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MoreLogic.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
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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>MoreLogic: More on Logic in Coq</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
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<div id="page">
<div id="header">
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<div id="main">
<h1 class="libtitle">MoreLogic<span class="subtitle">More on Logic in Coq</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> "Prop".<br/>
<br/>
</div>
<div class="doc">
<a name="lab286"></a><h1 class="section">Existential Quantification</h1>
<div class="paragraph"> </div>
Another critical logical connective is <i>existential
quantification</i>. We can express it with the following
definition:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ex</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">X</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">ex_intro</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">witness</span>:<span class="id" type="var">X</span>), <span class="id" type="var">P</span> <span class="id" type="var">witness</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ex</span> <span class="id" type="var">X</span> <span class="id" type="var">P</span>.<br/>
<br/>
</div>
<div class="doc">
That is, <span class="inlinecode"><span class="id" type="var">ex</span></span> is a family of propositions indexed by a type <span class="inlinecode"><span class="id" type="var">X</span></span>
and a property <span class="inlinecode"><span class="id" type="var">P</span></span> over <span class="inlinecode"><span class="id" type="var">X</span></span>. In order to give evidence for the
assertion "there exists an <span class="inlinecode"><span class="id" type="var">x</span></span> for which the property <span class="inlinecode"><span class="id" type="var">P</span></span> holds"
we must actually name a <i>witness</i> — a specific value <span class="inlinecode"><span class="id" type="var">x</span></span> — and
then give evidence for <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span>, i.e., evidence that <span class="inlinecode"><span class="id" type="var">x</span></span> has the
property <span class="inlinecode"><span class="id" type="var">P</span></span>.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<a name="lab287"></a><h3 class="section"> </h3>
Coq's <span class="inlinecode"><span class="id" type="keyword">Notation</span></span> facility can be used to introduce more
familiar notation for writing existentially quantified
propositions, exactly parallel to the built-in syntax for
universally quantified propositions. Instead of writing <span class="inlinecode"><span class="id" type="var">ex</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span>
<span class="inlinecode"><span class="id" type="var">ev</span></span> to express the proposition that there exists some number that
is even, for example, we can write <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">x</span>:<span class="id" type="var">nat</span>,</span> <span class="inlinecode"><span class="id" type="var">ev</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span>. (It is
not necessary to understand exactly how the <span class="inlinecode"><span class="id" type="keyword">Notation</span></span> definition
works.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "'exists' x , p" := (<span class="id" type="var">ex</span> <span class="id" type="var">_</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span> ⇒ <span class="id" type="var">p</span>))<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 200, <span class="id" type="var">x</span> <span class="id" type="var">ident</span>, <span class="id" type="var">right</span> <span class="id" type="var">associativity</span>) : <span class="id" type="var">type_scope</span>.<br/>
<span class="id" type="keyword">Notation</span> "'exists' x : X , p" := (<span class="id" type="var">ex</span> <span class="id" type="var">_</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">x</span>:<span class="id" type="var">X</span> ⇒ <span class="id" type="var">p</span>))<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 200, <span class="id" type="var">x</span> <span class="id" type="var">ident</span>, <span class="id" type="var">right</span> <span class="id" type="var">associativity</span>) : <span class="id" type="var">type_scope</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab288"></a><h3 class="section"> </h3>
We can use the usual set of tactics for
manipulating existentials. For example, to prove an
existential, we can <span class="inlinecode"><span class="id" type="tactic">apply</span></span> the constructor <span class="inlinecode"><span class="id" type="var">ex_intro</span></span>. Since the
premise of <span class="inlinecode"><span class="id" type="var">ex_intro</span></span> involves a variable (<span class="inlinecode"><span class="id" type="var">witness</span></span>) that does
not appear in its conclusion, we need to explicitly give its value
when we use <span class="inlinecode"><span class="id" type="tactic">apply</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">exists_example_1</span> : <span style="font-family: arial;">∃</span><span class="id" type="var">n</span>, <span class="id" type="var">n</span> + (<span class="id" type="var">n</span> × <span class="id" type="var">n</span>) = 6.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">ex_intro</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">witness</span>:=2).<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Note that we have to explicitly give the witness.
