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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=iso-8859-1"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Types: Type Systems</title>
</head>
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<body onLoad="hideAll('proofscript')">
<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="comment">(* $Date: 2011-06-03 13:58:55 -0400 (Fri, 03 Jun 2011) $ *)</span><br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Smallstep</span>.<br/>
<br/>
</div>
<div class="doc">
Our next topic, a large one, 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="lab507"></a><h1 class="section">More Automation</h1>
<div class="paragraph"> </div>
Before we start, let's spend a little time learning to use
some of Coq's more powerful automation features...
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab508"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="tactic">auto</span></span> and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> Tactics</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">auto</span></span> tactic solves goals that are solvable by any combination of
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="tactic">intros</span></span>,
</li>
<li> <span class="inlinecode"><span class="id" type="tactic">apply</span></span> (with a local hypothesis, by default), and
</li>
<li> <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> tactic works just like <span class="inlinecode"><span class="id" type="tactic">auto</span></span>, except that it uses
<span class="inlinecode"><span class="id" type="tactic">eapply</span></span> instead of <span class="inlinecode"><span class="id" type="tactic">apply</span></span>.
<div class="paragraph"> </div>
Using <span class="inlinecode"><span class="id" type="tactic">auto</span></span> is always "safe" in the sense that it will never fail
and will never change the proof state: either it completely solves
the current goal, or it does nothing.
<div class="paragraph"> </div>
Here is a contrived example:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">auto_example_1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">R</span> <span class="id" type="var">S</span> <span class="id" type="var">T</span> <span class="id" type="var">U</span> : <span class="id" type="keyword">Prop</span>,<br/>
(<span class="id" type="var">P</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span>) <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">P</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</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">R</span>) <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">S</span> <span style="font-family: arial;">→</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span> <span class="id" type="var">U</span>) <span style="font-family: arial;">→</span><br/>
((<span class="id" type="var">P</span><span style="font-family: arial;">→</span><span class="id" type="var">Q</span>) <span style="font-family: arial;">→</span> (<span class="id" type="var">P</span><span style="font-family: arial;">→</span><span class="id" type="var">S</span>)) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">U</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">auto</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
When searching for potential proofs of the current goal, <span class="inlinecode"><span class="id" type="tactic">auto</span></span>
and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> consider the hypotheses in the current context
together with a <i>hint database</i> of other lemmas and constructors.
Some of the lemmas and constructors we've already seen — e.g.,
<span class="inlinecode"><span class="id" type="var">conj</span></span>, <span class="inlinecode"><span class="id" type="var">or_introl</span></span>, and <span class="inlinecode"><span class="id" type="var">or_intror</span></span> — are installed in this hint
database by default.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">auto_example_2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">R</span> : <span class="id" type="keyword">Prop</span>,<br/>
<span class="id" type="var">Q</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">Q</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</span> <span style="font-family: arial;">∨</span> (<span class="id" type="var">Q</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">R</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">auto</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
We can extend the hint database just for the purposes of one
application of <span class="inlinecode"><span class="id" type="tactic">auto</span></span> or <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> by writing <span class="inlinecode"><span class="id" type="tactic">auto</span></span> <span class="inlinecode"><span class="id" type="keyword">using</span></span> <span class="inlinecode">...</span>.
E.g., if <span class="inlinecode"><span class="id" type="var">conj</span></span>, <span class="inlinecode"><span class="id" type="var">or_introl</span></span>, and <span class="inlinecode"><span class="id" type="var">or_intror</span></span> had <i>not</i> already
been in the hint database, we could have done this instead:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">auto_example_2a</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">R</span> : <span class="id" type="keyword">Prop</span>,<br/>
<span class="id" type="var">Q</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">Q</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</span> <span style="font-family: arial;">∨</span> (<span class="id" type="var">Q</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">R</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">auto</span> <span class="id" type="keyword">using</span> <span class="id" type="var">conj</span>, <span class="id" type="var">or_introl</span>, <span class="id" type="var">or_intror</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Of course, in any given development there will also be some of our
own specific constructors and lemmas that are used very often in
proofs. We can add these to the global hint database by writing
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Resolve</span> <span class="id" type="var">T</span>.
