tactic">auto</span></span> tactic can actually find this
proof automatically.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">has_type_not</span> : <br/>
¬ (<span style="font-family: arial;">⊢</span> <span class="id" type="var">tif</span> <span class="id" type="var">tfalse</span> <span class="id" type="var">tzero</span> <span class="id" type="var">ttrue</span> ∈ <span class="id" type="var">TBool</span>).<br/>
<div class="togglescript" id="proofcontrol7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')"><span class="show"></span></div>
<div class="proofscript" id="proof7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">Contra</span>. <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span> 2. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab635"></a><h4 class="section">Exercise: 1 star, optional (succ_hastype_nat__hastype_nat)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">succ_hastype_nat__hastype_nat</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">tsucc</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">TNat</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">TNat</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab636"></a><h2 class="section">Canonical forms</h2>
<div class="paragraph"> </div>
The following two lemmas capture the basic property that defines
the shape of well-typed values. They say that the definition of value
and the typing relation agree.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">bool_canonical</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">TBool</span> <span style="font-family: arial;">→</span> <span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bvalue</span> <span class="id" type="var">t</span>.<br/>
<div class="togglescript" id="proofcontrol8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')"><span class="show"></span></div>
<div class="proofscript" id="proof8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">HT</span> <span class="id" type="var">HV</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HV</span>; <span class="id" type="tactic">auto</span>.<br/>
<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">inversion</span> <span class="id" type="var">HT</span>; <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">nat_canonical</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">TNat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nvalue</span> <span class="id" type="var">t</span>.<br/>
<div class="togglescript" id="proofcontrol9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')"><span class="show"></span></div>
<div class="proofscript" id="proof9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">HT</span> <span class="id" type="var">HV</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HV</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">inversion</span> <span class="id" type="var">HT</span>.<br/>
<br/>
<span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab637"></a><h2 class="section">Progress</h2>
<div class="paragraph"> </div>
The typing relation enjoys two critical properties. The first is
that well-typed normal forms are values (i.e., not stuck).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="tactic">progress</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span>, <span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab638"></a><h4 class="section">Exercise: 3 stars (finish_progress)</h4>
Complete the formal proof of the <span class="inlinecode"><span class="id" type="tactic">progress</span></span> property. (Make sure
you understand the informal proof fragment in the following
exercise before starting — this will save you a lot of time.)
</div>
<div class="code code-tight">
<br/>
<div class="togglescript" id="proofcontrol10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')"><span class="show"></span></div>
<div class="proofscript" id="proof10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')">
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">HT</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">HT</span>...<br/>
<span class="comment">(* The cases that were obviously values, like T_True and<br/>
T_False, were eliminated immediately by auto *)</span><br/>
- <span class="comment">(* T_If *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">IHHT1</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">IHHT1</span>.<br/>
+ <span class="comment">(* t<sub>1</sub> is a value *)</span><br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">bool_canonical</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">HT1</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">t<sub>2</sub></span>...<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">t<sub>3</sub></span>...<br/>
+ <span class="comment">(* t<sub>1</sub> can take a step *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">H1</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">tif</span> <span class="id" type="var">t<sub>1</sub>'</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>3</sub></span>)...<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
</div>
<br/>
</div>
<div class="doc">
<a name="lab639"></a><h4 class="section">Exercise: 3 stars, advanced (finish_progress_informal)</h4>
Complete the corresponding informal proof:
<div class="paragraph"> </div>
<i>Theorem</i>: If <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>, then either <span class="inlinecode"><span class="id" type="var">t</span></span> is a value or else
<span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> for some <span class="inlinecode"><span class="id" type="var">t'</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on a derivation of <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last rule in the derivation is <span class="inlinecode"><span class="id" type="var">T_If</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, with <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">Bool</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</span></span> and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>
<span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>. By the IH, either <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value or else <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> can step
to some <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is a value, then by the canonical forms lemmas
and the fact that <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">Bool</span></span> we have that <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
is a <span class="inlinecode"><span class="id" type="var">bvalue</span></span> — i.e., it is either <span class="inlinecode"><span class="id" type="var">true</span></span> or <span class="inlinecode"><span class="id" type="var">false</span></span>.
If <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">true</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> steps to <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> by <span class="inlinecode"><span class="id" type="var">ST_IfTrue</span></span>,
while if <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">false</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> steps to <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span> by
<span class="inlinecode"><span class="id" type="var">ST_IfFalse</span></span>. Either way, <span class="inlinecode"><span class="id" type="var">t</span></span> can step, which is what
we wanted to show.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> itself can take a step, then, by <span class="inlinecode"><span class="id" type="var">ST_If</span></span>, so can
<span class="inlinecode"><span class="id" type="var">t</span></span>.
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
This is more interesting than the strong progress theorem that we
saw in the Smallstep chapter, where <i>all</i> normal forms were
values. Here, a term can be stuck, but only if it is ill
typed.
<div class="paragraph"> </div>
<a name="lab640"></a><h4 class="section">Exercise: 1 star (step_review)</h4>
Quick review. Answer <i>true</i> or <i>false</i>. In this language...
