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BasicTactics.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>BasicTactics: Additional Basic Coq Tactics</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">BasicTactics<span class="subtitle">Additional Basic Coq Tactics</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">Poly</span>.<br/>
<br/>
</div>
<div class="doc">
This chapter introduces several more proof strategies and
tactics that, together, allow us to prove theorems about the
functional programs we have been writing. In particular, we'll
reason about functions that work with natural numbers and
lists. We will see:
<div class="paragraph"> </div>
<ul class="doclist">
<li> how to use auxiliary lemmas, in both forwards and backwards reasoning;
</li>
<li> how to reason about data constructors, which are injective and disjoint;
</li>
<li> how to create a strong induction hypothesis (and when
strengthening is required); and
</li>
<li> how to reason by case analysis.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab160"></a><h1 class="section">The <span class="inlinecode"><span class="id" type="tactic">apply</span></span> Tactic</h1>
<div class="paragraph"> </div>
We often encounter situations where the goal to be proved is
exactly the same as some hypothesis in the context or some
previously proved lemma.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly1</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">p</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>;<span class="id" type="var">o</span>] = [<span class="id" type="var">n</span>;<span class="id" type="var">p</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>;<span class="id" type="var">o</span>] = [<span class="id" type="var">m</span>;<span class="id" type="var">p</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">p</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">eq1</span>.<br/>
<span class="comment">(* At this point, we could finish with <br/>
"<span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">eq2</span>.</span> <span class="inlinecode"><span class="id" type="tactic">reflexivity</span>.</span>" as we have <br/>
done several times above. But we can achieve the<br/>
same effect in a single step by using the <br/>
<span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic instead: *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">eq2</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic also works with <i>conditional</i> hypotheses
and lemmas: if the statement being applied is an implication, then
the premises of this implication will be added to the list of
subgoals needing to be proved.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly2</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">p</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">q</span> <span class="id" type="var">r</span> : <span class="id" type="var">nat</span>), <span class="id" type="var">q</span> = <span class="id" type="var">r</span> <span style="font-family: arial;">→</span> [<span class="id" type="var">q</span>;<span class="id" type="var">o</span>] = [<span class="id" type="var">r</span>;<span class="id" type="var">p</span>]) <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>;<span class="id" type="var">o</span>] = [<span class="id" type="var">m</span>;<span class="id" type="var">p</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">p</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">eq2</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">eq1</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
You may find it instructive to experiment with this proof
and see if there is a way to complete it using just <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span>
instead of <span class="inlinecode"><span class="id" type="tactic">apply</span></span>.
<div class="paragraph"> </div>
Typically, when we use <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span>, the statement <span class="inlinecode"><span class="id" type="var">H</span></span> will
begin with a <span class="inlinecode"><span style="font-family: arial;">∀</span></span> binding some <i>universal variables</i>. When
Coq matches the current goal against the conclusion of <span class="inlinecode"><span class="id" type="var">H</span></span>, it
will try to find appropriate values for these variables. For
example, when we do <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">eq2</span></span> in the following proof, the
universal variable <span class="inlinecode"><span class="id" type="var">q</span></span> in <span class="inlinecode"><span class="id" type="var">eq2</span></span> gets instantiated with <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">r</span></span>
gets instantiated with <span class="inlinecode"><span class="id" type="var">m</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly2a</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>),<br/>
(<span class="id" type="var">n</span>,<span class="id" type="var">n</span>) = (<span class="id" type="var">m</span>,<span class="id" type="var">m</span>) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">q</span> <span class="id" type="var">r</span> : <span class="id" type="var">nat</span>), (<span class="id" type="var">q</span>,<span class="id" type="var">q</span>) = (<span class="id" type="var">r</span>,<span class="id" type="var">r</span>) <span style="font-family: arial;">→</span> [<span class="id" type="var">q</span>] = [<span class="id" type="var">r</span>]) <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>] = [<span class="id" type="var">m</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">eq2</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">eq1</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab161"></a><h4 class="section">Exercise: 2 stars, optional (silly_ex)</h4>
