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Lecture2.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<html xmlns="http://www.w3.org/1999/xhtml">
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<title>Lecture2</title>
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<div id="page">
<div id="header">
<div id='logoinheader'><a href='https://softwarefoundations.cis.upenn.edu'>
<img src='common/media/image/sf_logo_sm.png' alt='Software Foundations Logo'></a></div>
<div class='booktitleinheader'><a href='index.html'>Volume 1: Logical Foundations</a></div>
<ul id='menu'>
<li class='section_name'><a href='toc.html'>Table of Contents</a></li>
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</ul>
</div>
<div id="main">
<h1 class="libtitle">Lecture2</h1>
<div class="doc">
<a id="lab55"></a><h1 class="section">Прежде чем мы начнём</h1>
<div class="paragraph"> </div>
На предыдущей лекции я забыл обратить Ваше внимание
на следующее важное упражнение:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">eqb_refl</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>,<br/>
(<span class="id" title="var">n</span> =? <span class="id" title="var">n</span>) = <span class="id" title="var">true</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
</div>
<div class="doc">
Оно содержится в полной версии предыдущей лекции
и наверняка пригодится Вам в Ваших будущих доказательствах. <a id="lab56"></a><h1 class="section">Списки и работа со структурами данных</h1>
</div>
<div class="doc">
<a id="lab57"></a><h1 class="section">Раздельная компиляция</h1>
<div class="paragraph"> </div>
Чтобы переиспользовать материал прошлой лекции,
мы сначала должны скомпилировать <span class="inlinecode"><span class="id" title="var">Lecture1.v</span></span> в <span class="inlinecode"><span class="id" title="var">Lecture1.vo</span></span> --
подробные инструкции можно найти в полной версии данного текста.
</div>
<div class="code">
<span class="id" title="keyword">From</span> <span class="id" title="var">Lectures</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <span class="id" title="var">Lecture1</span>.<br/>
</div>
<div class="doc">
<a id="lab58"></a><h1 class="section">Пары натуральных чисел</h1>
<div class="paragraph"> </div>
<span class="inlinecode">Индуктивное</span> определение пары натуральных чисел. Оно содержит
единственный конструктор, получающий два аргумента:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">natprod</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">pair</span> (<span class="id" title="var">n<sub>1</sub></span> <span class="id" title="var">n<sub>2</sub></span> : <span class="id" title="var">nat</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> (<span class="id" title="var">pair</span> 3 5) : <span class="id" title="var">natprod</span>.<br/>
</div>
<div class="doc">
<a id="lab59"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Функции для извлечения первой и второй компонент пары
можно определить через сопоставление с образцом.
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fst</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">p</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">pair</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> ⇒ <span class="id" title="var">x</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">snd</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">p</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">pair</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> ⇒ <span class="id" title="var">y</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Compute</span> (<span class="id" title="var">fst</span> (<span class="id" title="var">pair</span> 3 5)).<br/>
<span class="comment">(* ===> 3 *)</span><br/>
</div>
<div class="doc">
<a id="lab60"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Более удобная нотация для пар:
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> "( x , y )" := (<span class="id" title="var">pair</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
</div>
<div class="doc">
Нотации можно использовать как в выражениях,
так и в сравнении с образцом.
</div>
<div class="code">
<span class="id" title="keyword">Compute</span> (<span class="id" title="var">fst</span> (3,5)).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">fst'</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">p</span> <span class="id" title="keyword">with</span><br/>
| (<span class="id" title="var">x</span>,<span class="id" title="var">y</span>) ⇒ <span class="id" title="var">x</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">snd'</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">p</span> <span class="id" title="keyword">with</span><br/>
| (<span class="id" title="var">x</span>,<span class="id" title="var">y</span>) ⇒ <span class="id" title="var">y</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">swap_pair</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>) : <span class="id" title="var">natprod</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">p</span> <span class="id" title="keyword">with</span><br/>
| (<span class="id" title="var">x</span>,<span class="id" title="var">y</span>) ⇒ (<span class="id" title="var">y</span>,<span class="id" title="var">x</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
<a id="lab61"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Если мы будем формулировать свойства пар несколько обходным путём, мы иногда
сможем завершить их доказательства чисто с помощью <span class="inlinecode">рефлексивности</span> и
встроенных упрощений:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">surjective_pairing'</span> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">n</span> <span class="id" title="var">m</span> : <span class="id" title="var">nat</span>),<br/>
(<span class="id" title="var">n</span>,<span class="id" title="var">m</span>) = (<span class="id" title="var">fst</span> (<span class="id" title="var">n</span>,<span class="id" title="var">m</span>), <span class="id" title="var">snd</span> (<span class="id" title="var">n</span>,<span class="id" title="var">m</span>)).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab62"></a><h3 class="section"> </h3>
Однако просто <span class="inlinecode">рефлексивности</span> недостаточно, если мы сформулируем
утверждение леммы естественным образом:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">surjective_pairing_stuck</span> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>),<br/>
<span class="id" title="var">p</span> = (<span class="id" title="var">fst</span> <span class="id" title="var">p</span>, <span class="id" title="var">snd</span> <span class="id" title="var">p</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="comment">(* Ничего не упрощает! *)</span><br/>
<span class="id" title="keyword">Abort</span>.<br/>
</div>