<div class="paragraph"> </div>
<a name="lab289"></a><h3 class="section"> </h3>
Or, instead of writing <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">ex_intro</span></span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">(<span class="id" type="var">witness</span>:=<span class="id" type="var">e</span>)</span> all the
time, we can use the convenient shorthand <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">e</span></span>, which means
the same thing.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">exists_example_1'</span> : <span style="font-family: arial;">∃</span><span class="id" type="var">n</span>, <span class="id" type="var">n</span> + (<span class="id" type="var">n</span> × <span class="id" type="var">n</span>) = 6.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span style="font-family: arial;">∃</span>2.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab290"></a><h3 class="section"> </h3>
Conversely, if we have an existential hypothesis in the
context, we can eliminate it with <span class="inlinecode"><span class="id" type="tactic">inversion</span></span>. Note the use
of the <span class="inlinecode"><span class="id" type="keyword">as</span>...</span> pattern to name the variable that Coq
introduces to name the witness value and get evidence that
the hypothesis holds for the witness. (If we don't
explicitly choose one, Coq will just call it <span class="inlinecode"><span class="id" type="var">witness</span></span>, which
makes proofs confusing.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">exists_example_2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
(<span style="font-family: arial;">∃</span><span class="id" type="var">m</span>, <span class="id" type="var">n</span> = 4 + <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">o</span>, <span class="id" type="var">n</span> = 2 + <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">n</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">m</span> <span class="id" type="var">Hm</span>].<br/>
<span style="font-family: arial;">∃</span>(2 + <span class="id" type="var">m</span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">Hm</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Here is another example of how to work with existentials.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">exists_example_3</span> : <br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>), <span class="id" type="var">even</span> <span class="id" type="var">n</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span style="font-family: arial;">∃</span>8.<br/>
<span class="id" type="tactic">split</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">even</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_sum</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">n</span>:=3) (<span class="id" type="var">m</span>:=5).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_3</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">b_5</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab291"></a><h4 class="section">Exercise: 1 star, optional (english_exists)</h4>
In English, what does the proposition
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">ex</span> <span class="id" type="var">nat</span> (<span class="id" type="keyword">fun</span> <span class="id" type="var">n</span> ⇒ <span class="id" type="var">beautiful</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>))
<div class="paragraph"> </div>
</div>
mean?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
<br/>
<span class="comment">(*<br/>
*)</span><br/>
</div>
<div class="doc">
<a name="lab292"></a><h4 class="section">Exercise: 1 star (dist_not_exists)</h4>
Prove that "<span class="inlinecode"><span class="id" type="var">P</span></span> holds for all <span class="inlinecode"><span class="id" type="var">x</span></span>" implies "there is no <span class="inlinecode"><span class="id" type="var">x</span></span> for
which <span class="inlinecode"><span class="id" type="var">P</span></span> does not hold."