Complete the following proof without using <span class="inlinecode"><span class="id" type="tactic">simpl</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly_ex</span> : <br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">evenb</span> <span class="id" type="var">n</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">oddb</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) = <span class="id" type="var">true</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">evenb</span> 3 = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">oddb</span> 4 = <span class="id" type="var">true</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 class="paragraph"> </div>
To use the <span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic, the (conclusion of the) fact
being applied must match the goal <i>exactly</i> — for example, <span class="inlinecode"><span class="id" type="tactic">apply</span></span>
will not work if the left and right sides of the equality are
swapped.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly3_firsttry</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">true</span> = <span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 5 <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) 7 = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="comment">(* Here we cannot use <span class="inlinecode"><span class="id" type="tactic">apply</span></span> directly *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
In this case we can use the <span class="inlinecode"><span class="id" type="tactic">symmetry</span></span> tactic, which switches the
left and right sides of an equality in the goal.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly3</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">true</span> = <span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 5 <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) 7 = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">symmetry</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* Actually, this <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> is unnecessary, since <br/>
<span class="inlinecode"><span class="id" type="tactic">apply</span></span> will perform simplification first. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab162"></a><h4 class="section">Exercise: 3 stars (apply_exercise1)</h4>
Hint: you can use <span class="inlinecode"><span class="id" type="tactic">apply</span></span> with previously defined lemmas, not
just hypotheses in the context. Remember that <span class="inlinecode"><span class="id" type="var">SearchAbout</span></span> is
your friend.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">rev_exercise1</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">l</span> <span class="id" type="var">l'</span> : <span class="id" type="var">list</span> <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">l</span> = <span class="id" type="var">rev</span> <span class="id" type="var">l'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">l'</span> = <span class="id" type="var">rev</span> <span class="id" type="var">l</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 class="paragraph"> </div>
<a name="lab163"></a><h4 class="section">Exercise: 1 star, optional (apply_rewrite)</h4>
Briefly explain the difference between the tactics <span class="inlinecode"><span class="id" type="tactic">apply</span></span> and
<span class="inlinecode"><span class="id" type="tactic">rewrite</span></span>. Are there situations where both can usefully be
applied?
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab164"></a><h1 class="section">The <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">...</span> Tactic</h1>
<div class="paragraph"> </div>
The following silly example uses two rewrites in a row to
get from <span class="inlinecode">[<span class="id" type="var">a</span>,<span class="id" type="var">b</span>]</span> to <span class="inlinecode">[<span class="id" type="var">e</span>,<span class="id" type="var">f</span>]</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">trans_eq_example</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> <span class="id" type="var">e</span> <span class="id" type="var">f</span> : <span class="id" type="var">nat</span>),<br/>
[<span class="id" type="var">a</span>;<span class="id" type="var">b</span>] = [<span class="id" type="var">c</span>;<span class="id" type="var">d</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">c</span>;<span class="id" type="var">d</span>] = [<span class="id" type="var">e</span>;<span class="id" type="var">f</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">a</span>;<span class="id" type="var">b</span>] = [<span class="id" type="var">e</span>;<span class="id" type="var">f</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> <span class="id" type="var">e</span> <span class="id" type="var">f</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">eq1</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">eq2</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Since this is a common pattern, we might
abstract it out as a lemma recording once and for all