<div class="paragraph"> </div>
</div>
at the top level, where <span class="inlinecode"><span class="id" type="var">T</span></span> is a top-level theorem or a
constructor of an inductively defined proposition (i.e., anything
whose type is an implication). As a shorthand, we can write
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">c</span>.
<div class="paragraph"> </div>
</div>
to tell Coq to do a <span class="inlinecode"><span class="id" type="keyword">Hint</span></span> <span class="inlinecode"><span class="id" type="keyword">Resolve</span></span> for <i>all</i> of the constructors
from the inductive definition of <span class="inlinecode"><span class="id" type="var">c</span></span>.
<div class="paragraph"> </div>
It is also sometimes necessary to add
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Unfold</span> <span class="id" type="var">d</span>.
<div class="paragraph"> </div>
</div>
where <span class="inlinecode"><span class="id" type="var">d</span></span> is a defined symbol, so that <span class="inlinecode"><span class="id" type="tactic">auto</span></span> knows to expand
uses of <span class="inlinecode"><span class="id" type="var">d</span></span> and enable further possibilities for applying
lemmas that it knows about.
<div class="paragraph"> </div>
Here are some <span class="inlinecode"><span class="id" type="keyword">Hint</span></span>s we will find useful.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">multi</span>.<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="keyword">Resolve</span> <span class="id" type="var">beq_id_eq</span> <span class="id" type="var">beq_id_false_not_eq</span>.<br/>
<br/>
</div>
<div class="doc">
Warning: Just as with Coq's other automation facilities, it is
easy to overuse <span class="inlinecode"><span class="id" type="tactic">auto</span></span> and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> and wind up with proofs that
are impossible to understand later!
<div class="paragraph"> </div>
Also, overuse of <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> can make proof scripts very slow. Get in
the habit of using <span class="inlinecode"><span class="id" type="tactic">auto</span></span> most of the time and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span> only when
necessary.
<div class="paragraph"> </div>
For much more detailed information about using <span class="inlinecode"><span class="id" type="tactic">auto</span></span> and <span class="inlinecode"><span class="id" type="tactic">eauto</span></span>,
see the chapter <span class="inlinecode"><span class="id" type="var">UseAuto</span></span>.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab509"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="keyword">Proof</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> Tactic</h2>
<div class="paragraph"> </div>
If you start a proof by saying <span class="inlinecode"><span class="id" type="keyword">Proof</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">(<span class="id" type="var">tactic</span>)</span> instead of
just <span class="inlinecode"><span class="id" type="keyword">Proof</span></span>, then writing <span class="inlinecode">...</span> instead of <span class="inlinecode">.</span> after a tactic in
the body of the proof will try to solve all generated subgoals
with <span class="inlinecode"><span class="id" type="var">tactic</span></span> (and fail if this doesn't work).
<div class="paragraph"> </div>
One common use of this facility is "<span class="inlinecode"><span class="id" type="keyword">Proof</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode"><span class="id" type="tactic">auto</span></span>" (or
<span class="inlinecode"><span class="id" type="tactic">eauto</span></span>). We'll see many examples of this later in the file.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab510"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode"><span class="id" type="tactic">by</span></span> <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> Tactic</h2>
<div class="paragraph"> </div>
Here's another nice automation feature: it often arises that the
context contains a contradictory assumption and we want to use
<span class="inlinecode"><span class="id" type="tactic">inversion</span></span> on it to solve the goal. We'd like to be able to say
to Coq, "find a contradictory assumption and invert it" without
giving its name explicitly.
<div class="paragraph"> </div>
Doing <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode"><span class="id" type="tactic">by</span></span> <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> will find a hypothesis that can be
inverted to solve the goal, if there is one. The tactics <span class="inlinecode"><span class="id" type="var">solve</span></span>
<span class="inlinecode"><span class="id" type="tactic">by</span></span> <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode">2</span> and <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode"><span class="id" type="tactic">by</span></span> <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode">3</span> are slightly fancier
versions which will perform two or three inversions in a row, if
necessary, to solve the goal.
<div class="paragraph"> </div>
(These tactics are not actually built into Coq — their
definitions are in <span class="inlinecode"><span class="id" type="var">Sflib</span></span>.)