<div class="doc">
<a id="lab63"></a><h3 class="section"> </h3>
Решение: использовать <span class="inlinecode"><span class="id" title="tactic">destruct</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">surjective_pairing</span> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">p</span> : <span class="id" title="var">natprod</span>),<br/>
<span class="id" title="var">p</span> = (<span class="id" title="var">fst</span> <span class="id" title="var">p</span>, <span class="id" title="var">snd</span> <span class="id" title="var">p</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">p</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">n</span> <span class="id" title="var">m</span>]. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab64"></a><h1 class="section">Списки натуральных чисел</h1>
<div class="paragraph"> </div>
Индуктивное определение _списка_ чисел:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">natlist</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">cons</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mylist</span> := <span class="id" title="var">cons</span> 1 (<span class="id" title="var">cons</span> 2 (<span class="id" title="var">cons</span> 3 <span class="id" title="var">nil</span>)).<br/>
</div>
<div class="doc">
<a id="lab65"></a><h3 class="section"> </h3>
Щепотка нотации для списков, чтобы жилось попроще:
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> "x :: l" := (<span class="id" title="var">cons</span> <span class="id" title="var">x</span> <span class="id" title="var">l</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> "[ ]" := <span class="id" title="var">nil</span>.<br/>
<span class="id" title="keyword">Notation</span> "[ x ; .. ; y ]" := (<span class="id" title="var">cons</span> <span class="id" title="var">x</span> .. (<span class="id" title="var">cons</span> <span class="id" title="var">y</span> <span class="id" title="var">nil</span>) ..).<br/>
</div>
<div class="doc">
С нашей нотацией, все три определения ниже значат одно и то же:
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mylist1</span> := 1 :: (2 :: (3 :: <span class="id" title="var">nil</span>)).<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mylist2</span> := 1 :: 2 :: 3 :: <span class="id" title="var">nil</span>.<br/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">mylist3</span> := [1;2;3].<br/>
</div>
<div class="doc">
Немножко функций, полезных в работе со списками...
<div class="paragraph"> </div>
<a id="lab66"></a><h3 class="section">Повтор элемента</h3>
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="tactic">repeat</span> (<span class="id" title="var">n</span> <span class="id" title="var">count</span> : <span class="id" title="var">nat</span>) : <span class="id" title="var">natlist</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">count</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">O</span> ⇒ <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">count'</span> ⇒ <span class="id" title="var">n</span> :: (<span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">count'</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
<a id="lab67"></a><h3 class="section">Длина списка</h3>
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">length</span> (<span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">O</span><br/>
| <span class="id" title="var">h</span> :: <span class="id" title="var">t</span> ⇒ <span class="id" title="var">S</span> (<span class="id" title="var">length</span> <span class="id" title="var">t</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
<a id="lab68"></a><h3 class="section">Конкатенация</h3>
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">app</span> (<span class="id" title="var">l<sub>1</sub></span> <span class="id" title="var">l<sub>2</sub></span> : <span class="id" title="var">natlist</span>) : <span class="id" title="var">natlist</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l<sub>1</sub></span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">l<sub>2</sub></span><br/>
| <span class="id" title="var">h</span> :: <span class="id" title="var">t</span> ⇒ <span class="id" title="var">h</span> :: (<span class="id" title="var">app</span> <span class="id" title="var">t</span> <span class="id" title="var">l<sub>2</sub></span>)<br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> "x ++ y" := (<span class="id" title="var">app</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)<br/>
(<span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 60).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_app1</span>: [1;2;3] ++ [4;5] = [1;2;3;4;5].<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_app2</span>: <span class="id" title="var">nil</span> ++ [4;5] = [4;5].<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_app3</span>: [1;2;3] ++ <span class="id" title="var">nil</span> = [1;2;3].<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab69"></a><h3 class="section">Голова и хвост</h3>
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">hd</span> (<span class="id" title="var">default</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">default</span><br/>
| <span class="id" title="var">h</span> :: <span class="id" title="var">t</span> ⇒ <span class="id" title="var">h</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <span class="id" title="var">tl</span> (<span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>) : <span class="id" title="var">natlist</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">h</span> :: <span class="id" title="var">t</span> ⇒ <span class="id" title="var">t</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_hd<sub>1</sub></span>: <span class="id" title="var">hd</span> 0 [1;2;3] = 1.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_hd<sub>2</sub></span>: <span class="id" title="var">hd</span> 0 [] = 0.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_tl</span>: <span class="id" title="var">tl</span> [1;2;3] = [2;3].<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<!-- /quiz -->
</div><div class="quiz">
<div class="doc">
Что делает следующая функция?
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">foo</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>) : <span class="id" title="var">natlist</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span><br/>
| 0 ⇒ <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n'</span> ⇒ <span class="id" title="var">n</span> :: (<span class="id" title="var">foo</span> <span class="id" title="var">n'</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
</div></div> <!-- /quiz -->
<div class="doc">
<a id="lab70"></a><h1 class="section">Суждения о списках</h1>
<div class="paragraph"> </div>
Как было и с числами, для некоторых доказательств
о функциях на списках достаточно упростить выражения...