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">dist_not_exists</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">P</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> ¬ (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, ¬ <span class="id" type="var">P</span> <span class="id" type="var">x</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="lab293"></a><h4 class="section">Exercise: 3 stars, optional (not_exists_dist)</h4>
(The other direction of this theorem requires the classical "law
of the excluded middle".)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">not_exists_dist</span> :<br/>
<span class="id" type="var">excluded_middle</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
¬ (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, ¬ <span class="id" type="var">P</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> (<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</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="lab294"></a><h4 class="section">Exercise: 2 stars (dist_exists_or)</h4>
Prove that existential quantification distributes over
disjunction.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">dist_exists_or</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">P</span> <span class="id" type="var">Q</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">Q</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">↔</span> (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">∨</span> (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">Q</span> <span class="id" type="var">x</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="lab295"></a><h1 class="section">Evidence-Carrying Booleans</h1>
<div class="paragraph"> </div>
So far we've seen two different forms of equality predicates:
<span class="inlinecode"><span class="id" type="var">eq</span></span>, which produces a <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>, and the type-specific forms, like
<span class="inlinecode"><span class="id" type="var">beq_nat</span></span>, that produce <span class="inlinecode"><span class="id" type="var">boolean</span></span> values. The former are more
convenient to reason about, but we've relied on the latter to let
us use equality tests in <i>computations</i>. While it is
straightforward to write lemmas (e.g. <span class="inlinecode"><span class="id" type="var">beq_nat_true</span></span> and
<span class="inlinecode"><span class="id" type="var">beq_nat_false</span></span>) that connect the two forms, using these lemmas
quickly gets tedious.
<div class="paragraph"> </div>
<a name="lab296"></a><h3 class="section"> </h3>
It turns out that we can get the benefits of both forms at once by
using a construct called <span class="inlinecode"><span class="id" type="var">sumbool</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">sumbool</span> (<span class="id" type="var">A</span> <span class="id" type="var">B</span> : <span class="id" type="keyword">Prop</span>) : <span class="id" type="keyword">Set</span> :=<br/>
| <span class="id" type="var">left</span> : <span class="id" type="var">A</span> <span style="font-family: arial;">→</span> <span class="id" type="var">sumbool</span> <span class="id" type="var">A</span> <span class="id" type="var">B</span> <br/>
| <span class="id" type="var">right</span> : <span class="id" type="var">B</span> <span style="font-family: arial;">→</span> <span class="id" type="var">sumbool</span> <span class="id" type="var">A</span> <span class="id" type="var">B</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "{ A } + { B }" := (<span class="id" type="var">sumbool</span> <span class="id" type="var">A</span> <span class="id" type="var">B</span>) : <span class="id" type="var">type_scope</span>.<br/>
<br/>
</div>
<div class="doc">
Think of <span class="inlinecode"><span class="id" type="var">sumbool</span></span> as being like the <span class="inlinecode"><span class="id" type="var">boolean</span></span> type, but instead
of its values being just <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">false</span></span>, they carry <i>evidence</i>
of truth or falsity. This means that when we <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> them, we
are left with the relevant evidence as a hypothesis — just as
with <span class="inlinecode"><span class="id" type="var">or</span></span>. (In fact, the definition of <span class="inlinecode"><span class="id" type="var">sumbool</span></span> is almost the
same as for <span class="inlinecode"><span class="id" type="var">or</span></span>. The only difference is that values of <span class="inlinecode"><span class="id" type="var">sumbool</span></span>
are declared to be in <span class="inlinecode"><span class="id" type="keyword">Set</span></span> rather than in <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>; this is a
technical distinction that allows us to compute with them.)