the fact that equality is transitive.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">trans_eq</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> : <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">m</span> = <span class="id" type="var">o</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = <span class="id" type="var">o</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">X</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">eq1</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">eq2</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Now, we should be able to use <span class="inlinecode"><span class="id" type="var">trans_eq</span></span> to
prove the above example. However, to do this we need
a slight refinement of the <span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">trans_eq_example'</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> <span class="id" type="var">e</span> <span class="id" type="var">f</span> : <span class="id" type="var">nat</span>),<br/>
[<span class="id" type="var">a</span>;<span class="id" type="var">b</span>] = [<span class="id" type="var">c</span>;<span class="id" type="var">d</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">c</span>;<span class="id" type="var">d</span>] = [<span class="id" type="var">e</span>;<span class="id" type="var">f</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">a</span>;<span class="id" type="var">b</span>] = [<span class="id" type="var">e</span>;<span class="id" type="var">f</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> <span class="id" type="var">e</span> <span class="id" type="var">f</span> <span class="id" type="var">eq1</span> <span class="id" type="var">eq2</span>.<br/>
<span class="comment">(* If we simply tell Coq <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">trans_eq</span></span> at this point,<br/>
it can tell (by matching the goal against the<br/>
conclusion of the lemma) that it should instantiate <span class="inlinecode"><span class="id" type="var">X</span></span><br/>
with <span class="inlinecode">[<span class="id" type="var">nat</span>]</span>, <span class="inlinecode"><span class="id" type="var">n</span></span> with <span class="inlinecode">[<span class="id" type="var">a</span>,<span class="id" type="var">b</span>]</span>, and <span class="inlinecode"><span class="id" type="var">o</span></span> with <span class="inlinecode">[<span class="id" type="var">e</span>,<span class="id" type="var">f</span>]</span>.<br/>
However, the matching process doesn't determine an<br/>
instantiation for <span class="inlinecode"><span class="id" type="var">m</span></span>: we have to supply one explicitly<br/>
by adding <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">(<span class="id" type="var">m</span>:=[<span class="id" type="var">c</span>,<span class="id" type="var">d</span>])</span> to the invocation of<br/>
<span class="inlinecode"><span class="id" type="tactic">apply</span></span>. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">trans_eq</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">m</span>:=[<span class="id" type="var">c</span>;<span class="id" type="var">d</span>]). <span class="id" type="tactic">apply</span> <span class="id" type="var">eq1</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">eq2</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Actually, we usually don't have to include the name <span class="inlinecode"><span class="id" type="var">m</span></span>
in the <span class="inlinecode"><span class="id" type="keyword">with</span></span> clause; Coq is often smart enough to
figure out which instantiation we're giving. We could
instead write: <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">trans_eq</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">[<span class="id" type="var">c</span>,<span class="id" type="var">d</span>]</span>.
<div class="paragraph"> </div>
<a name="lab165"></a><h4 class="section">Exercise: 3 stars, optional (apply_with_exercise)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">trans_eq_exercise</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">p</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">m</span> = (<span class="id" type="var">minustwo</span> <span class="id" type="var">o</span>) <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">n</span> + <span class="id" type="var">p</span>) = <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">n</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">minustwo</span> <span class="id" type="var">o</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="lab166"></a><h1 class="section">The <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic</h1>
<div class="paragraph"> </div>
Recall the definition of natural numbers:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">nat</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">O</span> : <span class="id" type="var">nat</span><br/>
| <span class="id" type="var">S</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span>.
<div class="paragraph"> </div>
</div>
It is clear from this definition that every number has one of two
forms: either it is the constructor <span class="inlinecode"><span class="id" type="var">O</span></span> or it is built by applying
the constructor <span class="inlinecode"><span class="id" type="var">S</span></span> to another number. But there is more here than
meets the eye: implicit in the definition (and in our informal
understanding of how datatype declarations work in other
programming languages) are two other facts:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The constructor <span class="inlinecode"><span class="id" type="var">S</span></span> is <i>injective</i>. That is, the only way we can
have <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> is if <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>.