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">nil_app</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>,<br/>
[] ++ <span class="id" title="var">l</span> = <span class="id" title="var">l</span>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab71"></a><h3 class="section"> </h3>
...а для некоторых потребуется анализ случаев.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">tl_length_pred</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>,<br/>
<span class="id" title="var">pred</span> (<span class="id" title="var">length</span> <span class="id" title="var">l</span>) = <span class="id" title="var">length</span> (<span class="id" title="var">tl</span> <span class="id" title="var">l</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n</span> <span class="id" title="var">l'</span>].<br/>
- <span class="comment">(* l = nil *)</span><br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = cons n l' *)</span><br/>
<span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Правда, чаще всего, доказательство интересных теорем о списках требует
использования _индукции по спискам_. Сейчас мы увидим, как это делать.
</div>
<div class="doc">
<a id="lab72"></a><h2 class="section">Индукция на списках</h2>
<div class="paragraph"> </div>
Coq генерирует _принцип индукции_ для каждого <span class="inlinecode">Индуктивного</span>
определения, включая списки. Благодаря этому, например, мы можем
использовать тактику <span class="inlinecode"><span class="id" title="tactic">induction</span></span> на списках, чтобы доказать вещи вроде
ассоциативности конкатенации...
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">app_assoc</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l<sub>1</sub></span> <span class="id" title="var">l<sub>2</sub></span> <span class="id" title="var">l<sub>3</sub></span> : <span class="id" title="var">natlist</span>,<br/>
(<span class="id" title="var">l<sub>1</sub></span> ++ <span class="id" title="var">l<sub>2</sub></span>) ++ <span class="id" title="var">l<sub>3</sub></span> = <span class="id" title="var">l<sub>1</sub></span> ++ (<span class="id" title="var">l<sub>2</sub></span> ++ <span class="id" title="var">l<sub>3</sub></span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l<sub>1</sub></span> <span class="id" title="var">l<sub>2</sub></span> <span class="id" title="var">l<sub>3</sub></span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">l<sub>1</sub></span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n</span> <span class="id" title="var">l<sub>1</sub>'</span> <span class="id" title="var">IHl1'</span>].<br/>
- <span class="comment">(* l<sub>1</sub> = nil *)</span><br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l<sub>1</sub> = cons n l<sub>1</sub>' *)</span><br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> → <span class="id" title="var">IHl1'</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab73"></a><h3 class="section"> </h3>
Для сравнения, вот неформальное (но строгое) доказательство
той же самой теоремы.
<div class="paragraph"> </div>
_Теорема_: Для всяких списков <span class="inlinecode"><span class="id" title="var">l<sub>1</sub></span></span>, <span class="inlinecode"><span class="id" title="var">l<sub>2</sub></span></span> и <span class="inlinecode"><span class="id" title="var">l<sub>3</sub></span></span> верно, что
<span class="inlinecode">(<span class="id" title="var">l<sub>1</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" title="var">l<sub>2</sub></span>)</span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" title="var">l<sub>3</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">l<sub>1</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode">(<span class="id" title="var">l<sub>2</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" title="var">l<sub>3</sub></span>)</span>.
<div class="paragraph"> </div>
_Доказательство_: Индукцией по <span class="inlinecode"><span class="id" title="var">l<sub>1</sub></span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> База: <span class="inlinecode"><span class="id" title="var">l<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode">[]</span>. Необходимо показать, что
<br/>
<span class="inlinecode"> ([] ++ <span class="id" title="var">l<sub>2</sub></span>) ++ <span class="id" title="var">l<sub>3</sub></span> = [] ++ (<span class="id" title="var">l<sub>2</sub></span> ++ <span class="id" title="var">l<sub>3</sub></span>),
</span> что напрямую следует из определения <span class="inlinecode">++</span>.
<div class="paragraph"> </div>
</li>
<li> Переход: <span class="inlinecode"><span class="id" title="var">l<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">n</span>::<span class="id" title="var">l<sub>1</sub>'</span></span>, причём
<br/>
<span class="inlinecode"> (<span class="id" title="var">l<sub>1</sub>'</span> ++ <span class="id" title="var">l<sub>2</sub></span>) ++ <span class="id" title="var">l<sub>3</sub></span> = <span class="id" title="var">l<sub>1</sub>'</span> ++ (<span class="id" title="var">l<sub>2</sub></span> ++ <span class="id" title="var">l<sub>3</sub></span>)
</span> (по предположению индукции). Необходимо показать, что
<br/>
<span class="inlinecode"> ((<span class="id" title="var">n</span> :: <span class="id" title="var">l<sub>1</sub>'</span>) ++ <span class="id" title="var">l<sub>2</sub></span>) ++ <span class="id" title="var">l<sub>3</sub></span> = (<span class="id" title="var">n</span> :: <span class="id" title="var">l<sub>1</sub>'</span>) ++ (<span class="id" title="var">l<sub>2</sub></span> ++ <span class="id" title="var">l<sub>3</sub></span>).