<div class="paragraph"> </div>
<a name="lab297"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Here's how we can define a <span class="inlinecode"><span class="id" type="var">sumbool</span></span> for equality on <span class="inlinecode"><span class="id" type="var">nat</span></span>s
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">eq_nat_dec</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">n</span> = <span class="id" type="var">m</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">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>.<br/>
<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 = 0 *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">m</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 = 0 *)</span><br/>
<span class="id" type="var">left</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="comment">(* m = S m' *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">contra</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">contra</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">m</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 = 0 *)</span><br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">contra</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">contra</span>.<br/>
+ <span class="comment">(* m = S m' *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHn'</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">m</span> := <span class="id" type="var">m'</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">eq</span> | <span class="id" type="var">neq</span>].<br/>
<span class="id" type="var">left</span>. <span class="id" type="tactic">apply</span> <span class="id" type="tactic">f_equal</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">eq</span>.<br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">Heq</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">Heq</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Heq'</span>]. <span class="id" type="tactic">apply</span> <span class="id" type="var">neq</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">Heq'</span>.<br/>
<span class="id" type="keyword">Defined</span>.<br/>
<br/>
</div>
<div class="doc">
Read as a theorem, this says that equality on <span class="inlinecode"><span class="id" type="var">nat</span></span>s is decidable:
that is, given two <span class="inlinecode"><span class="id" type="var">nat</span></span> values, we can always produce either
evidence that they are equal or evidence that they are not. Read
computationally, <span class="inlinecode"><span class="id" type="var">eq_nat_dec</span></span> takes two <span class="inlinecode"><span class="id" type="var">nat</span></span> values and returns a
<span class="inlinecode"><span class="id" type="var">sumbool</span></span> constructed with <span class="inlinecode"><span class="id" type="var">left</span></span> if they are equal and <span class="inlinecode"><span class="id" type="var">right</span></span> if
they are not; this result can be tested with a <span class="inlinecode"><span class="id" type="keyword">match</span></span> or, better,
with an <span class="inlinecode"><span class="id" type="keyword">if</span>-<span class="id" type="keyword">then</span>-<span class="id" type="keyword">else</span></span>, just like a regular <span class="inlinecode"><span class="id" type="var">boolean</span></span>. (Notice
that we ended this proof with <span class="inlinecode"><span class="id" type="keyword">Defined</span></span> rather than <span class="inlinecode"><span class="id" type="keyword">Qed</span></span>. The
only difference this makes is that the proof becomes
<i>transparent</i>, meaning that its definition is available when Coq
tries to do reductions, which is important for the computational
interpretation.)
<div class="paragraph"> </div>
<a name="lab298"></a><h3 class="section"> </h3>
Here's a simple example illustrating the advantages of the
<span class="inlinecode"><span class="id" type="var">sumbool</span></span> form.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">override'</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">f</span>: <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span>) (<span class="id" type="var">k</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">x</span>:<span class="id" type="var">X</span>) : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span>:=<br/>
<span class="id" type="keyword">fun</span> (<span class="id" type="var">k'</span>:<span class="id" type="var">nat</span>) ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">eq_nat_dec</span> <span class="id" type="var">k</span> <span class="id" type="var">k'</span> <span class="id" type="keyword">then</span> <span class="id" type="var">x</span> <span class="id" type="keyword">else</span> <span class="id" type="var">f</span> <span class="id" type="var">k'</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">override_same'</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">x1</span> <span class="id" type="var">k1</span> <span class="id" type="var">k2</span> (<span class="id" type="var">f</span> : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span>),<br/>
<span class="id" type="var">f</span> <span class="id" type="var">k1</span> = <span class="id" type="var">x1</span> <span style="font-family: arial;">→</span> <br/>
(<span class="id" type="var">override'</span> <span class="id" type="var">f</span> <span class="id" type="var">k1</span> <span class="id" type="var">x1</span>) <span class="id" type="var">k2</span> = <span class="id" type="var">f</span> <span class="id" type="var">k2</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">x1</span> <span class="id" type="var">k1</span> <span class="id" type="var">k2</span> <span class="id" type="var">f</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">Hx1</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">override'</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">eq_nat_dec</span> <span class="id" type="var">k1</span> <span class="id" type="var">k2</span>). <span class="comment">(* observe what appears as a hypothesis *)</span><br/>
- <span class="comment">(* k1 = k2 *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">e</span>.<br/>
<span class="id" type="tactic">symmetry</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">Hx1</span>.<br/>
- <span class="comment">(* k1 <> k2 *)</span><br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Compare this to the more laborious proof (in BasicTactics.v) for the
version of <span class="inlinecode"><span class="id" type="var">override</span></span> defined using <span class="inlinecode"><span class="id" type="var">beq_nat</span></span>, where we had to use
the auxiliary lemma <span class="inlinecode"><span class="id" type="var">beq_nat_true</span></span> to convert a fact about
booleans to a Prop.
