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Smallstep.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>Smallstep: Small-step Operational Semantics</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">Smallstep<span class="subtitle">Small-step Operational Semantics</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">Imp</span>.<br/>
<br/>
</div>
<div class="doc">
The evaluators we have seen so far (e.g., the ones for
<span class="inlinecode"><span class="id" type="var">aexp</span></span>s, <span class="inlinecode"><span class="id" type="var">bexp</span></span>s, and commands) have been formulated in a
"big-step" style — they specify how a given expression can be
evaluated to its final value (or a command plus a store to a final
store) "all in one big step."
<div class="paragraph"> </div>
This style is simple and natural for many purposes — indeed,
Gilles Kahn, who popularized its use, called it <i>natural
semantics</i>. But there are some things it does not do well. In
particular, it does not give us a natural way of talking about
<i>concurrent</i> programming languages, where the "semantics" of a
program — i.e., the essence of how it behaves — is not just
which input states get mapped to which output states, but also
includes the intermediate states that it passes through along the
way, since these states can also be observed by concurrently
executing code.
<div class="paragraph"> </div>
Another shortcoming of the big-step style is more technical, but
critical in some situations. To see the issue, suppose we wanted
to define a variant of Imp where variables could hold <i>either</i>
numbers <i>or</i> lists of numbers (see the <span class="inlinecode"><span class="id" type="var">HoareList</span></span> chapter for
details). In the syntax of this extended language, it will be
possible to write strange expressions like <span class="inlinecode">2</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">nil</span></span>, and our
semantics for arithmetic expressions will then need to say
something about how such expressions behave. One
possibility (explored in the <span class="inlinecode"><span class="id" type="var">HoareList</span></span> chapter) is to maintain
the convention that every arithmetic expressions evaluates to some
number by choosing some way of viewing a list as a number — e.g.,
by specifying that a list should be interpreted as <span class="inlinecode">0</span> when it
occurs in a context expecting a number. But this is really a bit
of a hack.
<div class="paragraph"> </div>
A much more natural approach is simply to say that the behavior of
an expression like <span class="inlinecode">2+<span class="id" type="var">nil</span></span> is <i>undefined</i> — it doesn't evaluate
to any result at all. And we can easily do this: we just have to
formulate <span class="inlinecode"><span class="id" type="var">aeval</span></span> and <span class="inlinecode"><span class="id" type="var">beval</span></span> as <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> propositions rather
than Fixpoints, so that we can make them partial functions instead
of total ones.
<div class="paragraph"> </div>
However, now we encounter a serious deficiency. In this language,
a command might <i>fail</i> to map a given starting state to any ending
state for two quite different reasons: either because the
execution gets into an infinite loop or because, at some point,
the program tries to do an operation that makes no sense, such as
adding a number to a list, and none of the evaluation rules can be
applied.
<div class="paragraph"> </div>
These two outcomes — nontermination vs. getting stuck in an
erroneous configuration — are quite different. In particular, we
want to allow the first (permitting the possibility of infinite
loops is the price we pay for the convenience of programming with
general looping constructs like <span class="inlinecode"><span class="id" type="var">while</span></span>) but prevent the
second (which is just wrong), for example by adding some form of
<i>typechecking</i> to the language. Indeed, this will be a major
topic for the rest of the course. As a first step, we need a
different way of presenting the semantics that allows us to
distinguish nontermination from erroneous "stuck states."
<div class="paragraph"> </div>
So, for lots of reasons, we'd like to have a finer-grained way of
defining and reasoning about program behaviors. This is the topic
of the present chapter. We replace the "big-step" <span class="inlinecode"><span class="id" type="var">eval</span></span> relation
with a "small-step" relation that specifies, for a given program,
how the "atomic steps" of computation are performed.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab581"></a><h1 class="section">A Toy Language</h1>
<div class="paragraph"> </div>
To save space in the discussion, let's go back to an
incredibly simple language containing just constants and
addition. (We use single letters — <span class="inlinecode"><span class="id" type="var">C</span></span> and <span class="inlinecode"><span class="id" type="var">P</span></span> — for the
constructor names, for brevity.) At the end of the chapter, we'll
see how to apply the same techniques to the full Imp language.