</span> По определению <span class="inlinecode">++</span>, это следует из
<br/>
<span class="inlinecode"> <span class="id" title="var">n</span> :: ((<span class="id" title="var">l<sub>1</sub>'</span> ++ <span class="id" title="var">l<sub>2</sub></span>) ++ <span class="id" title="var">l<sub>3</sub></span>) = <span class="id" title="var">n</span> :: (<span class="id" title="var">l<sub>1</sub>'</span> ++ (<span class="id" title="var">l<sub>2</sub></span> ++ <span class="id" title="var">l<sub>3</sub></span>)),
</span> что также верно благодаря предположению индукции. <font size=-2>☐</font>
</li>
</ul>
<div class="paragraph"> </div>
<a id="lab74"></a><h3 class="section">Усиление предположения индукции</h3>
<div class="paragraph"> </div>
Иногда утверждения нужно усилить, чтобы доказать по индукции:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">repeat_double_firsttry</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">c</span> <span class="id" title="var">n</span>: <span class="id" title="var">nat</span>,<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> ++ <span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">c</span> = <span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> (<span class="id" title="var">c</span> + <span class="id" title="var">c</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IHc'</span>].<br/>
- <span class="comment">(* c = 0 *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* c = S c' *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* Здесь-то мы, кажется, и застряли. Мы не можем использовать IH,<br/>
чтобы переписать <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">(<span class="id" title="var">c'</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">c'</span>)</span>: гипотеза работает только для<br/>
<span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">(<span class="id" title="var">c'</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">c'</span>)</span>. Если бы IH была более гибкой<br/>
(например, если бы она работала для произвольного второго слагаемого),<br/>
доказательство бы прошло. *)</span><br/>
<span class="id" title="keyword">Abort</span>.<br/>
</div>
<div class="doc">
Усиление утверждения, которое приводит к более общему
предположению индукции:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">repeat_plus</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">c<sub>1</sub></span> <span class="id" title="var">c<sub>2</sub></span> <span class="id" title="var">n</span>: <span class="id" title="var">nat</span>,<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">c<sub>1</sub></span> ++ <span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">c<sub>2</sub></span> = <span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> (<span class="id" title="var">c<sub>1</sub></span> + <span class="id" title="var">c<sub>2</sub></span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">c<sub>1</sub></span> <span class="id" title="var">c<sub>2</sub></span> <span class="id" title="var">n</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">c<sub>1</sub></span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c<sub>1</sub>'</span> <span class="id" title="var">IHc1'</span>].<br/>
- <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">IHc1'</span>.<br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab75"></a><h3 class="section">Разворот списка</h3>
<div class="paragraph"> </div>
Более интересный подопытный для индукции на списках:
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">rev</span> (<span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>) : <span class="id" title="var">natlist</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">h</span> :: <span class="id" title="var">t</span> ⇒ <span class="id" title="var">rev</span> <span class="id" title="var">t</span> ++ [<span class="id" title="var">h</span>]<br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_rev1</span>: <span class="id" title="var">rev</span> [1;2;3] = [3;2;1].<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_rev2</span>: <span class="id" title="var">rev</span> <span class="id" title="var">nil</span> = <span class="id" title="var">nil</span>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab76"></a><h3 class="section"> </h3>
Давайте попробуем доказать, что <span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode">(<span class="id" title="var">rev</span></span> <span class="inlinecode"><span class="id" title="var">l</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode"><span class="id" title="var">l</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">rev_length_firsttry</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="var">rev</span> <span class="id" title="var">l</span>) = <span class="id" title="var">length</span> <span class="id" title="var">l</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n</span> <span class="id" title="var">l'</span> <span class="id" title="var">IHl'</span>].<br/>
- <span class="comment">(* l = nil *)</span><br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = n :: l' *)</span><br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">IHl'</span>.<br/>
<span class="id" title="keyword">Abort</span>.<br/>
</div>
<div class="doc">
Первая попытка:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">app_rev_length_S_firsttry</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> <span class="id" title="var">n</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="var">rev</span> <span class="id" title="var">l</span> ++ [<span class="id" title="var">n</span>]) = <span class="id" title="var">S</span> (<span class="id" title="var">length</span> (<span class="id" title="var">rev</span> <span class="id" title="var">l</span>)).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">m</span> <span class="id" title="var">l'</span> <span class="id" title="var">IHl'</span>].<br/>
- <span class="comment">(* l = <span class="inlinecode"></span> *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = m:: l' *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="comment">(* IHl' не применима. *)</span><br/>