<div class="paragraph"> </div>
</li>
<li> The constructors <span class="inlinecode"><span class="id" type="var">O</span></span> and <span class="inlinecode"><span class="id" type="var">S</span></span> are <i>disjoint</i>. That is, <span class="inlinecode"><span class="id" type="var">O</span></span> is not
equal to <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> for any <span class="inlinecode"><span class="id" type="var">n</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
Similar principles apply to all inductively defined types: all
constructors are injective, and the values built from distinct
constructors are never equal. For lists, the <span class="inlinecode"><span class="id" type="var">cons</span></span> constructor is
injective and <span class="inlinecode"><span class="id" type="var">nil</span></span> is different from every non-empty list. For
booleans, <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span> are unequal. (Since neither <span class="inlinecode"><span class="id" type="var">true</span></span>
nor <span class="inlinecode"><span class="id" type="var">false</span></span> take any arguments, their injectivity is not an issue.)
<div class="paragraph"> </div>
Coq provides a tactic called <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> that allows us to exploit
these principles in proofs.
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic is used like this. Suppose <span class="inlinecode"><span class="id" type="var">H</span></span> is a
hypothesis in the context (or a previously proven lemma) of the
form
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">c</span> <span class="id" type="var">a1</span> <span class="id" type="var">a2</span> ... <span class="id" type="var">an</span> = <span class="id" type="var">d</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> ... <span class="id" type="var">bm</span>
<div class="paragraph"> </div>
</div>
for some constructors <span class="inlinecode"><span class="id" type="var">c</span></span> and <span class="inlinecode"><span class="id" type="var">d</span></span> and arguments <span class="inlinecode"><span class="id" type="var">a1</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">an</span></span> and
<span class="inlinecode"><span class="id" type="var">b1</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">bm</span></span>. Then <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> instructs Coq to "invert" this
equality to extract the information it contains about these terms:
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">c</span></span> and <span class="inlinecode"><span class="id" type="var">d</span></span> are the same constructor, then we know, by the
injectivity of this constructor, that <span class="inlinecode"><span class="id" type="var">a1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">b1</span></span>, <span class="inlinecode"><span class="id" type="var">a2</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">b2</span></span>,
etc.; <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> adds these facts to the context, and tries
to use them to rewrite the goal.
<div class="paragraph"> </div>
</li>
<li> If <span class="inlinecode"><span class="id" type="var">c</span></span> and <span class="inlinecode"><span class="id" type="var">d</span></span> are different constructors, then the hypothesis
<span class="inlinecode"><span class="id" type="var">H</span></span> is contradictory. That is, a false assumption has crept
into the context, and this means that any goal whatsoever is
provable! In this case, <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> marks the current goal as
completed and pops it off the goal stack.
</li>
</ul>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic is probably easier to understand by
seeing it in action than from general descriptions like the above.
Below you will find example theorems that demonstrate the use of
<span class="inlinecode"><span class="id" type="tactic">inversion</span></span> and exercises to test your understanding.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">eq_add_S</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">S</span> <span class="id" type="var">n</span> = <span class="id" type="var">S</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly4</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>),<br/>
[<span class="id" type="var">n</span>] = [<span class="id" type="var">m</span>] <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
As a convenience, the <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic can also
destruct equalities between complex values, binding
multiple variables as it goes.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly5</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> : <span class="id" type="var">nat</span>),<br/>
[<span class="id" type="var">n</span>;<span class="id" type="var">m</span>] = [<span class="id" type="var">o</span>;<span class="id" type="var">o</span>] <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>] = [<span class="id" type="var">m</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab167"></a><h4 class="section">Exercise: 1 star (sillyex1)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">sillyex1</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span> : <span class="id" type="var">X</span>) (<span class="id" type="var">l</span> <span class="id" type="var">j</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">x</span> :: <span class="id" type="var">y</span> :: <span class="id" type="var">l</span> = <span class="id" type="var">z</span> :: <span class="id" type="var">j</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">y</span> :: <span class="id" type="var">l</span> = <span class="id" type="var">x</span> :: <span class="id" type="var">j</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">x</span> = <span class="id" type="var">y</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/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly6</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">S</span> <span class="id" type="var">n</span> = <span class="id" type="var">O</span> <span style="font-family: arial;">→</span><br/>