</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">C</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tm</span> <span class="comment">(* Constant *)</span><br/>
| <span class="id" type="var">P</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 class="comment">(* Plus *)</span><br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "tm_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> "C" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "P" ].<br/>
<br/>
</div>
<div class="doc">
Here is a standard evaluator for this language, written in the
same (big-step) style as we've been using up to this point.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">evalF</span> (<span class="id" type="var">t</span> : <span class="id" type="var">tm</span>) : <span class="id" type="var">nat</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">C</span> <span class="id" type="var">n</span> ⇒ <span class="id" type="var">n</span><br/>
| <span class="id" type="var">P</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> ⇒ <span class="id" type="var">evalF</span> <span class="id" type="var">a1</span> + <span class="id" type="var">evalF</span> <span class="id" type="var">a2</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Now, here is the same evaluator, written in exactly the same
style, but formulated as an inductively defined relation. Again,
we use the notation <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">n</span></span> for "<span class="inlinecode"><span class="id" type="var">t</span></span> evaluates to <span class="inlinecode"><span class="id" type="var">n</span></span>."
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(E_Const)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">C n <span style="font-family: arial;">⇓</span> n</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> n1</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇓</span> n2</td>
<td class="infrulenamecol" rowspan="3">
(E_Plus)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P t<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇓</span> C (n1 + n2)</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> " t '<span style="font-family: arial;">⇓</span>' n " (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 50, <span class="id" type="var">left</span> <span class="id" type="var">associativity</span>).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">eval</span> : <span class="id" type="var">tm</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<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="var">E_Const</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">C</span> <span class="id" type="var">n</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">n</span><br/>
| <span class="id" type="var">E_Plus</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">n1</span> <span class="id" type="var">n2</span>, <br/>
<span class="id" type="var">t<sub>1</sub></span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">n1</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">n2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</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">n1</span> + <span class="id" type="var">n2</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> " t '<span style="font-family: arial;">⇓</span>' n " := (<span class="id" type="var">eval</span> <span class="id" type="var">t</span> <span class="id" type="var">n</span>).<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "eval_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> "E_Const" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "E_Plus" ].<br/>
</div>
<br/>
<br/>
</div>
<div class="doc">
Now, here is a small-step version.
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_PlusConstConst)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P (C n1) (C n2) <span style="font-family: arial;">⇒</span> C (n1 + n2)</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_Plus1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P t<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> P t<sub>1</sub>' t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Plus2)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P (C n1) t<sub>2</sub> <span style="font-family: arial;">⇒</span> P (C n1) t<sub>2</sub>'</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> " t '<span style="font-family: arial;">⇒</span>' 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">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_PlusConstConst</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n1</span> <span class="id" type="var">n2</span>,<br/>
<span class="id" type="var">P</span> (<span class="id" type="var">C</span> <span class="id" type="var">n1</span>) (<span class="id" type="var">C</span> <span class="id" type="var">n2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">C</span> (<span class="id" type="var">n1</span> + <span class="id" type="var">n2</span>)<br/>
| <span class="id" type="var">ST_Plus1</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">P</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">P</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_Plus2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n1</span> <span class="id" type="var">t<sub>2</sub></span> <span class="id" type="var">t<sub>2</sub>'</span>,<br/>
<span class="id" type="var">t<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">P</span> (<span class="id" type="var">C</span> <span class="id" type="var">n1</span>) <span class="id" type="var">t<sub>2</sub></span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">P</span> (<span class="id" type="var">C</span> <span class="id" type="var">n1</span>) <span class="id" type="var">t<sub>2</sub>'</span><br/>
<br/>
<span class="id" type="keyword">where</span> " t '<span style="font-family: arial;">⇒</span>' t' " := (<span class="id" type="var">step</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</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">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_PlusConstConst"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Plus1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Plus2" ].<br/>
</div>
<br/>
</div>
<div class="doc">
Things to notice:
<div class="paragraph"> </div>
<ul class="doclist">
<li> We are defining just a single reduction step, in which
one <span class="inlinecode"><span class="id" type="var">P</span></span> node is replaced by its value.
<div class="paragraph"> </div>
</li>
<li> Each step finds the <i>leftmost</i> <span class="inlinecode"><span class="id" type="var">P</span></span> node that is ready to
go (both of its operands are constants) and rewrites it in
place. The first rule tells how to rewrite this <span class="inlinecode"><span class="id" type="var">P</span></span> node
itself; the other two rules tell how to find it.
<div class="paragraph"> </div>
</li>
<li> A term that is just a constant cannot take a step.
