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<h1 class="title">Aula 3 - Recursão Primitiva em Coq</h1>
<div id="outline-container-org3084f6b" class="outline-2">
<h2 id="org3084f6b"><span class="section-number-2">1</span> Recursão Primitiva</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org3a5f74d" class="outline-3">
<h3 id="org3a5f74d"><span class="section-number-3">1.1</span> Terminação em Coq</h3>
<div class="outline-text-3" id="text-1-1">
<ul class="org-ul">
<li>Gallina é total. Logo, terminação é uma propriedade crucial da linguagem Gallina.</li>
<li>Programas que não terminam são equivalentes à proposições falsa.</li>
<li>Escrever um programa que não termina em Coq é provar o falso, é introduzir inconsistência.</li>
</ul>
</div>
</div>
<div id="outline-container-org80357fb" class="outline-3">
<h3 id="org80357fb"><span class="section-number-3">1.2</span> Recursão Primitiva</h3>
<div class="outline-text-3" id="text-1-2">
<ul class="org-ul">
<li>Recursão Estrita, Recursão primitiva e Recursão estrutural significam a mesma coisa.</li>
<li>Um tipo especial de recursão.</li>
<li>Todas as funções com recursão estrita terminam, portanto são totais.</li>
<li>Mas nem todas as funções totais são de recursão primitiva!</li>
<li>São como laços contados em linguagens imperativas, com limite de iteração fixo (<i>for</i>, etc…).</li>
<li>As funções da Aritmética de Peano são exatamente essas com recursão primitiva (adinção, subtração, multiplicação…).</li>
</ul>
</div>
</div>
<div id="outline-container-org6751725" class="outline-3">
<h3 id="org6751725"><span class="section-number-3">1.3</span> Recursão Primitiva em Coq</h3>
<div class="outline-text-3" id="text-1-3">
<ul class="org-ul">
<li>Em Gallina, funções com recursão primitiva são aquelas cuja chamada recursiva acontece apenas em subtermos sintáticos do argumento original.</li>
<li>Ou seja, a chamada recursiva ocorre num subtermo que pode ser capturado por um casamento de padrão.</li>
<li>Isso é feito por meio do comando <i>Fixpoint</i>, que é similar ao <i>Definition</i> mas permite recursão primitiva.</li>
<li>Por exemplo, a função abaixo testa se um número é par.</li>
<li>Note que <i>n'</i> é subtermo de <i>n</i>.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">evenb</span> (<span style="color: #DFAF8F;">n</span>:nat) : bool :=
<span style="color: #7CB8BB;">match</span> n <span style="color: #7CB8BB;">with</span>
| O => true
| S O => false
| S (S n') => evenb n'
<span style="color: #7CB8BB;">end</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org7e007ec" class="outline-3">
<h3 id="org7e007ec"><span class="section-number-3">1.4</span> Apenas alguns testes</h3>
<div class="outline-text-3" id="text-1-4">
<ul class="org-ul">
<li>Vamos aproveitar para definir ímpar como sendo não par.</li>
<li>Vamos testar!</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Definition</span> <span style="color: #93E0E3;">oddb</span> (<span style="color: #DFAF8F;">n</span>:nat) : bool := negb (evenb n).
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_oddb1</span>: oddb 1 = true.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_oddb2</span>: oddb 4 = false.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org8aec026" class="outline-2">
<h2 id="org8aec026"><span class="section-number-2">2</span> Funções da A.P.</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-orgb4071ad" class="outline-3">
<h3 id="orgb4071ad"><span class="section-number-3">2.1</span> Função soma de Peano</h3>
<div class="outline-text-3" id="text-2-1">
<ul class="org-ul">
<li>A função soma abaixo recebe dois números.</li>
<li>É feito o casamento de padrão no primeiro argumento:
<ol class="org-ol">
<li>Se for zero, então o resultado é o valor do segundo número.</li>
<li>Se for um sucessor, então o resultado é o sucessor da soma de <i>n'</i> com <i>m</i>. Adiciona um sucessor no resultado para cada sucessor do primeiro argumento.</li>
</ol></li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Module</span> <span style="color: #93E0E3;">NatPlayground2</span>.
<span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">plus</span> (<span style="color: #DFAF8F;">n</span> : nat) (<span style="color: #DFAF8F;">m</span> : nat) : nat :=
<span style="color: #7CB8BB;">match</span> n <span style="color: #7CB8BB;">with</span>
| O => m
| S n' => S (plus n' m)
<span style="color: #7CB8BB;">end</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-orga2e856c" class="outline-3">
<h3 id="orga2e856c"><span class="section-number-3">2.2</span> Exemplo de soma</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-coq">Compute (plus 3 2).
