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<div id="content">
<h1 class="title">Aula 13 - Lógica em Coq 1</h1>
<div id="outline-container-orgc4cd701" class="outline-2">
<h2 id="orgc4cd701"><span class="section-number-2">1</span> Proposições em Coq</h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-org58c27e6" class="outline-3">
<h3 id="org58c27e6"><span class="section-number-3">1.1</span> Revisão dos elementos de lógica vistos</h3>
<div class="outline-text-3" id="text-1-1">
<ul class="org-ul">
<li>Implicação (<i>-></i>).</li>
<li>Quantificação universal (<i>forall</i>).</li>
<li>Operador de igualdade (<i>=</i>).</li>
<li>Operadores booleanos (<i>orb</i>, <i>andb</i>).</li>
</ul>
</div>
</div>
<div id="outline-container-orgafde3e1" class="outline-3">
<h3 id="orgafde3e1"><span class="section-number-3">1.2</span> Proposições em Coq</h3>
<div class="outline-text-3" id="text-1-2">
<ul class="org-ul">
<li>Coq é uma linguagem tipada: toda expressão (bem tipada) tem um tipo.</li>
<li>Proposições são expressões!</li>
<li>Qual é o tipo de uma proposição como <i>3 = 3</i>?</li>
<li>O tipo <i>Prop</i> é o tipo das proposições!</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Check</span> 3 = 3.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> Prop </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #F0DFAF; font-weight: bold;">Check</span> <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n m</span> : nat, n + m = m + n.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> Prop </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #F0DFAF; font-weight: bold;">Check</span> 3 = 4.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> Prop </span><span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orged9ff08" class="outline-3">
<h3 id="orged9ff08"><span class="section-number-3">1.3</span> Proposições são <i>first class citizens</i>!</h3>
<div class="outline-text-3" id="text-1-3">
<ul class="org-ul">
<li>Proposições são <i>first class citizens</i> (valores de primeira classe).</li>
<li>Podem ser argumentos e retornos de funções.</li>
<li>A função abaixo retorna uma proposição.</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;">plus_fact</span> : <span style="color: #7CB8BB;">Prop</span> := 2 + 2 = 4.
<span style="color: #F0DFAF; font-weight: bold;">Check</span> plus_fact.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> plus_fact : Prop </span><span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org62ea608" class="outline-3">
<h3 id="org62ea608"><span class="section-number-3">1.4</span> Proposições parametrizadas</h3>
<div class="outline-text-3" id="text-1-4">
<ul class="org-ul">
<li>Podemos escrever funções com argumentos e que retornam uma proposição sobre os mesmos.</li>
<li>Em Coq, chamamos funções essas funções de propriedades.</li>
<li>A função abaixo define a propriedade de <i>f</i> ser injetiva.</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;">injective</span> {A B} (<span style="color: #DFAF8F;">f</span> : A -> B) :=
<span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">x y</span> : A, f x = f y -> x = y.
</pre>
</div>
</div>
</div>
<div id="outline-container-org6e94fcf" class="outline-3">
<h3 id="org6e94fcf"><span class="section-number-3">1.5</span> O operador de igualdade</h3>
<div class="outline-text-3" id="text-1-5">
<ul class="org-ul">
<li>O operador de igualdade (<i>=</i>) também é uma função.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Check</span> <span style="color: #BFEBBF;">@</span>eq.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> forall A : Type, A -> A -> Prop </span><span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org84ecd0d" class="outline-2">
<h2 id="org84ecd0d"><span class="section-number-2">2</span> Conectivos lógicos de proposições</h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org4c9dd7d" class="outline-3">
<h3 id="org4c9dd7d"><span class="section-number-3">2.1</span> Conjunção de proposições</h3>
<div class="outline-text-3" id="text-2-1">
<ul class="org-ul">
<li>A conjunção entre duas proprosições <i>A</i> e <i>B</i> é escrita como A /\ B .</li>
<li><i>\ é apenas açúcar sintático para /and</i>.</li>
<li>Para provar um objetivo com uma conjunção é necessário provar ambas as proposições.</li>
<li>A tática <i>split</i> separa uma conjunção no objetivo em dois sub-objetivos.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Example</span> <span style="color: #93E0E3;">and_example</span> : 3 + 4 = 7 /\ 2 * 2 = 4.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">split</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">3 + 4 = 7 </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">2 + 2 = 4 </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Check</span> and.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> and : Prop -> Prop -> Prop </span><span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgccc93be" class="outline-3">
<h3 id="orgccc93be"><span class="section-number-3">2.2</span> Introdução da conjunção</h3>
<div class="outline-text-3" id="text-2-2">
<ul class="org-ul">
<li>Se assumirmos as proposições <i>A</i> e <i>B</i>, então podemos concluir A /\ B.</li>
<li>Onde você já viu isso antes?</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">and_intro</span> : <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">A B</span> : <span style="color: #7CB8BB;">Prop</span>, A -> B -> A /\ B.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> A B HA HB. <span style="color: #BFEBBF; background-color: #3F3F3F;">split</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> HA.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> HB.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org0f6577d" class="outline-3">
<h3 id="org0f6577d"><span class="section-number-3">2.3</span> Conjunção numa das hipóteses</h3>
<div class="outline-text-3" id="text-2-3">
<ul class="org-ul">
<li>Uma conjunção numa das hipóteses pode ser separada em duas hipóteses.</li>
<li>A tática <i>destruct</i> na hipótese faz exatamente isso.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">and_example2</span> :
<span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n m</span> : nat, n = 0 /\ m = 0 -> n + m = 0.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m H.