2 + 2 = 5.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</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="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly7</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">false</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span><br/>
[<span class="id" type="var">n</span>] = [<span class="id" type="var">m</span>].<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</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="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab168"></a><h4 class="section">Exercise: 1 star (sillyex2)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Example</span> <span class="id" type="var">sillyex2</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span> : <span class="id" type="var">X</span>) (<span class="id" type="var">l</span> <span class="id" type="var">j</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">x</span> :: <span class="id" type="var">y</span> :: <span class="id" type="var">l</span> = [] <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">y</span> :: <span class="id" type="var">l</span> = <span class="id" type="var">z</span> :: <span class="id" type="var">j</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">x</span> = <span class="id" type="var">z</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 class="paragraph"> </div>
While the injectivity of constructors allows us to reason
<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">nat</span>),</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>, the reverse direction of
the implication is an instance of a more general fact about
constructors and functions, which we will often find useful:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="tactic">f_equal</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">A</span> <span class="id" type="var">B</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">f</span>: <span class="id" type="var">A</span> <span style="font-family: arial;">→</span> <span class="id" type="var">B</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span>: <span class="id" type="var">A</span>), <br/>
<span class="id" type="var">x</span> = <span class="id" type="var">y</span> <span style="font-family: arial;">→</span> <span class="id" type="var">f</span> <span class="id" type="var">x</span> = <span class="id" type="var">f</span> <span class="id" type="var">y</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">B</span> <span class="id" type="var">f</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab169"></a><h4 class="section">Exercise: 2 stars, optional (practice)</h4>
A couple more nontrivial but not-too-complicated proofs to work
together in class, or for you to work as exercises.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beq_nat_0_l</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">beq_nat</span> 0 <span class="id" type="var">n</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beq_nat_0_r</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 0 = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = 0.<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="lab170"></a><h1 class="section">Using Tactics on Hypotheses</h1>
<div class="paragraph"> </div>
By default, most tactics work on the goal formula and leave
the context unchanged. However, most tactics also have a variant
that performs a similar operation on a statement in the context.
<div class="paragraph"> </div>
For example, the tactic <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> performs simplification in
the hypothesis named <span class="inlinecode"><span class="id" type="var">H</span></span> in the context.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">S_inj</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">b</span> : <span class="id" type="var">bool</span>),<br/>
<span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) (<span class="id" type="var">S</span> <span class="id" type="var">m</span>) = <span class="id" type="var">b</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">b</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">b</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Similarly, the tactic <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">L</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> matches some
conditional statement <span class="inlinecode"><span class="id" type="var">L</span></span> (of the form <span class="inlinecode"><span class="id" type="var">L1</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">L2</span></span>, say) against a
hypothesis <span class="inlinecode"><span class="id" type="var">H</span></span> in the context. However, unlike ordinary
<span class="inlinecode"><span class="id" type="tactic">apply</span></span> (which rewrites a goal matching <span class="inlinecode"><span class="id" type="var">L2</span></span> into a subgoal <span class="inlinecode"><span class="id" type="var">L1</span></span>),
<span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">L</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> matches <span class="inlinecode"><span class="id" type="var">H</span></span> against <span class="inlinecode"><span class="id" type="var">L1</span></span> and, if successful,
replaces it with <span class="inlinecode"><span class="id" type="var">L2</span></span>.
<div class="paragraph"> </div>
In other words, <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">L</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> gives us a form of "forward
reasoning" — from <span class="inlinecode"><span class="id" type="var">L1</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">L2</span></span> and a hypothesis matching <span class="inlinecode"><span class="id" type="var">L1</span></span>, it
gives us a hypothesis matching <span class="inlinecode"><span class="id" type="var">L2</span></span>. By contrast, <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">L</span></span> is
"backward reasoning" — it says that if we know <span class="inlinecode"><span class="id" type="var">L1</span><span style="font-family: arial;">→</span><span class="id" type="var">L2</span></span> and we
are trying to prove <span class="inlinecode"><span class="id" type="var">L2</span></span>, it suffices to prove <span class="inlinecode"><span class="id" type="var">L1</span></span>.