</li>
</ul>
<div class="paragraph"> </div>
Let's pause and check a couple of examples of reasoning with
the <span class="inlinecode"><span class="id" type="var">step</span></span> relation...
<div class="paragraph"> </div>
If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> can take a step to <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>, then <span class="inlinecode"><span class="id" type="var">P</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> steps
to <span class="inlinecode"><span class="id" type="var">P</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>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_step_1</span> : <br/>
<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">P</span> (<span class="id" type="var">C</span> 0) (<span class="id" type="var">C</span> 3))<br/>
(<span class="id" type="var">P</span> (<span class="id" type="var">C</span> 2) (<span class="id" type="var">C</span> 4))<br/>
<span style="font-family: arial;">⇒</span><br/>
<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">C</span> (0 + 3))<br/>
(<span class="id" type="var">P</span> (<span class="id" type="var">C</span> 2) (<span class="id" type="var">C</span> 4)).<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="id" type="tactic">apply</span> <span class="id" type="var">ST_Plus1</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">ST_PlusConstConst</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab582"></a><h4 class="section">Exercise: 1 star (test_step_2)</h4>
Right-hand sides of sums can take a step only when the
left-hand side is finished: if <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> can take a step to <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span>,
then <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> steps to <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>
<span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_step_2</span> : <br/>
<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">C</span> 0)<br/>
(<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">C</span> 2) <br/>
(<span class="id" type="var">P</span> (<span class="id" type="var">C</span> 0) (<span class="id" type="var">C</span> 3)))<br/>
<span style="font-family: arial;">⇒</span><br/>
<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">C</span> 0)<br/>
(<span class="id" type="var">P</span> <br/>
(<span class="id" type="var">C</span> 2) <br/>
(<span class="id" type="var">C</span> (0 + 3))).<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="lab583"></a><h1 class="section">Relations</h1>
<div class="paragraph"> </div>
We will be using several different step relations, so it is
helpful to generalize a bit...
<div class="paragraph"> </div>
A (binary) <i>relation</i> on a set <span class="inlinecode"><span class="id" type="var">X</span></span> is a family of propositions
parameterized by two elements of <span class="inlinecode"><span class="id" type="var">X</span></span> — i.e., a proposition about
pairs of elements of <span class="inlinecode"><span class="id" type="var">X</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">relation</span> (<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>) := <span class="id" type="var">X</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>.<br/>
<br/>
</div>
<div class="doc">
Our main examples of such relations in this chapter will be
the single-step and multi-step reduction relations on terms, <span class="inlinecode"><span style="font-family: arial;">⇒</span></span>
and <span class="inlinecode"><span style="font-family: arial;">⇒*</span></span>, but there are many other examples — some that come to
mind are the "equals," "less than," "less than or equal to," and
"is the square of" relations on numbers, and the "prefix of"
relation on lists and strings.
<div class="paragraph"> </div>
One simple property of the <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> relation is that, like the
evaluation relation for our language of Imp programs, it is
<i>deterministic</i>.
<div class="paragraph"> </div>
<i>Theorem</i>: For each <span class="inlinecode"><span class="id" type="var">t</span></span>, there is at most one <span class="inlinecode"><span class="id" type="var">t'</span></span> such that <span class="inlinecode"><span class="id" type="var">t</span></span>
steps to <span class="inlinecode"><span class="id" type="var">t'</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">t'</span></span> is provable). Formally, this is the
same as saying that <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> is deterministic.
<div class="paragraph"> </div>
<i>Proof sketch</i>: We show that if <span class="inlinecode"><span class="id" type="var">x</span></span> steps to both <span class="inlinecode"><span class="id" type="var">y1</span></span> and <span class="inlinecode"><span class="id" type="var">y2</span></span>
then <span class="inlinecode"><span class="id" type="var">y1</span></span> and <span class="inlinecode"><span class="id" type="var">y2</span></span> are equal, by induction on a derivation of
<span class="inlinecode"><span class="id" type="var">step</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y1</span></span>. There are several cases to consider, depending on
the last rule used in this derivation and in the given derivation
of <span class="inlinecode"><span class="id" type="var">step</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y2</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If both are <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span>, the result is immediate.
<div class="paragraph"> </div>
</li>
<li> The cases when both derivations end with <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> or
<span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span> follow by the induction hypothesis.