<span style="color: #BFEBBF; background-color: #3F3F3F;">destruct</span> H <span style="color: #7CB8BB;">as</span> [Hn Hm].
<span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> Hn. <span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> Hm.
<span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org05efdfe" class="outline-3">
<h3 id="org05efdfe"><span class="section-number-3">2.4</span> Conjunção é comutativa</h3>
<div class="outline-text-3" id="text-2-4">
<ul class="org-ul">
<li>Algumas vezes pode ser necessário rearranjar a ordem de uma conjunção.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">and_commut</span> : <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">P Q</span> : <span style="color: #7CB8BB;">Prop</span>,
P /\ Q -> Q /\ P.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> P Q [HP HQ].
<span style="color: #BFEBBF; background-color: #3F3F3F;">split</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">left </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> HQ.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">right </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> HP. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org803c01a" class="outline-3">
<h3 id="org803c01a"><span class="section-number-3">2.5</span> Disjunção de proposições</h3>
<div class="outline-text-3" id="text-2-5">
<ul class="org-ul">
<li>Uma disjunção de proposições A \/ B é verdadeira quando ao menos uma delas é verdadeira.</li>
<li>\/ é açúcar sintático para <i>or</i>.</li>
<li>Se uma hipótese é uma disjução, então pode-se fazer análise de caso nela por meio de <i>destruct</i>.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">or_example</span> :
<span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n m</span> : nat, n = 0 \/ m = 0 -> n * m = 0.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m Hnm.
<span style="color: #BFEBBF; background-color: #3F3F3F;">destruct</span> Hnm <span style="color: #7CB8BB;">as</span> [Hn | Hm].
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">Here, [n = 0] </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> Hn. <span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">Here, [m = 0] </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> Hm. <span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> <- mult_n_O.
<span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Check</span> or.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> or : Prop -> Prop -> Prop </span><span style="color: #5F7F5F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org7d56a15" class="outline-3">
<h3 id="org7d56a15"><span class="section-number-3">2.6</span> O objetivo é uma disjunção</h3>
<div class="outline-text-3" id="text-2-6">
<ul class="org-ul">
<li>Para provar uma disjunção é preciso demostrar apenas um lado da mesma.</li>
<li>As táticas <i>left</i> e <i>right</i> servem para fazer essa escolha.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">or_intro</span> : <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">A B</span> : <span style="color: #7CB8BB;">Prop</span>, A -> A \/ B.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> A B HA.
<span style="color: #BFEBBF; background-color: #3F3F3F;">left</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> HA.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-orgfc8eba2" class="outline-3">
<h3 id="orgfc8eba2"><span class="section-number-3">2.7</span> Exemplo com left e right</h3>
<div class="outline-text-3" id="text-2-7">
<ul class="org-ul">
<li>O exemplo abaixo faz uma análise de caso em <i>n</i>.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">zero_or_succ</span> :
<span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n</span> : nat, n = 0 \/ n = S (pred n).
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">WORKED IN CLASS </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n.
<span style="color: #BFEBBF; background-color: #3F3F3F;">destruct</span> n <span style="color: #7CB8BB;">as</span> [|n'].