<div class="paragraph"> </div>
Here is a variant of a proof from above, using forward reasoning
throughout instead of backward reasoning.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">silly3'</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
(<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 5 = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) 7 = <span class="id" type="var">true</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">true</span> = <span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> 5 <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">true</span> = <span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) 7.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">eq</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">eq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Forward reasoning starts from what is <i>given</i> (premises,
previously proven theorems) and iteratively draws conclusions from
them until the goal is reached. Backward reasoning starts from
the <i>goal</i>, and iteratively reasons about what would imply the
goal, until premises or previously proven theorems are reached.
If you've seen informal proofs before (for example, in a math or
computer science class), they probably used forward reasoning. In
general, Coq tends to favor backward reasoning, but in some
situations the forward style can be easier to use or to think
about.
<div class="paragraph"> </div>
<a name="lab171"></a><h4 class="section">Exercise: 3 stars (plus_n_n_injective)</h4>
Practice using "in" variants in this exercise.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_n_n_injective</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">n</span> + <span class="id" type="var">n</span> = <span class="id" type="var">m</span> + <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="comment">(* Hint: use the plus_n_Sm lemma *)</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="lab172"></a><h1 class="section">Varying the Induction Hypothesis</h1>
<div class="paragraph"> </div>
Sometimes it is important to control the exact form of the
induction hypothesis when carrying out inductive proofs in Coq.
In particular, we need to be careful about which of the
assumptions we move (using <span class="inlinecode"><span class="id" type="tactic">intros</span></span>) from the goal to the context
before invoking the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic. For example, suppose
we want to show that the <span class="inlinecode"><span class="id" type="var">double</span></span> function is injective — i.e.,
that it always maps different arguments to different results:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_injective</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>, <span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">double</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = <span class="id" type="var">m</span>.
<div class="paragraph"> </div>
</div>
The way we <i>start</i> this proof is a little bit delicate: if we
begin it with
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span>.
<div class="paragraph"> </div>
</div>
all is well. But if we begin it with
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span>.
<div class="paragraph"> </div>
</div>
we get stuck in the middle of the inductive case...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_injective_FAILED</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">double</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
+ <span class="comment">(* m = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
+ <span class="comment">(* m = O *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
+ <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="tactic">f_equal</span>.<br/>
<span class="comment">(* Here we are stuck. The induction hypothesis, <span class="inlinecode"><span class="id" type="var">IHn'</span></span>, does<br/>
not give us <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m'</span></span> -- there is an extra <span class="inlinecode"><span class="id" type="var">S</span></span> in the<br/>
way -- so the goal is not provable. *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
What went wrong?
<div class="paragraph"> </div>
The problem is that, at the point we invoke the induction
hypothesis, we have already introduced <span class="inlinecode"><span class="id" type="var">m</span></span> into the context —
intuitively, we have told Coq, "Let's consider some particular
<span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span>..." and we now have to prove that, if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> for <i>these particular</i> <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span>, then <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>.
<div class="paragraph"> </div>
The next tactic, <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> says to Coq: We are going to show
the goal by induction on <span class="inlinecode"><span class="id" type="var">n</span></span>. That is, we are going to prove that
the proposition
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> = "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>, then <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>"
</li>
</ul>
<div class="paragraph"> </div>
holds for all <span class="inlinecode"><span class="id" type="var">n</span></span> by showing
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span>
<div class="paragraph"> </div>
(i.e., "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> then <span class="inlinecode"><span class="id" type="var">O</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>")
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>
<div class="paragraph"> </div>
(i.e., "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> then <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>" implies "if
<span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> then <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>").