<div class="paragraph"> </div>
Caution: Overuse of <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode"><span class="id" type="tactic">by</span></span> <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> can lead to slow proof
scripts.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab511"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="tactic">try</span></span> <span class="inlinecode"><span class="id" type="var">solve</span></span> Tactic</h2>
<div class="paragraph"> </div>
If <span class="inlinecode"><span class="id" type="var">t</span></span> is a tactic, then <span class="inlinecode"><span class="id" type="tactic">try</span></span> <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode">[<span class="id" type="var">t</span>]</span> is a tactic that
<div class="paragraph"> </div>
<ul class="doclist">
<li> if <span class="inlinecode"><span class="id" type="var">t</span></span> solves the goal, behaves just like <span class="inlinecode"><span class="id" type="var">t</span></span>, or
</li>
<li> if <span class="inlinecode"><span class="id" type="var">t</span></span> cannot completely solve the goal, does
nothing.
</li>
</ul>
<div class="paragraph"> </div>
More generally, <span class="inlinecode"><span class="id" type="tactic">try</span></span> <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode">[<span class="id" type="var">t1</span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" type="var">t2</span></span> <span class="inlinecode">|</span> <span class="inlinecode">...]</span> will try to solve the
goal by using <span class="inlinecode"><span class="id" type="var">t1</span></span>, <span class="inlinecode"><span class="id" type="var">t2</span></span>, etc. If none of them succeeds in
completely solving the goal, then <span class="inlinecode"><span class="id" type="tactic">try</span></span> <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode">[<span class="id" type="var">t1</span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" type="var">t2</span></span> <span class="inlinecode">|</span> <span class="inlinecode">...]</span> does
nothing.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab512"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="tactic">f_equal</span></span> Tactic</h2>
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" type="tactic">f_equal</span></span> replaces a goal of the form <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">x1</span></span> <span class="inlinecode"><span class="id" type="var">x2</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">xn</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">y1</span></span> <span class="inlinecode"><span class="id" type="var">y2</span></span>
<span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">yn</span></span>, where <span class="inlinecode"><span class="id" type="var">f</span></span> is some function, with the subgoals <span class="inlinecode"><span class="id" type="var">x1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">y1</span></span>,
<span class="inlinecode"><span class="id" type="var">x2</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">y2</span></span>,...,<span class="inlinecode"><span class="id" type="var">xn</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">yn</span></span>. It is useful for avoiding explicit
rewriting steps, and often the generated subgoals can be quickly
cleared by <span class="inlinecode"><span class="id" type="tactic">auto</span></span>. This tactic is not fundamental, in the sense
that it can always be replaced by a sequence of <span class="inlinecode"><span class="id" type="tactic">assert</span></span>s.
However in some cases it can be very handy.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab513"></a><h2 class="section">The <span class="inlinecode"><span class="id" type="var">normalize</span></span> Tactic</h2>
<div class="paragraph"> </div>
When experimenting with definitions of programming languages in
Coq, we often want to see what a particular concrete term steps
to — i.e., we want to find proofs for goals of the form <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>, where <span class="inlinecode"><span class="id" type="var">t</span></span> is a completely concrete term and <span class="inlinecode"><span class="id" type="var">t'</span></span> is unknown.