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">plus (S (S (S O))) (S (S O))</span>
<span style="color: #7F9F7F;">==> S (plus (S (S O)) (S (S O)))</span>
<span style="color: #7F9F7F;"> by the second clause of the match</span>
<span style="color: #7F9F7F;">==> S (S (plus (S O) (S (S O))))</span>
<span style="color: #7F9F7F;"> by the second clause of the match</span>
<span style="color: #7F9F7F;">==> S (S (S (plus O (S (S O)))))</span>
<span style="color: #7F9F7F;"> by the second clause of the match</span>
<span style="color: #7F9F7F;">==> S (S (S (S (S O))))</span>
<span style="color: #7F9F7F;"> by the first clause of the match</span>
<span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orge0ce0e8" class="outline-3">
<h3 id="orge0ce0e8"><span class="section-number-3">2.3</span> Multiplicação de Peano</h3>
<div class="outline-text-3" id="text-2-3">
<ul class="org-ul">
<li>Se os argumentos tiverem o mesmo tipo eles podem ser agrupados como <i>(n m : nat)</i>.</li>
<li>A função multiplicação soma o segundo argumento uma vez para cada sucessor do primeiro argumento.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">mult</span> (<span style="color: #DFAF8F;">n m</span> : nat) : nat :=
<span style="color: #7CB8BB;">match</span> n <span style="color: #7CB8BB;">with</span>
| O => O
| S n' => plus m (mult n' m)
<span style="color: #7CB8BB;">end</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_mult1</span>: (mult 3 3) = 9.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org800d4b0" class="outline-3">
<h3 id="org800d4b0"><span class="section-number-3">2.4</span> Subtração de Peano</h3>
<div class="outline-text-3" id="text-2-4">
<ul class="org-ul">
<li>É possível casar dois valores ao mesmo tempo ao colocar uma virgula entre eles.</li>
<li><i>_</i> é um padrão coringa.</li>
<li>A função subtração funciona assim:
<ol class="org-ol">
<li>Zero menos qualquer coisa dá zero. (Lembre-se que não tem negativo e a função precisa ser total).</li>
<li>Um número (sucessor) menos zero coisa resulta nesse número.</li>
<li>Um número (sucessor) menos um número (sucessor) resulta na subtração dos predecessores.</li>
</ol></li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">minus</span> (<span style="color: #DFAF8F;">n m</span>:nat) : nat :=
<span style="color: #7CB8BB;">match</span> n, m <span style="color: #7CB8BB;">with</span>
| O , _ => O
| S _ , O => n
| S n', S m' => minus n' m'
<span style="color: #7CB8BB;">end</span>.
<span style="color: #F0DFAF; font-weight: bold;">End</span> <span style="color: #93E0E3;">NatPlayground2</span>.
</pre>
</div>
</div>
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<div id="outline-container-orgb42688a" class="outline-3">
<h3 id="orgb42688a"><span class="section-number-3">2.5</span> Algumas Notações para A.P.</h3>
<div class="outline-text-3" id="text-2-5">
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Notation</span> <span style="color: #CC9393;">"x + y"</span> := (plus x y)
(<span style="color: #7CB8BB;">at</span> level 50, <span style="color: #BFEBBF; background-color: #3F3F3F;">left</span> associativity)
: nat_scope.
<span style="color: #F0DFAF; font-weight: bold;">Notation</span> <span style="color: #CC9393;">"x - y"</span> := (minus x y)
(<span style="color: #7CB8BB;">at</span> level 50, <span style="color: #BFEBBF; background-color: #3F3F3F;">left</span> associativity)
: nat_scope.
<span style="color: #F0DFAF; font-weight: bold;">Notation</span> <span style="color: #CC9393;">"x * y"</span> := (mult x y)
(<span style="color: #7CB8BB;">at</span> level 40, <span style="color: #BFEBBF; background-color: #3F3F3F;">left</span> associativity)
: nat_scope.
<span style="color: #F0DFAF; font-weight: bold;">Check</span> ((0 + 1) + 1).