</li>
</ul>
<div class="paragraph"> </div>
If we look closely at the second statement, it is saying something
rather strange: it says that, for a <i>particular</i> <span class="inlinecode"><span class="id" type="var">m</span></span>, if we know
<div class="paragraph"> </div>
<ul class="doclist">
<li> "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> then <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>"
</li>
</ul>
<div class="paragraph"> </div>
then we can prove
<div class="paragraph"> </div>
<ul class="doclist">
<li> "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> then <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>".
</li>
</ul>
<div class="paragraph"> </div>
To see why this is strange, let's think of a particular <span class="inlinecode"><span class="id" type="var">m</span></span> —
say, <span class="inlinecode">5</span>. The statement is then saying that, if we know
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">Q</span></span> = "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">10</span> then <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">5</span>"
</li>
</ul>
<div class="paragraph"> </div>
then we can prove
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">R</span></span> = "if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">10</span> then <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">5</span>".
</li>
</ul>
<div class="paragraph"> </div>
But knowing <span class="inlinecode"><span class="id" type="var">Q</span></span> doesn't give us any help with proving <span class="inlinecode"><span class="id" type="var">R</span></span>! (If we
tried to prove <span class="inlinecode"><span class="id" type="var">R</span></span> from <span class="inlinecode"><span class="id" type="var">Q</span></span>, we would say something like "Suppose
<span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">10</span>..." but then we'd be stuck: knowing that
<span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> is <span class="inlinecode">10</span> tells us nothing about whether <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>
is <span class="inlinecode">10</span>, so <span class="inlinecode"><span class="id" type="var">Q</span></span> is useless at this point.)
<div class="paragraph"> </div>
To summarize: Trying to carry out this proof by induction on <span class="inlinecode"><span class="id" type="var">n</span></span>
when <span class="inlinecode"><span class="id" type="var">m</span></span> is already in the context doesn't work because we are
trying to prove a relation involving <i>every</i> <span class="inlinecode"><span class="id" type="var">n</span></span> but just a
<i>single</i> <span class="inlinecode"><span class="id" type="var">m</span></span>.
<div class="paragraph"> </div>
The good proof of <span class="inlinecode"><span class="id" type="var">double_injective</span></span> leaves <span class="inlinecode"><span class="id" type="var">m</span></span> in the goal
statement at the point where the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic is invoked on
<span class="inlinecode"><span class="id" type="var">n</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_injective</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">double</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">m</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
+ <span class="comment">(* m = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="comment">(* Notice that both the goal and the induction<br/>
hypothesis have changed: the goal asks us to prove<br/>
something more general (i.e., to prove the<br/>
statement for _every_ <span class="inlinecode"><span class="id" type="var">m</span></span>), but the IH is<br/>
correspondingly more flexible, allowing us to<br/>
choose any <span class="inlinecode"><span class="id" type="var">m</span></span> we like when we apply the IH. *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">m</span> <span class="id" type="var">eq</span>.<br/>
<span class="comment">(* Now we choose a particular <span class="inlinecode"><span class="id" type="var">m</span></span> and introduce the<br/>
assumption that <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>. Since we<br/>
are doing a case analysis on <span class="inlinecode"><span class="id" type="var">n</span></span>, we need a case<br/>
analysis on <span class="inlinecode"><span class="id" type="var">m</span></span> to keep the two "in sync." *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
+ <span class="comment">(* m = O *)</span><br/>
<span class="comment">(* The 0 case is trivial *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
+ <span class="comment">(* m = S m' *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="tactic">f_equal</span>.<br/>
<span class="comment">(* At this point, since we are in the second<br/>
branch of the <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>, the <span class="inlinecode"><span class="id" type="var">m'</span></span> mentioned<br/>
in the context at this point is actually the<br/>
predecessor of the one we started out talking<br/>
about. Since we are also in the <span class="inlinecode"><span class="id" type="var">S</span></span> branch of<br/>
the induction, this is perfect: if we<br/>
instantiate the generic <span class="inlinecode"><span class="id" type="var">m</span></span> in the IH with the<br/>
<span class="inlinecode"><span class="id" type="var">m'</span></span> that we are talking about right now (this<br/>
instantiation is performed automatically by<br/>
<span class="inlinecode"><span class="id" type="tactic">apply</span></span>), then <span class="inlinecode"><span class="id" type="var">IHn'</span></span> gives us exactly what we<br/>
need to finish the proof. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
What this teaches us is that we need to be careful about using
induction to try to prove something too specific: If we're proving
a property of <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span> by induction on <span class="inlinecode"><span class="id" type="var">n</span></span>, we may need to
leave <span class="inlinecode"><span class="id" type="var">m</span></span> generic.