These proofs are simple but repetitive to do by hand. Consider for
example reducing an arithmetic expression using the small-step
relation <span class="inlinecode"><span class="id" type="var">astep</span></span> defined in the previous chapter:
</div>
<div class="code code-tight">
<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">Definition</span> <span class="id" type="var">amultistep</span> <span class="id" type="var">st</span> := <span class="id" type="var">multi</span> (<span class="id" type="var">astep</span> <span class="id" type="var">st</span>).<br/>
<span class="id" type="keyword">Notation</span> " t '/' st '<span style="font-family: arial;">⇒</span>a*' t' " := (<span class="id" type="var">amultistep</span> <span class="id" type="var">st</span> <span class="id" type="var">t</span> <span class="id" type="var">t'</span>)<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 40, <span class="id" type="var">st</span> <span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 39).<br/>
</div>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">astep_example1</span> : <br/>
(<span class="id" type="var">APlus</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">ANum</span> 4))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* (<span class="id" type="var">ANum</span> 15).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_step</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">ANum</span> 12)).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">AS_Plus2</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">av_num</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">AS_Mult</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_step</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">ANum</span> 15).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">AS_Plus</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_refl</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
We repeatedly applied <span class="inlinecode"><span class="id" type="var">multi_step</span></span> until we got to a normal
form. The proofs that the intermediate steps are possible are
simple enough that <span class="inlinecode"><span class="id" type="tactic">auto</span></span>, with appropriate hints, can solve
them.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">astep</span> <span class="id" type="var">aval</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">astep_example1'</span> : <br/>
(<span class="id" type="var">APlus</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">ANum</span> 4))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* (<span class="id" type="var">ANum</span> 15).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span>. <span class="id" type="tactic">auto</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span>. <span class="id" type="tactic">auto</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_refl</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The following custom <span class="inlinecode"><span class="id" type="keyword">Tactic</span></span> <span class="inlinecode"><span class="id" type="keyword">Notation</span></span> definition captures this
pattern. In addition, before each <span class="inlinecode"><span class="id" type="var">multi_step</span></span> we print out the
current goal, so that the user can follow how the term is being
evaluated.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Tactic Notation</span> "print_goal" := <span class="id" type="keyword">match</span> <span class="id" type="var">goal</span> <span class="id" type="keyword">with</span> <span style="font-family: arial;">⊢</span> ?<span class="id" type="var">x</span> => <span class="id" type="var">idtac</span> <span class="id" type="var">x</span> <span class="id" type="keyword">end</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "normalize" := <br/>
<span class="id" type="tactic">repeat</span> (<span class="id" type="var">print_goal</span>; <span class="id" type="tactic">eapply</span> <span class="id" type="var">multi_step</span> ; <br/>
[ (<span class="id" type="tactic">eauto</span> 10; <span class="id" type="tactic">fail</span>) | (<span class="id" type="var">instantiate</span>; <span class="id" type="tactic">simpl</span>)]);<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">multi_refl</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">astep_example1''</span> : <br/>
(<span class="id" type="var">APlus</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">ANum</span> 4))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* (<span class="id" type="var">ANum</span> 15).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">normalize</span>.<br/>
<span class="comment">(* At this point in the proof script, the Coq response shows <br/>
a trace of how the expression evaluated. <br/>
<br/>
(APlus (ANum 3) (AMult (ANum 3) (ANum 4)) / empty_state ==>a* ANum 15)<br/>
(multi (astep empty_state) (APlus (ANum 3) (ANum 12)) (ANum 15))<br/>
(multi (astep empty_state) (ANum 15) (ANum 15))<br/>
*)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="var">normalize</span></span> tactic also provides a simple way to calculate
what the normal form of a term is, by proving a goal with an
existential variable in it.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">astep_example1'''</span> : <span style="font-family: arial;">∃</span><span class="id" type="var">e'</span>,<br/>
(<span class="id" type="var">APlus</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">ANum</span> 4))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* <span class="id" type="var">e'</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">eapply</span> <span class="id" type="var">ex_intro</span>. <span class="id" type="var">normalize</span>.<br/>
<span class="comment">(* This time the trace will be:<br/>
<br/>
(APlus (ANum 3) (AMult (ANum 3) (ANum 4)) / empty_state ==>a* ??)<br/>
(multi (astep empty_state) (APlus (ANum 3) (ANum 12)) ??)<br/>
(multi (astep empty_state) (ANum 15) ??)<br/>
<br/>
where ?? is the variable ``guessed'' by eapply.<br/>
*)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab514"></a><h4 class="section">Exercise: 1 star (normalize_ex)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">normalize_ex</span> : <span style="font-family: arial;">∃</span><span class="id" type="var">e'</span>,<br/>
(<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 2) (<span class="id" type="var">ANum</span> 1))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* <span class="id" type="var">e'</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab515"></a><h4 class="section">Exercise: 1 star, optional (normalize_ex')</h4>
This time prove it by using <span class="inlinecode"><span class="id" type="tactic">apply</span></span> instead of <span class="inlinecode"><span class="id" type="tactic">eapply</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">normalize_ex'</span> : <span style="font-family: arial;">∃</span><span class="id" type="var">e'</span>,<br/>
(<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 3) (<span class="id" type="var">AMult</span> (<span class="id" type="var">ANum</span> 2) (<span class="id" type="var">ANum</span> 1))) / <span class="id" type="var">empty_state</span> <br/>
<span style="font-family: arial;">⇒</span><span class="id" type="var">a</span>* <span class="id" type="var">e'</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="lab516"></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. The only thing to
notice is that we are <i>not</i> using the <span class="inlinecode"><span class="id" type="var">asnum</span></span>/<span class="inlinecode"><span class="id" type="var">aslist</span></span> trick that
we used in chapter <span class="inlinecode"><span class="id" type="var">ImpList</span></span> to make all the operations total by
forcibly coercing the arguments to <span class="inlinecode">+</span> (for example) into numbers.