<div class="paragraph"> </div>
</li>
<li> It cannot happen that one is <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span> and the other
is <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> or <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span>, since this would imply that <span class="inlinecode"><span class="id" type="var">x</span></span> has
the form <span class="inlinecode"><span class="id" type="var">P</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> where 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
constants (by <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span>) <i>and</i> one of <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> or <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> has
the form <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">...</span>.
<div class="paragraph"> </div>
</li>
<li> Similarly, it cannot happen that one is <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> and the other
is <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span>, since this would imply that <span class="inlinecode"><span class="id" type="var">x</span></span> has the form
<span class="inlinecode"><span class="id" type="var">P</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> where <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> has both the form <span class="inlinecode"><span class="id" type="var">P</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> and
the form <span class="inlinecode"><span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>. <font size=-2>☐</font>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">deterministic</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">y1</span> <span class="id" type="var">y2</span> : <span class="id" type="var">X</span>, <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y1</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y2</span> <span style="font-family: arial;">→</span> <span class="id" type="var">y1</span> = <span class="id" type="var">y2</span>.<br/>
<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/>
<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="id" type="tactic">unfold</span> <span class="id" type="var">deterministic</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">x</span> <span class="id" type="var">y1</span> <span class="id" type="var">y2</span> <span class="id" type="var">Hy1</span> <span class="id" type="var">Hy2</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">y2</span>.<br/>
<span class="id" type="var">step_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Hy1</span>) <span class="id" type="var">Case</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">y2</span> <span class="id" type="var">Hy2</span>.<br/>
<span class="id" type="var">Case</span> "ST_PlusConstConst". <span class="id" type="var">step_cases</span> (<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy2</span>) <span class="id" type="var">SCase</span>.<br/>
<span class="id" type="var">SCase</span> "ST_PlusConstConst". <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus1". <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus2". <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">Case</span> "ST_Plus1". <span class="id" type="var">step_cases</span> (<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy2</span>) <span class="id" type="var">SCase</span>.<br/>
<span class="id" type="var">SCase</span> "ST_PlusConstConst". <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">H0</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hy1</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy1</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus1".<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> (<span class="id" type="var">IHHy1</span> <span class="id" type="var">t<sub>1</sub>'0</span>).<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus2". <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hy1</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy1</span>.<br/>
<span class="id" type="var">Case</span> "ST_Plus2". <span class="id" type="var">step_cases</span> (<span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy2</span>) <span class="id" type="var">SCase</span>.<br/>
<span class="id" type="var">SCase</span> "ST_PlusConstConst". <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">H1</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hy1</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Hy1</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus1". <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">SCase</span> "ST_Plus2".<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> (<span class="id" type="var">IHHy1</span> <span class="id" type="var">t<sub>2</sub>'0</span>).<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">SimpleArith1</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab584"></a><h2 class="section">Values</h2>
<div class="paragraph"> </div>
Let's take a moment to slightly generalize the way we state the
definition of single-step reduction.
<div class="paragraph"> </div>
It is useful to think of the <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> relation as defining an
<i>abstract machine</i>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> At any moment, the <i>state</i> of the machine is a term.
<div class="paragraph"> </div>
</li>
<li> A <i>step</i> of the machine is an atomic unit of computation —
here, a single "add" operation.
<div class="paragraph"> </div>
</li>
<li> The <i>halting states</i> of the machine are ones where there is no
more computation to be done.
</li>
</ul>
<div class="paragraph"> </div>
We can then execute a term <span class="inlinecode"><span class="id" type="var">t</span></span> as follows:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Take <span class="inlinecode"><span class="id" type="var">t</span></span> as the starting state of the machine.
<div class="paragraph"> </div>
</li>
<li> Repeatedly use the <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> relation to find a sequence of
machine states, starting with <span class="inlinecode"><span class="id" type="var">t</span></span>, where each state steps to
the next.
<div class="paragraph"> </div>
</li>
<li> When no more reduction is possible, "read out" the final state
of the machine as the result of execution.
</li>
</ul>
<div class="paragraph"> </div>
Intuitively, it is clear that the final states of the
machine are always terms of the form <span class="inlinecode"><span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> for some <span class="inlinecode"><span class="id" type="var">n</span></span>.
We call such terms <i>values</i>.