<span class="id" title="keyword">Abort</span>.<br/>
</div>
<div class="doc">
Слегка усиленное утверждение:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">app_length_S</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> <span class="id" title="var">n</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="var">l</span> ++ [<span class="id" title="var">n</span>]) = <span class="id" title="var">S</span> (<span class="id" title="var">length</span> <span class="id" title="var">l</span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">m</span> <span class="id" title="var">l'</span> <span class="id" title="var">IHl'</span>].<br/>
- <span class="comment">(* l = <span class="inlinecode"></span> *)</span><br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = m:: l' *)</span><br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHl'</span>.<br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab77"></a><h3 class="section"> </h3>
Теперь мы можем закончить исходное доказательство.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">rev_length</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> : <span class="id" title="var">natlist</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="var">rev</span> <span class="id" title="var">l</span>) = <span class="id" title="var">length</span> <span class="id" title="var">l</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n</span> <span class="id" title="var">l'</span> <span class="id" title="var">IHl'</span>].<br/>
- <span class="comment">(* l = nil *)</span><br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = cons *)</span><br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> → <span class="id" title="var">app_length_S</span>.<br/>
<span class="id" title="tactic">rewrite</span> → <span class="id" title="var">IHl'</span>.<br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab78"></a><h3 class="section"> </h3>
На самом деле, мы можем ещё больше усилить app_length_S, и доказать
утверждение про длину двух любых склеенных вместе списков.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">app_length</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">l<sub>1</sub></span> <span class="id" title="var">l<sub>2</sub></span> : <span class="id" title="var">natlist</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="var">l<sub>1</sub></span> ++ <span class="id" title="var">l<sub>2</sub></span>) = (<span class="id" title="var">length</span> <span class="id" title="var">l<sub>1</sub></span>) + (<span class="id" title="var">length</span> <span class="id" title="var">l<sub>2</sub></span>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* WORK IN CLASS *)</span> <span class="id" title="var">Admitted</span>.<br/><hr class='doublespaceincode'/>
<!-- /quiz -->
</div><div class="quiz">
<div class="doc">
Чтобы доказать следующую теорему, какие тактики потребуются, кроме
<span class="inlinecode"><span class="id" title="tactic">intros</span></span>, <span class="inlinecode"><span class="id" title="tactic">simpl</span></span>, <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> и <span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span>?
<div class="paragraph"> </div>
<br/>
<span class="inlinecode"> <span class="id" title="keyword">Theorem</span> <span class="id" title="var">foo1</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">n</span>:<span class="id" title="var">nat</span>, <span class="id" title="keyword">∀</span> <span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>,<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> 0 = <span class="id" title="var">l</span> → <span class="id" title="var">length</span> <span class="id" title="var">l</span> = 0.
</span>
<div class="paragraph"> </div>
(1) никаких?
<div class="paragraph"> </div>
(2) <span class="inlinecode"><span class="id" title="tactic">destruct</span></span>?
<div class="paragraph"> </div>
(3) <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">on</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>?
<div class="paragraph"> </div>
(4) <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">on</span></span> <span class="inlinecode"><span class="id" title="var">l</span></span>? Или
<div class="paragraph"> </div>
(5) не было ещё таких тактик.
</div>
</div> <!-- /quiz -->
<div class="quiz">
<div class="doc">
Как насчёт этой?
<div class="paragraph"> </div>
<br/>
<span class="inlinecode"> <span class="id" title="keyword">Theorem</span> <span class="id" title="var">foo2</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span> : <span class="id" title="var">nat</span>,<br/>
<span class="id" title="var">length</span> (<span class="id" title="tactic">repeat</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span>) = <span class="id" title="var">m</span>.
</span>
<div class="paragraph"> </div>
Какие тактики потребуются кроме <span class="inlinecode"><span class="id" title="tactic">intros</span></span>, <span class="inlinecode"><span class="id" title="tactic">simpl</span></span>, <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> и
<span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span>?
<div class="paragraph"> </div>
(1) никаких?
<div class="paragraph"> </div>
(2) <span class="inlinecode"><span class="id" title="tactic">destruct</span></span>?
<div class="paragraph"> </div>
(3) <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">on</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>?
<div class="paragraph"> </div>
(4) <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">on</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span>? Или
<div class="paragraph"> </div>
(5) не было ещё таких тактик.
</div>
</div> <!-- /quiz -->
<div class="doc">
<a id="lab79"></a><h1 class="section">Частичные значения</h1>
<div class="paragraph"> </div>
Пусть нам хочется функцию, возвращающую <span class="inlinecode"><span class="id" title="var">n</span></span>-й элемент списка.
Но что делать, если список слишком короткий?