<div class="paragraph"> </div>
<a name="lab299"></a><h4 class="section">Exercise: 1 star (override_shadow')</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">override_shadow'</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">x1</span> <span class="id" type="var">x2</span> <span class="id" type="var">k1</span> <span class="id" type="var">k2</span> (<span class="id" type="var">f</span> : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span>),<br/>
(<span class="id" type="var">override'</span> (<span class="id" type="var">override'</span> <span class="id" type="var">f</span> <span class="id" type="var">k1</span> <span class="id" type="var">x2</span>) <span class="id" type="var">k1</span> <span class="id" type="var">x1</span>) <span class="id" type="var">k2</span> = (<span class="id" type="var">override'</span> <span class="id" type="var">f</span> <span class="id" type="var">k1</span> <span class="id" type="var">x1</span>) <span class="id" type="var">k2</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="lab300"></a><h1 class="section">Additional Exercises</h1>
<div class="paragraph"> </div>
<a name="lab301"></a><h4 class="section">Exercise: 3 stars (all_forallb)</h4>
Inductively define a property <span class="inlinecode"><span class="id" type="var">all</span></span> of lists, parameterized by a
type <span class="inlinecode"><span class="id" type="var">X</span></span> and a property <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>, such that <span class="inlinecode"><span class="id" type="var">all</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span>
asserts that <span class="inlinecode"><span class="id" type="var">P</span></span> is true for every element of the list <span class="inlinecode"><span class="id" type="var">l</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">all</span> (<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
<br/>
</div>
<div class="doc">
Recall the function <span class="inlinecode"><span class="id" type="var">forallb</span></span>, from the exercise
<span class="inlinecode"><span class="id" type="var">forall_exists_challenge</span></span> in chapter <span class="inlinecode"><span class="id" type="var">Poly</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">forallb</span> {<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">test</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bool</span>) (<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">bool</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ <span class="id" type="var">true</span><br/>
| <span class="id" type="var">x</span> :: <span class="id" type="var">l'</span> ⇒ <span class="id" type="var">andb</span> (<span class="id" type="var">test</span> <span class="id" type="var">x</span>) (<span class="id" type="var">forallb</span> <span class="id" type="var">test</span> <span class="id" type="var">l'</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Using the property <span class="inlinecode"><span class="id" type="var">all</span></span>, write down a specification for <span class="inlinecode"><span class="id" type="var">forallb</span></span>,
and prove that it satisfies the specification. Try to make your
specification as precise as possible.
<div class="paragraph"> </div>
Are there any important properties of the function <span class="inlinecode"><span class="id" type="var">forallb</span></span> which
are not captured by your specification?
</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>
<a name="lab302"></a><h4 class="section">Exercise: 4 stars, advanced (filter_challenge)</h4>
One of the main purposes of Coq is to prove that programs match
their specifications. To this end, let's prove that our
definition of <span class="inlinecode"><span class="id" type="var">filter</span></span> matches a specification. Here is the
specification, written out informally in English.
<div class="paragraph"> </div>
Suppose we have a set <span class="inlinecode"><span class="id" type="var">X</span></span>, a function <span class="inlinecode"><span class="id" type="var">test</span>:</span> <span class="inlinecode"><span class="id" type="var">X</span><span style="font-family: arial;">→</span><span class="id" type="var">bool</span></span>, and a list
<span class="inlinecode"><span class="id" type="var">l</span></span> of type <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>. Suppose further that <span class="inlinecode"><span class="id" type="var">l</span></span> is an "in-order
merge" of two lists, <span class="inlinecode"><span class="id" type="var">l1</span></span> and <span class="inlinecode"><span class="id" type="var">l2</span></span>, such that every item in <span class="inlinecode"><span class="id" type="var">l1</span></span>
satisfies <span class="inlinecode"><span class="id" type="var">test</span></span> and no item in <span class="inlinecode"><span class="id" type="var">l2</span></span> satisfies test. Then <span class="inlinecode"><span class="id" type="var">filter</span></span>
<span class="inlinecode"><span class="id" type="var">test</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">l1</span></span>.