</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_const</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">value</span> (<span class="id" type="var">C</span> <span class="id" type="var">n</span>).<br/>
<br/>
</div>
<div class="doc">
Having introduced the idea of values, we can use it in the
definition of the <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> relation to write <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span> rule in a
slightly more elegant way:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(ST_PlusConstConst)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P (C n1) (C n2) <span style="font-family: arial;">⇒</span> C (n1 + n2)</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_Plus1)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P t<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> P t<sub>1</sub>' t<sub>2</sub></td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">value v<sub>1</sub></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">t<sub>2</sub> <span style="font-family: arial;">⇒</span> t<sub>2</sub>'</td>
<td class="infrulenamecol" rowspan="3">
(ST_Plus2)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">P v<sub>1</sub> t<sub>2</sub> <span style="font-family: arial;">⇒</span> P v<sub>1</sub> t<sub>2</sub>'</td>
<td></td>
</td>
</table></center> Again, the variable names here carry important information:
by convention, <span class="inlinecode"><span class="id" type="var">v<sub>1</sub></span></span> ranges only over values, while <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>
range over arbitrary terms. (Given this convention, the explicit
<span class="inlinecode"><span class="id" type="var">value</span></span> hypothesis is arguably redundant. We'll keep it for now,
to maintain a close correspondence between the informal and Coq
versions of the rules, but later on we'll drop it in informal
rules, for the sake of brevity.)
<div class="paragraph"> </div>
Here are the formal rules:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> " t '<span style="font-family: arial;">⇒</span>' 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">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_PlusConstConst</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n1</span> <span class="id" type="var">n2</span>,<br/>
<span class="id" type="var">P</span> (<span class="id" type="var">C</span> <span class="id" type="var">n1</span>) (<span class="id" type="var">C</span> <span class="id" type="var">n2</span>)<br/>
<span style="font-family: arial;">⇒</span> <span class="id" type="var">C</span> (<span class="id" type="var">n1</span> + <span class="id" type="var">n2</span>)<br/>
| <span class="id" type="var">ST_Plus1</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">P</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">P</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_Plus2</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> <span class="comment">(* <----- n.b. *)</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">P</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">P</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span><br/>
<br/>
<span class="id" type="keyword">where</span> " t '<span style="font-family: arial;">⇒</span>' t' " := (<span class="id" type="var">step</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">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_PlusConstConst"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Plus1" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Plus2" ].<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab585"></a><h4 class="section">Exercise: 3 stars (redo_determinism)</h4>
As a sanity check on this change, let's re-verify determinism
<div class="paragraph"> </div>
Proof sketch: We must show that if <span class="inlinecode"><span class="id" type="var">x</span></span> steps to both <span class="inlinecode"><span class="id" type="var">y1</span></span> and <span class="inlinecode"><span class="id" type="var">y2</span></span>
then <span class="inlinecode"><span class="id" type="var">y1</span></span> and <span class="inlinecode"><span class="id" type="var">y2</span></span> are equal. Consider the final rules used in
the derivations of <span class="inlinecode"><span class="id" type="var">step</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y1</span></span> and <span class="inlinecode"><span class="id" type="var">step</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y2</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> If both are <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span>, the result is immediate.
<div class="paragraph"> </div>
</li>
<li> It cannot happen that one is <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span> and the other
is <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> or <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span>, since this would imply that <span class="inlinecode"><span class="id" type="var">x</span></span> has
the form <span class="inlinecode"><span class="id" type="var">P</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> where 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
constants (by <span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span>) AND one of <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> or <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> has
the form <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">...</span>.
<div class="paragraph"> </div>
</li>
<li> Similarly, it cannot happen that one is <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> and the other
is <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span>, since this would imply that <span class="inlinecode"><span class="id" type="var">x</span></span> has the form
<span class="inlinecode"><span class="id" type="var">P</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> where <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> both has the form <span class="inlinecode"><span class="id" type="var">P</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> and
is a value (hence has the form <span class="inlinecode"><span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>).
<div class="paragraph"> </div>
</li>
<li> The cases when both derivations end with <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span> or
<span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span> follow by the induction hypothesis. <font size=-2>☐</font>
</li>
</ul>
<div class="paragraph"> </div>
Most of this proof is the same as the one above. But to get
maximum benefit from the exercise you should try to write it from
scratch and just use the earlier one if you get stuck.