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">nth_bad</span> (<span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) : <span class="id" title="var">nat</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ 42<br/>
| <span class="id" title="var">a</span> :: <span class="id" title="var">l'</span> ⇒ <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span><br/>
| 0 ⇒ <span class="id" title="var">a</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n'</span> ⇒ <span class="id" title="var">nth_bad</span> <span class="id" title="var">l'</span> <span class="id" title="var">n'</span><br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
<a id="lab80"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Решение: <span class="inlinecode"><span class="id" title="var">natoption</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">natoption</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">Some</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>)<br/>
| <span class="id" title="var">None</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">nth_error</span> (<span class="id" title="var">l</span>:<span class="id" title="var">natlist</span>) (<span class="id" title="var">n</span>:<span class="id" title="var">nat</span>) : <span class="id" title="var">natoption</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">nil</span> ⇒ <span class="id" title="var">None</span><br/>
| <span class="id" title="var">a</span> :: <span class="id" title="var">l'</span> ⇒ <span class="id" title="keyword">match</span> <span class="id" title="var">n</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">O</span> ⇒ <span class="id" title="var">Some</span> <span class="id" title="var">a</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">n'</span> ⇒ <span class="id" title="var">nth_error</span> <span class="id" title="var">l'</span> <span class="id" title="var">n'</span><br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_nth_error1</span> : <span class="id" title="var">nth_error</span> [4;5;6;7] 0 = <span class="id" title="var">Some</span> 4.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_nth_error2</span> : <span class="id" title="var">nth_error</span> [4;5;6;7] 3 = <span class="id" title="var">Some</span> 7.<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_nth_error3</span> : <span class="id" title="var">nth_error</span> [4;5;6;7] 9 = <span class="id" title="var">None</span>.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a id="lab81"></a><h3 class="section"> </h3>
</div>
<div class="code">
<span class="id" title="keyword">End</span> <span class="id" title="var">NatList</span>.<br/>
</div>
<div class="doc">
<a id="lab82"></a><h1 class="section">Частичные функции</h1>
<div class="paragraph"> </div>
В качестве последней иллюстрации того, как можно определять типы данных в
Coq, ниже мы введём тип данных _частичных функций_, аналогичный
ассоциативным массивам (aka словарям), которые легко найти в любом другом
языке программирования.
<div class="paragraph"> </div>
Во-первых, определим новый индуктивный тип данных <span class="inlinecode"><span class="id" title="var">id</span></span>, который будет
выступать в качестве "ключей" в наших словарях.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">id</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">Id</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>).<br/>
</div>
<div class="doc">
Внутри <span class="inlinecode"><span class="id" title="var">id</span></span> лежит просто число. Введение отдельного типа данных,
оборачивающего каждое число тэгом <span class="inlinecode"><span class="id" title="var">Id</span></span>, делает определения более читаемыми
и дарит нам гибкость в позднейшем изменении внутреннего представления <span class="inlinecode"><span class="id" title="var">Id</span></span>,
если впоследствии нам это потребуется.
<div class="paragraph"> </div>
<a id="lab83"></a><h3 class="section"> </h3>
Также мы захотим проверять <span class="inlinecode"><span class="id" title="var">id</span></span> на равенство:
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">eqb_id</span> (<span class="id" title="var">x<sub>1</sub></span> <span class="id" title="var">x<sub>2</sub></span> : <span class="id" title="var">id</span>) :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">x<sub>1</sub></span>, <span class="id" title="var">x<sub>2</sub></span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">Id</span> <span class="id" title="var">n<sub>1</sub></span>, <span class="id" title="var">Id</span> <span class="id" title="var">n<sub>2</sub></span> ⇒ <span class="id" title="var">n<sub>1</sub></span> =? <span class="id" title="var">n<sub>2</sub></span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
</div>
<div class="doc">
<a id="lab85"></a><h3 class="section"> </h3>
Далее, определим тип частичных функций:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">partial_map</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">empty</span><br/>
| <span class="id" title="var">record</span> (<span class="id" title="var">i</span> : <span class="id" title="var">id</span>) (<span class="id" title="var">v</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">m</span> : <span class="id" title="var">partial_map</span>).<br/>
</div>
<div class="doc">
<a id="lab86"></a><h3 class="section"> </h3>
Функция <span class="inlinecode"><span class="id" title="var">update</span></span> перегружает запись для данного ключа в словаре с помощью
перекрытия старого определения новым (либо просто добавляет новую запись,
если такого ключа ещё нет).
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <span class="id" title="var">update</span> (<span class="id" title="var">d</span> : <span class="id" title="var">partial_map</span>)<br/>
(<span class="id" title="var">x</span> : <span class="id" title="var">id</span>) (<span class="id" title="var">value</span> : <span class="id" title="var">nat</span>)<br/>
: <span class="id" title="var">partial_map</span> :=<br/>
<span class="id" title="var">record</span> <span class="id" title="var">x</span> <span class="id" title="var">value</span> <span class="id" title="var">d</span>.<br/>
</div>
<div class="doc">
<a id="lab87"></a><h3 class="section"> </h3>
Мы можем определять функции на <span class="inlinecode"><span class="id" title="var">partial_map</span></span> с помощью сопоставления
с образцом.
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="var">find</span> (<span class="id" title="var">x</span> : <span class="id" title="var">id</span>) (<span class="id" title="var">d</span> : <span class="id" title="var">partial_map</span>) : <span class="id" title="var">natoption</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">d</span> <span class="id" title="keyword">with</span><br/>
| <span class="id" title="var">empty</span> ⇒ <span class="id" title="var">None</span><br/>
| <span class="id" title="var">record</span> <span class="id" title="var">y</span> <span class="id" title="var">v</span> <span class="id" title="var">d'</span> ⇒ <span class="id" title="keyword">if</span> <span class="id" title="var">eqb_id</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
<span class="id" title="keyword">then</span> <span class="id" title="var">Some</span> <span class="id" title="var">v</span><br/>
<span class="id" title="keyword">else</span> <span class="id" title="var">find</span> <span class="id" title="var">x</span> <span class="id" title="var">d'</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<!-- /quiz -->
</div><div class="quiz">
<div class="doc">
Верно ли следующее утверждение?