<div class="paragraph"> </div>
A list <span class="inlinecode"><span class="id" type="var">l</span></span> is an "in-order merge" of <span class="inlinecode"><span class="id" type="var">l1</span></span> and <span class="inlinecode"><span class="id" type="var">l2</span></span> if it contains
all the same elements as <span class="inlinecode"><span class="id" type="var">l1</span></span> and <span class="inlinecode"><span class="id" type="var">l2</span></span>, in the same order as <span class="inlinecode"><span class="id" type="var">l1</span></span>
and <span class="inlinecode"><span class="id" type="var">l2</span></span>, but possibly interleaved. For example,
<div class="paragraph"> </div>
<div class="code code-tight">
[1,4,6,2,3]
<div class="paragraph"> </div>
</div>
is an in-order merge of
<div class="paragraph"> </div>
<div class="code code-tight">
[1,6,2]
<div class="paragraph"> </div>
</div>
and
<div class="paragraph"> </div>
<div class="code code-tight">
[4,3].
<div class="paragraph"> </div>
</div>
Your job is to translate this specification into a Coq theorem and
prove it. (Hint: You'll need to begin by defining what it means
for one list to be a merge of two others. Do this with an
inductive relation, not a <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>.)
</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>
<a name="lab303"></a><h4 class="section">Exercise: 5 stars, advanced, optional (filter_challenge_2)</h4>
A different way to formally characterize the behavior of <span class="inlinecode"><span class="id" type="var">filter</span></span>
goes like this: Among all subsequences of <span class="inlinecode"><span class="id" type="var">l</span></span> with the property
that <span class="inlinecode"><span class="id" type="var">test</span></span> evaluates to <span class="inlinecode"><span class="id" type="var">true</span></span> on all their members, <span class="inlinecode"><span class="id" type="var">filter</span></span> <span class="inlinecode"><span class="id" type="var">test</span></span>
<span class="inlinecode"><span class="id" type="var">l</span></span> is the longest. Express this claim formally and prove it.
</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>
<a name="lab304"></a><h4 class="section">Exercise: 4 stars, advanced (no_repeats)</h4>
The following inductively defined proposition...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">appears_in</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">a</span>:<span class="id" type="var">X</span>) : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ai_here</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>, <span class="id" type="var">appears_in</span> <span class="id" type="var">a</span> (<span class="id" type="var">a</span>::<span class="id" type="var">l</span>)<br/>
| <span class="id" type="var">ai_later</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">l</span>, <span class="id" type="var">appears_in</span> <span class="id" type="var">a</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">a</span> (<span class="id" type="var">b</span>::<span class="id" type="var">l</span>).<br/>
<br/>
</div>
<div class="doc">
...gives us a precise way of saying that a value <span class="inlinecode"><span class="id" type="var">a</span></span> appears at
least once as a member of a list <span class="inlinecode"><span class="id" type="var">l</span></span>.