</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>.<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="lab586"></a><h2 class="section">Strong Progress and Normal Forms</h2>
<div class="paragraph"> </div>
The definition of single-step reduction for our toy language is
fairly simple, but for a larger language it would be pretty easy
to forget one of the rules and create a situation where some term
cannot take a step even though it has not been completely reduced
to a value. The following theorem shows that we did not, in fact,
make such a mistake here.
<div class="paragraph"> </div>
<i>Theorem</i> (<i>Strong Progress</i>): If <span class="inlinecode"><span class="id" type="var">t</span></span> is a term, then either <span class="inlinecode"><span class="id" type="var">t</span></span>
is a value, or there exists a term <span class="inlinecode"><span class="id" type="var">t'</span></span> such that <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">t</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">C</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>. Then <span class="inlinecode"><span class="id" type="var">t</span></span> is a <span class="inlinecode"><span class="id" type="var">value</span></span>.
<div class="paragraph"> </div>
</li>
<li> Suppose <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">P</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>, where (by the IH) <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> is either a
value or can step to some <span class="inlinecode"><span class="id" type="var">t<sub>1</sub>'</span></span>, and where <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> is either a
value or can step to some <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span></span>. We must show <span class="inlinecode"><span class="id" type="var">P</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> is
either a value or steps to some <span class="inlinecode"><span class="id" type="var">t'</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> and <span class="inlinecode"><span class="id" type="var">t<sub>2</sub></span></span> are both values, then <span class="inlinecode"><span class="id" type="var">t</span></span> can take a step, by
<span class="inlinecode"><span class="id" type="var">ST_PlusConstConst</span></span>.
<div class="paragraph"> </div>
</li>
<li> 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> can take a step, then so can <span class="inlinecode"><span class="id" type="var">t</span></span>,
by <span class="inlinecode"><span class="id" type="var">ST_Plus2</span></span>.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">t<sub>1</sub></span></span> can take a step, then so can <span class="inlinecode"><span class="id" type="var">t</span></span>, by <span class="inlinecode"><span class="id" type="var">ST_Plus1</span></span>. <font size=-2>☐</font>
</li>
</ul>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">strong_progress</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 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/>
<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="var">tm_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">t</span>) <span class="id" type="var">Case</span>.<br/>
<span class="id" type="var">Case</span> "C". <span class="id" type="var">left</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">v_const</span>.<br/>
<span class="id" type="var">Case</span> "P". <span class="id" type="var">right</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">IHt1</span>.<br/>
<span class="id" type="var">SCase</span> "l". <span class="id" type="tactic">inversion</span> <span class="id" type="var">IHt2</span>.<br/>
<span class="id" type="var">SSCase</span> "l". <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>.<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">C</span> (<span class="id" type="var">n</span> + <span class="id" type="var">n0</span>)).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_PlusConstConst</span>.<br/>
<span class="id" type="var">SSCase</span> "r". <span class="id" type="tactic">inversion</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">H1</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">P</span> <span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t'</span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_Plus2</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="var">SCase</span> "r". <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'</span> <span class="id" type="var">H0</span>].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">P</span> <span class="id" type="var">t'</span> <span class="id" type="var">t<sub>2</sub></span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ST_Plus1</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H0</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
This important property is called <i>strong progress</i>, because
every term either is a value or can "make progress" by stepping to
some other term. (The qualifier "strong" distinguishes it from a
more refined version that we'll see in later chapters, called
simply "progress.")
<div class="paragraph"> </div>
The idea of "making progress" can be extended to tell us something
interesting about <span class="inlinecode"><span class="id" type="var">value</span></span>s: in this language <span class="inlinecode"><span class="id" type="var">value</span></span>s are exactly
the terms that <i>cannot</i> make progress in this sense.
<div class="paragraph"> </div>
To state this observation formally, let's begin by giving a name
to terms that cannot make progress. We'll call them <i>normal
forms</i>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">normal_form</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>:<span class="id" type="var">relation</span> <span class="id" type="var">X</span>) (<span class="id" type="var">t</span>:<span class="id" type="var">X</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
¬ <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span>, <span class="id" type="var">R</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span>.<br/>
<br/>
</div>
<div class="doc">
This definition actually specifies what it is to be a normal form
for an <i>arbitrary</i> relation <span class="inlinecode"><span class="id" type="var">R</span></span> over an arbitrary set <span class="inlinecode"><span class="id" type="var">X</span></span>, not
just for the particular single-step reduction relation over terms
that we are interested in at the moment. We'll re-use the same
terminology for talking about other relations later in the
course.