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">quiz1</span> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">d</span> : <span class="id" title="var">partial_map</span>)<br/>
(<span class="id" title="var">x</span> : <span class="id" title="var">id</span>) (<span class="id" title="var">v</span>: <span class="id" title="var">nat</span>),<br/>
<span class="id" title="var">find</span> <span class="id" title="var">x</span> (<span class="id" title="var">update</span> <span class="id" title="var">d</span> <span class="id" title="var">x</span> <span class="id" title="var">v</span>) = <span class="id" title="var">Some</span> <span class="id" title="var">v</span>.<br/><hr class='doublespaceincode'/>
</div>
<div class="doc">
(1) Да
<div class="paragraph"> </div>
(2) Нет
<div class="paragraph"> </div>
(3) Не уверен
</div>
</div> <!-- /quiz -->
<div class="quiz">
<div class="doc">
Верно ли следующее утверждение?
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <span class="id" title="var">quiz2</span> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">d</span> : <span class="id" title="var">partial_map</span>)<br/>
(<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <span class="id" title="var">id</span>) (<span class="id" title="var">o</span>: <span class="id" title="var">nat</span>),<br/>
<span class="id" title="var">eqb_id</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> = <span class="id" title="var">false</span> →<br/>
<span class="id" title="var">find</span> <span class="id" title="var">x</span> (<span class="id" title="var">update</span> <span class="id" title="var">d</span> <span class="id" title="var">y</span> <span class="id" title="var">o</span>) = <span class="id" title="var">find</span> <span class="id" title="var">x</span> <span class="id" title="var">d</span>.<br/>
</div>
<div class="doc">
(1) Да
<div class="paragraph"> </div>
(2) Нет
<div class="paragraph"> </div>
(3) Не уверен
</div>
</div> <!-- /quiz -->
<div class="doc">
<a id="lab88"></a><h1 class="section">Poly: полиморфизм и функции высшего порядка</h1>
</div>
<div class="doc">
<a id="lab89"></a><h2 class="section">Полиморфные списки</h2>
<div class="paragraph"> </div>
Вместо того, чтобы определять по новому тип списков для каждого типа
элементов, т.е. как-то так...
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">boollist</span> : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">bool_nil</span><br/>
| <span class="id" title="var">bool_cons</span> (<span class="id" title="var">b</span> : <span class="id" title="var">bool</span>) (<span class="id" title="var">l</span> : <span class="id" title="var">boollist</span>).<br/>
</div>
<div class="doc">
<a id="lab90"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
...Coq позволяет давать _полиморфные_ определения, позволяющие, в
частности, выбрать произвольный тип элементов списка:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <span class="id" title="var">list</span> (<span class="id" title="var">X</span>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">nil</span><br/>
| <span class="id" title="var">cons</span> (<span class="id" title="var">x</span> : <span class="id" title="var">X</span>) (<span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span>).<br/>
</div>
<div class="doc">
Сам <span class="inlinecode"><span class="id" title="var">list</span></span> -- это _функция_ из типов в типы.
</div>
<div class="code">
<span class="id" title="keyword">Check</span> <span class="id" title="var">list</span> : <span class="id" title="keyword">Type</span> → <span class="id" title="keyword">Type</span>.<br/>
</div>
<div class="doc">
<a id="lab91"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" title="var">X</span></span> из определения <span class="inlinecode"><span class="id" title="var">list</span></span>
становится параметром конструкторов <span class="inlinecode"><span class="id" title="var">nil</span></span> и <span class="inlinecode"><span class="id" title="var">cons</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Check</span> (<span class="id" title="var">nil</span> <span class="id" title="var">nat</span>) : <span class="id" title="var">list</span> <span class="id" title="var">nat</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> (<span class="id" title="var">cons</span> <span class="id" title="var">nat</span> 3 (<span class="id" title="var">nil</span> <span class="id" title="var">nat</span>)) : <span class="id" title="var">list</span> <span class="id" title="var">nat</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <span class="id" title="var">nil</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>, <span class="id" title="var">list</span> <span class="id" title="var">X</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <span class="id" title="var">cons</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>, <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span>.<br/>
</div>
<div class="doc">
Заметка о нотации: в файлах .v, квантор "forall" пишется побуквенно.
В соответствующих файлах HTML, он обычно пишется как стандартное
математическое "перевёрнутое A."
</div>
<div class="code">
<span class="id" title="keyword">Check</span> (<span class="id" title="var">cons</span> <span class="id" title="var">nat</span> 2 (<span class="id" title="var">cons</span> <span class="id" title="var">nat</span> 1 (<span class="id" title="var">nil</span> <span class="id" title="var">nat</span>)))<br/>
: <span class="id" title="var">list</span> <span class="id" title="var">nat</span>.<br/>
</div>
<div class="doc">
<a id="lab92"></a><h3 class="section"> </h3>
Теперь мы можем определить полиморфные версии функций, которые мы уже
видели ранее...