<div class="paragraph"> </div>
The proof of this theorem (left as an exercise) has to be treated similarly:
<div class="paragraph"> </div>
<a name="lab173"></a><h4 class="section">Exercise: 2 stars (beq_nat_true)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beq_nat_true</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = <span class="id" type="var">m</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 class="paragraph"> </div>
<a name="lab174"></a><h4 class="section">Exercise: 2 stars, advanced (beq_nat_true_informal)</h4>
Give a careful informal proof of <span class="inlinecode"><span class="id" type="var">beq_nat_true</span></span>, being as explicit
as possible about quantifiers.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
The strategy of doing fewer <span class="inlinecode"><span class="id" type="tactic">intros</span></span> before an <span class="inlinecode"><span class="id" type="tactic">induction</span></span> doesn't
always work directly; sometimes a little <i>rearrangement</i> of
quantified variables is needed. Suppose, for example, that we
wanted to prove <span class="inlinecode"><span class="id" type="var">double_injective</span></span> by induction on <span class="inlinecode"><span class="id" type="var">m</span></span> instead of
<span class="inlinecode"><span class="id" type="var">n</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_injective_take2_FAILED</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">double</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
- <span class="comment">(* m = O *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
+ <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
- <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
+ <span class="comment">(* n = O *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
+ <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="tactic">f_equal</span>.<br/>
<span class="comment">(* Stuck again here, just like before. *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
The problem is that, to do induction on <span class="inlinecode"><span class="id" type="var">m</span></span>, we must first
introduce <span class="inlinecode"><span class="id" type="var">n</span></span>. (If we simply say <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> without
introducing anything first, Coq will automatically introduce
<span class="inlinecode"><span class="id" type="var">n</span></span> for us!)
<div class="paragraph"> </div>
What can we do about this? One possibility is to rewrite the
statement of the lemma so that <span class="inlinecode"><span class="id" type="var">m</span></span> is quantified before <span class="inlinecode"><span class="id" type="var">n</span></span>. This
will work, but it's not nice: We don't want to have to mangle the
statements of lemmas to fit the needs of a particular strategy for
proving them — we want to state them in the most clear and
natural way.
<div class="paragraph"> </div>
What we can do instead is to first introduce all the
quantified variables and then <i>re-generalize</i> one or more of
them, taking them out of the context and putting them back at
the beginning of the goal. The <span class="inlinecode"><span class="id" type="tactic">generalize</span></span> <span class="inlinecode"><span class="id" type="tactic">dependent</span></span> tactic
does this.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_injective_take2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">double</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> = <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span> are both in the context *)</span><br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">n</span>.<br/>
<span class="comment">(* Now <span class="inlinecode"><span class="id" type="var">n</span></span> is back in the goal and we can do induction on<br/>
<span class="inlinecode"><span class="id" type="var">m</span></span> and get a sufficiently general IH. *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
- <span class="comment">(* m = O *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
+ <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
- <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
+ <span class="comment">(* n = O *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>.<br/>
+ <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="tactic">f_equal</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHm'</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Let's look at an informal proof of this theorem. Note that
the proposition we prove by induction leaves <span class="inlinecode"><span class="id" type="var">n</span></span> quantified,
corresponding to the use of generalize dependent in our formal
proof.
<div class="paragraph"> </div>
<i>Theorem</i>: For any nats <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span>, if <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">double</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>, then
<span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span>.