<div class="paragraph"> </div>
We can use this terminology to generalize the observation we made
in the strong progress theorem: in this language, normal forms and
values are actually the same thing.
</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">v</span>,<br/>
<span class="id" type="var">value</span> <span class="id" type="var">v</span> <span style="font-family: arial;">→</span> <span class="id" type="var">normal_form</span> <span class="id" type="var">step</span> <span class="id" type="var">v</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">unfold</span> <span class="id" type="var">normal_form</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">v</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">contra</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">contra</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">nf_is_value</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span class="id" type="var">normal_form</span> <span class="id" type="var">step</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/>
<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>. <span class="comment">(* a corollary of <span class="inlinecode"><span class="id" type="var">strong_progress</span></span>... *)</span><br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">normal_form</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">t</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">G</span> : <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="var">SCase</span> "Proof of assertion". <span class="id" type="tactic">apply</span> <span class="id" type="var">strong_progress</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">G</span>.<br/>
<span class="id" type="var">SCase</span> "l". <span class="id" type="tactic">apply</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="var">SCase</span> "r". <span class="id" type="tactic">apply</span> <span class="id" type="var">ex_falso_quodlibet</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Corollary</span> <span class="id" type="var">nf_same_as_value</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t</span>,<br/>
<span class="id" type="var">normal_form</span> <span class="id" type="var">step</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/>
<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">split</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">nf_is_value</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">value_is_nf</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
Why is this interesting?
<div class="paragraph"> </div>
Because <span class="inlinecode"><span class="id" type="var">value</span></span> is a syntactic concept — it is defined by looking
at the form of a term — while <span class="inlinecode"><span class="id" type="var">normal_form</span></span> is a semantic one —
it is defined by looking at how the term steps. It is not obvious
that these concepts should coincide!
<div class="paragraph"> </div>
Indeed, we could easily have written the definitions so that they
would not coincide...
</div>
<div class="code code-tight">
<br/>
<br/>
</div>
<div class="doc">
We might, for example, mistakenly define <span class="inlinecode"><span class="id" type="var">value</span></span> so that it
includes some terms that are not finished reducing.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Temp1</span>.<br/>
<span class="comment">(* Open an inner module so we can redefine value and step. *)</span><br/>
<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_const</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">value</span> (<span class="id" type="var">C</span> <span class="id" type="var">n</span>)<br/>
| <span class="id" type="var">v_funny</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">n2</span>, <span class="comment">(* <---- *)</span><br/>
<span class="id" type="var">value</span> (<span class="id" type="var">P</span> <span class="id" type="var">t<sub>1</sub></span> (<span class="id" type="var">C</span> <span class="id" type="var">n2</span>)).<br/>
<br/>
<span class="id" type="keyword">Reserved Notation</span> " t '<span style="font-family: arial;">⇒</span>' 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">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_PlusConstConst</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n1</span> <span class="id" type="var">n2</span>,<br/>
<span class="id" type="var">P</span> (<span class="id" type="var">C</span> <span class="id" type="var">n1</span>) (<span class="id" type="var">C</span> <span class="id" type="var">n2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">C</span> (<span class="id" type="var">n1</span> + <span class="id" type="var">n2</span>)<br/>
| <span class="id" type="var">ST_Plus1</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">P</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">P</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_Plus2</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">P</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">P</span> <span class="id" type="var">v<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub>'</span><br/>
<br/>
<span class="id" type="keyword">where</span> " t '<span style="font-family: arial;">⇒</span>' t' " := (<span class="id" type="var">step</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab587"></a><h4 class="section">Exercise: 3 stars, advanced (value_not_same_as_normal_form)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">value_not_same_as_normal_form</span> :<br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">v</span>, <span class="id" type="var">value</span> <span class="id" type="var">v</span> <span style="font-family: arial;">∧</span> ¬ <span class="id" type="var">normal_form</span> <span class="id" type="var">step</span> <span class="id" type="var">v</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">
Alternatively, we might mistakenly define <span class="inlinecode"><span class="id" type="var">step</span></span> so that it
permits something designated as a value to reduce further.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">Temp2</span>.<br/>
<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_const</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">value</span> (<span class="id" type="var">C</span> <span class="id" type="var">n</span>).<br/>