</pre>
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<div id="outline-container-orgab9ecd4" class="outline-2">
<h2 id="orgab9ecd4"><span class="section-number-2">3</span> Exemplos de funções usando A.P.</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgc3287ac" class="outline-3">
<h3 id="orgc3287ac"><span class="section-number-3">3.1</span> Função fatorial</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<b>Exercício realizado em aula</b>
</p>
<p>
<i>factorial(0) = 1</i>
</p>
<p>
<i>factorial(n) = n * factorial(n-1)</i>
</p>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">factorial</span> (<span style="color: #DFAF8F;">n</span>:nat) : nat
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">REPLACE THIS LINE WITH ":= _your_definition_ ." </span><span style="color: #5F7F5F;">*)</span>. <span style="background-color: #ff0000;">Admitted</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_factorial1</span>: (factorial 3) = 6.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">FILL IN HERE </span><span style="color: #5F7F5F;">*)</span> <span style="background-color: #ff0000;">Admitted</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_factorial2</span>: (factorial 5) = (mult 10 12).
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">FILL IN HERE </span><span style="color: #5F7F5F;">*)</span> <span style="background-color: #ff0000;">Admitted</span>
</pre>
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<div id="outline-container-org4e97b9e" class="outline-3">
<h3 id="org4e97b9e"><span class="section-number-3">3.2</span> Função teste de igualdade booleano</h3>
<div class="outline-text-3" id="text-3-2">
<ul class="org-ul">
<li>Como funciona a seguinte função de igualdade booleana?</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">beq_nat</span> (<span style="color: #DFAF8F;">n m</span> : nat) : bool :=
<span style="color: #7CB8BB;">match</span> n <span style="color: #7CB8BB;">with</span>
| O => <span style="color: #7CB8BB;">match</span> m <span style="color: #7CB8BB;">with</span>
| O => true
| S m' => false
<span style="color: #7CB8BB;">end</span>
| S n' => <span style="color: #7CB8BB;">match</span> m <span style="color: #7CB8BB;">with</span>
| O => false
| S m' => beq_nat n' m'
<span style="color: #7CB8BB;">end</span>
<span style="color: #7CB8BB;">end</span>.
</pre>
</div>
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<div id="outline-container-orgdbe1507" class="outline-3">
<h3 id="orgdbe1507"><span class="section-number-3">3.3</span> Função teste de menor ou igual booleano</h3>
<div class="outline-text-3" id="text-3-3">
<ul class="org-ul">
<li>Como funciona a seguinte função de menor/igual bololeana?</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Fixpoint</span> <span style="color: #93E0E3;">leb_nat</span> (<span style="color: #DFAF8F;">n m</span> : nat) : bool :=
<span style="color: #7CB8BB;">match</span> n <span style="color: #7CB8BB;">with</span>
| O => true
| S n' =>
<span style="color: #7CB8BB;">match</span> m <span style="color: #7CB8BB;">with</span>
| O => false
| S m' => leb_nat n' m'
<span style="color: #7CB8BB;">end</span>
<span style="color: #7CB8BB;">end</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_leb1</span>: (leb_nat 2 2) = true.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_leb2</span>: (leb_nat 2 4) = true.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_leb3</span>: (leb_nat 4 2) = false.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org64a7052" class="outline-3">
<h3 id="org64a7052"><span class="section-number-3">3.4</span> Função teste menor booleano</h3>
<div class="outline-text-3" id="text-3-4">
<p>
<b>Exercício realizado em sala</b>
</p>
<ul class="org-ul">
<li>Dica: <i>n < m <-> ? <= ?</i></li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Definition</span> <span style="color: #93E0E3;">ltb_nat</span> (<span style="color: #DFAF8F;">n m</span> : nat) : bool
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">REPLACE THIS LINE WITH ":= _your_definition_ ." </span><span style="color: #5F7F5F;">*)</span>. <span style="background-color: #ff0000;">Admitted</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_ltb_nat1</span>: (ltb_nat 2 2) = false.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">FILL IN HERE </span><span style="color: #5F7F5F;">*)</span> <span style="background-color: #ff0000;">Admitted</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_ltb_nat2</span>: (ltb_nat 2 4) = true.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">FILL IN HERE </span><span style="color: #5F7F5F;">*)</span> <span style="background-color: #ff0000;">Admitted</span>.
<span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">test_ltb_nat3</span>: (ltb_nat 4 2) = false.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">FILL IN HERE </span><span style="color: #5F7F5F;">*)</span> <span style="background-color: #ff0000;">Admitted</span>.
</pre>
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<div id="postamble" class="status">
<p class="date">Date: 07/05/2018</p>
<p class="author">Author: Rafael Castro - rafaelcgs10.github.io/coq</p>
<p class="date">Created: 2018-05-12 sáb 11:45</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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