<div class="paragraph"> </div>
Here's a pair of warm-ups about <span class="inlinecode"><span class="id" type="var">appears_in</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">appears_in_app</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">xs</span> <span class="id" type="var">ys</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">x</span>:<span class="id" type="var">X</span>), <br/>
<span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">xs</span> ++ <span class="id" type="var">ys</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">xs</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">ys</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">app_appears_in</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">xs</span> <span class="id" type="var">ys</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">x</span>:<span class="id" type="var">X</span>), <br/>
<span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">xs</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">ys</span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> (<span class="id" type="var">xs</span> ++ <span class="id" type="var">ys</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">
Now use <span class="inlinecode"><span class="id" type="var">appears_in</span></span> to define a proposition <span class="inlinecode"><span class="id" type="var">disjoint</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">l1</span></span> <span class="inlinecode"><span class="id" type="var">l2</span></span>,
which should be provable exactly when <span class="inlinecode"><span class="id" type="var">l1</span></span> and <span class="inlinecode"><span class="id" type="var">l2</span></span> are
lists (with elements of type X) that have no elements in common.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
<br/>
</div>
<div class="doc">
Next, use <span class="inlinecode"><span class="id" type="var">appears_in</span></span> to define an inductive proposition
<span class="inlinecode"><span class="id" type="var">no_repeats</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span>, which should be provable exactly when <span class="inlinecode"><span class="id" type="var">l</span></span> is a
list (with elements of type <span class="inlinecode"><span class="id" type="var">X</span></span>) where every member is different
from every other. For example, <span class="inlinecode"><span class="id" type="var">no_repeats</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span> <span class="inlinecode">[1,2,3,4]</span> and
<span class="inlinecode"><span class="id" type="var">no_repeats</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode">[]</span> should be provable, while <span class="inlinecode"><span class="id" type="var">no_repeats</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span>
<span class="inlinecode">[1,2,1]</span> and <span class="inlinecode"><span class="id" type="var">no_repeats</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode">[<span class="id" type="var">true</span>,<span class="id" type="var">true</span>]</span> should not be.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
<br/>
</div>
<div class="doc">
Finally, state and prove one or more interesting theorems relating
<span class="inlinecode"><span class="id" type="var">disjoint</span></span>, <span class="inlinecode"><span class="id" type="var">no_repeats</span></span> and <span class="inlinecode">++</span> (list append).
</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>
<a name="lab305"></a><h4 class="section">Exercise: 3 stars (nostutter)</h4>
Formulating inductive definitions of predicates is an important
skill you'll need in this course. Try to solve this exercise
without any help at all.
<div class="paragraph"> </div>
We say that a list of numbers "stutters" if it repeats the same
number consecutively. The predicate "<span class="inlinecode"><span class="id" type="var">nostutter</span></span> <span class="inlinecode"><span class="id" type="var">mylist</span></span>" means
that <span class="inlinecode"><span class="id" type="var">mylist</span></span> does not stutter. Formulate an inductive definition
for <span class="inlinecode"><span class="id" type="var">nostutter</span></span>. (This is different from the <span class="inlinecode"><span class="id" type="var">no_repeats</span></span>
predicate in the exercise above; the sequence <span class="inlinecode">1;4;1</span> repeats but
does not stutter.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">nostutter</span>: <span class="id" type="var">list</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
<br/>
</div>
<div class="doc">
Make sure each of these tests succeeds, but you are free
to change the proof if the given one doesn't work for you.
Your definition might be different from mine and still correct,
in which case the examples might need a different proof.
<div class="paragraph"> </div>
The suggested proofs for the examples (in comments) use a number
of tactics we haven't talked about, to try to make them robust
with respect to different possible ways of defining <span class="inlinecode"><span class="id" type="var">nostutter</span></span>.
You should be able to just uncomment and use them as-is, but if
you prefer you can also prove each example with more basic
tactics.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nostutter_1</span>: <span class="id" type="var">nostutter</span> [3;1;4;1;5;6].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="comment">(* <br/>
Proof. repeat constructor; apply beq_nat_false; auto. Qed.<br/>
*)</span><br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nostutter_2</span>: <span class="id" type="var">nostutter</span> [].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="comment">(* <br/>
Proof. repeat constructor; apply beq_nat_false; auto. Qed.<br/>
*)</span><br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nostutter_3</span>: <span class="id" type="var">nostutter</span> [5].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="comment">(* <br/>
Proof. repeat constructor; apply beq_nat_false; auto. Qed.<br/>
*)</span><br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_nostutter_4</span>: <span class="id" type="var">not</span> (<span class="id" type="var">nostutter</span> [3;1;1;4]).<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="comment">(* <br/>
Proof. intro.<br/>
repeat match goal with <br/>
h: nostutter _ |- _ => inversion h; clear h; subst <br/>
end.<br/>
contradiction H1; auto. Qed.<br/>
*)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab306"></a><h4 class="section">Exercise: 4 stars, advanced (pigeonhole principle)</h4>
The "pigeonhole principle" states a basic fact about counting:
if you distribute more than <span class="inlinecode"><span class="id" type="var">n</span></span> items into <span class="inlinecode"><span class="id" type="var">n</span></span> pigeonholes, some
pigeonhole must contain at least two items. As is often the case,
this apparently trivial fact about numbers requires non-trivial
machinery to prove, but we now have enough...