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <span class="id" title="tactic">repeat</span> (<span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">x</span> : <span class="id" title="var">X</span>) (<span class="id" title="var">count</span> : <span class="id" title="var">nat</span>) : <span class="id" title="var">list</span> <span class="id" title="var">X</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">count</span> <span class="id" title="keyword">with</span><br/>
| 0 ⇒ <span class="id" title="var">nil</span> <span class="id" title="var">X</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">count'</span> ⇒ <span class="id" title="var">cons</span> <span class="id" title="var">X</span> <span class="id" title="var">x</span> (<span class="id" title="tactic">repeat</span> <span class="id" title="var">X</span> <span class="id" title="var">x</span> <span class="id" title="var">count'</span>)<br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_repeat1</span> :<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="var">nat</span> 4 2 = <span class="id" title="var">cons</span> <span class="id" title="var">nat</span> 4 (<span class="id" title="var">cons</span> <span class="id" title="var">nat</span> 4 (<span class="id" title="var">nil</span> <span class="id" title="var">nat</span>)).<br/>
<div class="togglescript" id="proofcontrol4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')"><span class="show"></span></div>
<div class="proofscript" id="proof4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')">
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" title="keyword">Example</span> <span class="id" title="var">test_repeat2</span> :<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="var">bool</span> <span class="id" title="var">false</span> 1 = <span class="id" title="var">cons</span> <span class="id" title="var">bool</span> <span class="id" title="var">false</span> (<span class="id" title="var">nil</span> <span class="id" title="var">bool</span>).<br/>
<div class="togglescript" id="proofcontrol5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')"><span class="show"></span></div>
<div class="proofscript" id="proof5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')">
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<!-- /quiz -->
</div><div class="quiz">
<div class="doc">
Вспомним определения <span class="inlinecode"><span class="id" title="var">nil</span></span> и <span class="inlinecode"><span class="id" title="var">cons</span></span>:
<br/>
<span class="inlinecode"> <span class="id" title="var">nil</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>, <span class="id" title="var">list</span> <span class="id" title="var">X</span><br/>
<span class="id" title="var">cons</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>, <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span>
</span> Каков тип у <span class="inlinecode"><span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">bool</span></span> <span class="inlinecode"><span class="id" title="var">true</span></span> <span class="inlinecode">(<span class="id" title="var">cons</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">3</span> <span class="inlinecode">(<span class="id" title="var">nil</span></span> <span class="inlinecode"><span class="id" title="var">nat</span>))</span>?
<div class="paragraph"> </div>
(1) <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span>
<div class="paragraph"> </div>
(2) <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode">(<span class="id" title="var">X</span>:<span class="id" title="keyword">Type</span>),</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>
<div class="paragraph"> </div>
(3) <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span>
<div class="paragraph"> </div>
(4) <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">bool</span></span>
<div class="paragraph"> </div>
(5) Никакого
</div>
</div> <!-- /quiz -->
<div class="quiz">
<div class="doc">
Вспомним определение <span class="inlinecode"><span class="id" title="tactic">repeat</span></span>:
<br/>
<span class="inlinecode"> <span class="id" title="keyword">Fixpoint</span> <span class="id" title="tactic">repeat</span> (<span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">x</span> : <span class="id" title="var">X</span>) (<span class="id" title="var">count</span> : <span class="id" title="var">nat</span>)<br/>
: <span class="id" title="var">list</span> <span class="id" title="var">X</span> :=<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">count</span> <span class="id" title="keyword">with</span><br/>
| 0 ⇒ <span class="id" title="var">nil</span> <span class="id" title="var">X</span><br/>
| <span class="id" title="var">S</span> <span class="id" title="var">count'</span> ⇒ <span class="id" title="var">cons</span> <span class="id" title="var">X</span> <span class="id" title="var">x</span> (<span class="id" title="tactic">repeat</span> <span class="id" title="var">X</span> <span class="id" title="var">x</span> <span class="id" title="var">count'</span>)<br/>
<span class="id" title="keyword">end</span>.
</span> Каков тип <span class="inlinecode"><span class="id" title="tactic">repeat</span></span>?
<div class="paragraph"> </div>
(1) <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span>
<div class="paragraph"> </div>
(2) <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>
<div class="paragraph"> </div>
(3) <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode">(<span class="id" title="var">X</span></span> <span class="inlinecode"><span class="id" title="var">Y</span>:</span> <span class="inlinecode"><span class="id" title="keyword">Type</span>),</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">Y</span></span>
<div class="paragraph"> </div>
(4) <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode">(<span class="id" title="var">X</span>:<span class="id" title="keyword">Type</span>),</span> <span class="inlinecode"><span class="id" title="var">X</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>
<div class="paragraph"> </div>
(5) Никакого
</div>
</div> <!-- /quiz -->
<div class="quiz">
<div class="doc">
Каков тип <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">1</span> <span class="inlinecode">2</span>?
<div class="paragraph"> </div>
(1) <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span>