<div class="paragraph"> </div>
<ul class="doclist">
<li> Every well-typed normal form is a value.
<div class="paragraph"> </div>
</li>
<li> Every value is a normal form.
<div class="paragraph"> </div>
</li>
<li> The single-step evaluation relation is
a partial function (i.e., it is deterministic).
<div class="paragraph"> </div>
</li>
<li> The single-step evaluation relation is a <i>total</i> function.
</li>
</ul>
<div class="paragraph"> </div>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab641"></a><h2 class="section">Type Preservation</h2>
<div class="paragraph"> </div>
The second critical property of typing is that, when a well-typed
term takes a step, the result is also a well-typed term.
<div class="paragraph"> </div>
This theorem is often called the <i>subject reduction</i> property,
because it tells us what happens when the "subject" of the typing
relation is reduced. This terminology comes from thinking of
typing statements as sentences, where the term is the subject and
the type is the predicate.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">preservation</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span>,<br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t</span> ∈ <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">t</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">⊢</span> <span class="id" type="var">t'</span> ∈ <span class="id" type="var">T</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab642"></a><h4 class="section">Exercise: 2 stars (finish_preservation)</h4>
Complete the formal proof of the <span class="inlinecode"><span class="id" type="var">preservation</span></span> property. (Again,
make sure you understand the informal proof fragment in the
following exercise first.)
</div>
<div class="code code-tight">
<br/>
<div class="togglescript" id="proofcontrol11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')"><span class="show"></span></div>
<div class="proofscript" id="proof11" onclick="toggleDisplay('proof11');toggleDisplay('proofcontrol11')">
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span> <span class="id" type="var">T</span> <span class="id" type="var">HT</span> <span class="id" type="var">HE</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">t'</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">HT</span>; <br/>
<span class="comment">(* every case needs to introduce a couple of things *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">t'</span> <span class="id" type="var">HE</span>; <br/>
<span class="comment">(* and we can deal with several impossible<br/>
cases all at once *)</span><br/>
<span class="id" type="tactic">try</span> (<span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>).<br/>
- <span class="comment">(* T_If *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">HE</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">HE</span>.<br/>
+ <span class="comment">(* ST_IFTrue *)</span> <span class="id" type="tactic">assumption</span>.<br/>
+ <span class="comment">(* ST_IfFalse *)</span> <span class="id" type="tactic">assumption</span>.<br/>
+ <span class="comment">(* ST_If *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">T_If</span>; <span class="id" type="tactic">try</span> <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHHT1</span>; <span class="id" type="tactic">assumption</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab643"></a><h4 class="section">Exercise: 3 stars, advanced (finish_preservation_informal)</h4>
Complete the following proof:
<div class="paragraph"> </div>
<i>Theorem</i>: If <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span> and <span class="inlinecode"><span 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 style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on a derivation of <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last rule in the derivation is <span class="inlinecode"><span class="id" type="var">T_If</span></span>, then <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span>
<span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, with <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">Bool</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</span></span> and <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>
<span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>.
<div class="paragraph"> </div>
Inspecting the rules for the small-step reduction relation and
remembering that <span class="inlinecode"><span class="id" type="var">t</span></span> has the form <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode">...</span>, we see that the
only ones that could have been used to prove <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> are
<span class="inlinecode"><span class="id" type="var">ST_IfTrue</span></span>, <span class="inlinecode"><span class="id" type="var">ST_IfFalse</span></span>, or <span class="inlinecode"><span class="id" type="var">ST_If</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If the last rule was <span class="inlinecode"><span class="id" type="var">ST_IfTrue</span></span>, then <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>. But we
know that <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</span></span>, so we are done.
<div class="paragraph"> </div>
</li>
<li> If the last rule was <span class="inlinecode"><span class="id" type="var">ST_IfFalse</span></span>, then <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>. But we
know that <span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>, so we are done.
<div class="paragraph"> </div>
</li>
<li> If the last rule was <span class="inlinecode"><span class="id" type="var">ST_If</span></span>, then <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span>
<span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span>, where <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>. We know <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">Bool</span></span> so,
by the IH, <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">Bool</span></span>. The <span class="inlinecode"><span class="id" type="var">T_If</span></span> rule then gives us
<span class="inlinecode"><span style="font-family: arial;">⊢</span></span> <span class="inlinecode"><span class="id" type="keyword">if</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span> <span class="inlinecode"><span class="id" type="keyword">then</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> <span class="inlinecode"><span class="id" type="keyword">else</span></span> <span class="inlinecode"><span class="id" type="var">t<sub>3</sub></span></span> <span class="inlinecode">∈</span> <span class="inlinecode"><span class="id" type="var">T</span></span>, as required.
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab644"></a><h4 class="section">Exercise: 3 stars (preservation_alternate_proof)</h4>
Now prove the same property again by induction on the
<i>evaluation</i> derivation instead of on the typing derivation.
Begin by carefully reading and thinking about the first few
lines of the above proof to make sure you understand what
each one is doing. The set-up for this proof is similar, but