<div class="paragraph"> </div>
First a pair of useful lemmas (we already proved these for lists
of naturals, but not for arbitrary lists).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">app_length</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">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">length</span> (<span class="id" type="var">l1</span> ++ <span class="id" type="var">l2</span>) = <span class="id" type="var">length</span> <span class="id" type="var">l1</span> + <span class="id" type="var">length</span> <span class="id" type="var">l2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">appears_in_app_split</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">X</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <br/>
<span style="font-family: arial;">∃</span><span class="id" type="var">l1</span>, <span style="font-family: arial;">∃</span><span class="id" type="var">l2</span>, <span class="id" type="var">l</span> = <span class="id" type="var">l1</span> ++ (<span class="id" type="var">x</span>::<span class="id" type="var">l2</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">
Now define a predicate <span class="inlinecode"><span class="id" type="var">repeats</span></span> (analogous to <span class="inlinecode"><span class="id" type="var">no_repeats</span></span> in the
exercise above), such that <span class="inlinecode"><span class="id" type="var">repeats</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span> asserts that <span class="inlinecode"><span class="id" type="var">l</span></span> contains
at least one repeated element (of type <span class="inlinecode"><span class="id" type="var">X</span></span>).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">repeats</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
<br/>
</div>
<div class="doc">
Now here's a way to formalize the pigeonhole principle. List <span class="inlinecode"><span class="id" type="var">l2</span></span>
represents a list of pigeonhole labels, and list <span class="inlinecode"><span class="id" type="var">l1</span></span> represents
the labels assigned to a list of items: if there are more items
than labels, at least two items must have the same label. This
proof is much easier if you use the <span class="inlinecode"><span class="id" type="var">excluded_middle</span></span> hypothesis
to show that <span class="inlinecode"><span class="id" type="var">appears_in</span></span> is decidable, i.e. <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span>
<span class="inlinecode"><span class="id" type="var">l</span>,</span> <span class="inlinecode">(<span class="id" type="var">appears_in</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">l</span>)</span> <span class="inlinecode"><span style="font-family: arial;">∨</span></span> <span class="inlinecode">¬</span> <span class="inlinecode">(<span class="id" type="var">appears_in</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">l</span>)</span>. However, it is also
possible to make the proof go through <i>without</i> assuming that
<span class="inlinecode"><span class="id" type="var">appears_in</span></span> is decidable; if you can manage to do this, you will
not need the <span class="inlinecode"><span class="id" type="var">excluded_middle</span></span> hypothesis.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">pigeonhole_principle</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">l1</span> <span class="id" type="var">l2</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>), <br/>
<span class="id" type="var">excluded_middle</span> <span style="font-family: arial;">→</span> <br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">l1</span> <span style="font-family: arial;">→</span> <span class="id" type="var">appears_in</span> <span class="id" type="var">x</span> <span class="id" type="var">l2</span>) <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">length</span> <span class="id" type="var">l2</span> < <span class="id" type="var">length</span> <span class="id" type="var">l1</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">repeats</span> <span class="id" type="var">l1</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">l1</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">l1</span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">x</span> <span class="id" type="var">l1'</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="comment">(* FILL IN HERE *)</span><br/>
<br/>
</div>
<div class="doc">
</div>
<div class="code code-tight">
</div>
</div>
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