Instead, we simply let terms get stuck if they try to use an
operator with the wrong kind of operands: the <span class="inlinecode"><span class="id" type="var">step</span></span> relation
doesn't relate them to anything.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab517"></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="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">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/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab518"></a><h2 class="section">Operational Semantics</h2>
<div class="paragraph"> </div>
Informally:
<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 t1 else t2 <span style="font-family: arial;">⇒</span> t1</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 t1 else t2 <span style="font-family: arial;">⇒</span> t2</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 then t2 else t3 <span style="font-family: arial;">⇒</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">if t1' then t2 else t3</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> succ t1'</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 v1</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 v1) <span style="font-family: arial;">⇒</span> v1</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> pred t1'</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 v1</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 v1) <span style="font-family: arial;">⇒</span> false</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">t1 <span style="font-family: arial;">⇒</span> t1'</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 t1 <span style="font-family: arial;">⇒</span> iszero t1'</td>
<td></td>
</td>
</table></center> Formally:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Reserved Notation</span> "t1 '<span style="font-family: arial;">⇒</span>' t2" (<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">t1</span> <span class="id" type="var">t2</span>,<br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">ttrue</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1</span><br/>
| <span class="id" type="var">ST_IfFalse</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t2</span>,<br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">tfalse</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>) <span style="font-family: arial;">⇒</span> <span class="id" type="var">t2</span><br/>
| <span class="id" type="var">ST_If</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tif</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tif</span> <span class="id" type="var">t1'</span> <span class="id" type="var">t2</span> <span class="id" type="var">t3</span>)<br/>
| <span class="id" type="var">ST_Succ</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t1'</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tsucc</span> <span class="id" type="var">t1</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tsucc</span> <span class="id" type="var">t1'</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">t1</span>,<br/>
<span class="id" type="var">nvalue</span> <span class="id" type="var">t1</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">t1</span>)) <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1</span><br/>
| <span class="id" type="var">ST_Pred</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">t1</span> <span class="id" type="var">t1'</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tpred</span> <span class="id" type="var">t1</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tpred</span> <span class="id" type="var">t1'</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">t1</span>,<br/>
<span class="id" type="var">nvalue</span> <span class="id" type="var">t1</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">t1</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">t1</span> <span class="id" type="var">t1'</span>,<br/>
<span class="id" type="var">t1</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t1'</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">tiszero</span> <span class="id" type="var">t1</span>) <span style="font-family: arial;">⇒</span> (<span class="id" type="var">tiszero</span> <span class="id" type="var">t1'</span>)<br/>
<br/>
<span class="id" type="keyword">where</span> "t1 '<span style="font-family: arial;">⇒</span>' t2" := (<span class="id" type="var">step</span> <span class="id" type="var">t1</span> <span class="id" type="var">t2</span>).<br/>
<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">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_IfTrue" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_IfFalse" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_If" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Succ" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_PredZero"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_PredSucc" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Pred" <br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_IszeroZero" | <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_IszeroSucc"<br/>
| <span class="id" type="var">Case_aux</span> <span class="id" type="var">c</span> "ST_Iszero" ].<br/>
<br/>
<span class="id" type="keyword">Hint</span> <span class="id" type="var">Constructors</span> <span class="id" type="var">step</span>.<br/>
</div>
<br/>
</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. 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 <i>one</i> step.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab519"></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="lab520"></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="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>
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="lab521"></a><h4 class="section">Exercise: 3 stars, optional (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="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">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="lab522"></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="lab523"></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> t1 : Bool <span style="font-family: arial;">⊢</span> t2 : T <span style="font-family: arial;">⊢</span> t3 : 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 t1 then t2 else t3 : T</td>
<td></td>