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">left</span>. <span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold; text-decoration: underline;">-</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">right</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-orgd408fc1" class="outline-2">
<h2 id="orgd408fc1"><span class="section-number-2">3</span> Negação</h2>
<div class="outline-text-2" id="text-3">
</div>
<div id="outline-container-orgd515782" class="outline-3">
<h3 id="orgd515782"><span class="section-number-3">3.1</span> Negação e falsidade</h3>
<div class="outline-text-3" id="text-3-1">
<ul class="org-ul">
<li>Até o momento provamos coisas serem verdadeiras.</li>
<li>Em Coq também é possível provar que coisas são falsidades.</li>
<li>A negação é representada pelo operador unário <i>~</i>, que é um açucar sintático para <i>not</i>.</li>
<li>Coq define <i>not</i> como uma função de Prop para o tipo bottom. O tipo bottom (<i>False</i>) é definido como sempre falso, nada é uma prova do mesmo .</li>
<li>O tipo bottom também pode ser visto como uma contradição <i>0 = 1</i>.</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;">MyNot</span>.
<span style="color: #F0DFAF; font-weight: bold;">Definition</span> <span style="color: #93E0E3;">not</span> (<span style="color: #DFAF8F;">P</span>:<span style="color: #7CB8BB;">Prop</span>) := P -> <span style="color: #7CB8BB;">False</span>.
<span style="color: #F0DFAF; font-weight: bold;">Notation</span> <span style="color: #CC9393;">"~ x"</span> := (not x) : type_scope.
<span style="color: #F0DFAF; font-weight: bold;">Check</span> not.
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> Prop -> Prop </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #F0DFAF; font-weight: bold;">End</span> <span style="color: #93E0E3;">MyNot</span>.
<span style="color: #F0DFAF; font-weight: bold;">Print</span> <span style="color: #7CB8BB;">False</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org8c04d69" class="outline-3">
<h3 id="org8c04d69"><span class="section-number-3">3.2</span> Do falso tudo se prova</h3>
<div class="outline-text-3" id="text-3-2">
<ul class="org-ul">
<li>A teorema abaixo afirma que do falso segue-se que qualquer proposição <i>P</i> é verdade.</li>
<li><i>destruct</i> pode ser utilizado como <i>inversion</i> na hipótese falsa.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">ex_falso_quodlibet</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">P</span>:<span style="color: #7CB8BB;">Prop</span>),
<span style="color: #7CB8BB;">False</span> -> P.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> P contra.
<span style="color: #BFEBBF; background-color: #3F3F3F;">destruct</span> contra. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-orgf9cb13d" class="outline-3">
<h3 id="orgf9cb13d"><span class="section-number-3">3.3</span> Exemplo de coisa falsa</h3>
<div class="outline-text-3" id="text-3-3">
<ul class="org-ul">
<li>Como bem estabelicido pela ciência: zero e um são diferentes!</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">zero_not_one</span> : ~(0 = 1).
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> contra. <span style="color: #BFEBBF; background-color: #3F3F3F;">inversion</span> contra.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org56eafde" class="outline-3">
<h3 id="org56eafde"><span class="section-number-3">3.4</span> Desigualdade em Coq</h3>
<div class="outline-text-3" id="text-3-4">
<ul class="org-ul">
<li>Ao invés de negar uma igualdade, pode-se utilizar o operador de desigualdade <>.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Check</span> (0 <> 1).
<span style="color: #F0DFAF; font-weight: bold;">Locate</span> <span style="color: #CC9393;">"<>"</span> .
<span style="color: #5F7F5F;">(* </span><span style="color: #7F9F7F;">===> Prop </span><span style="color: #5F7F5F;">*)</span>
<span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">zero_not_one'</span> : 0 <> 1.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> H. <span style="color: #BFEBBF; background-color: #3F3F3F;">inversion</span> H.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org8e8f59d" class="outline-3">
<h3 id="org8e8f59d"><span class="section-number-3">3.5</span> Negação é apenas uma função para falso!</h3>
<div class="outline-text-3" id="text-3-5">
<ul class="org-ul">
<li>Negação é apenas uma função para falso. Podemos utilizar <i>unfold</i> em funções.</li>
<li>Falso implica em Falso:</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">not_False</span> :
~ <span style="color: #7CB8BB;">False</span>.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">unfold</span> not. <span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> H. <span style="color: #BFEBBF; background-color: #3F3F3F;">destruct</span> H. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org2acc99a" class="outline-3">
<h3 id="org2acc99a"><span class="section-number-3">3.6</span> Introdução da dupla negação</h3>
<div class="outline-text-3" id="text-3-6">
<ul class="org-ul">
<li>Introdução da dupla negação pode ser provado.</li>
<li>Tente fazer a eliminação da dupla negação!</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">double_neg</span> : <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">P</span> : <span style="color: #7CB8BB;">Prop</span>,
P -> ~~P.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> P H. <span style="color: #BFEBBF; background-color: #3F3F3F;">unfold</span> not. <span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> G. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> G. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> H. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org8906fef" class="outline-3">
<h3 id="org8906fef"><span class="section-number-3">3.7</span> Tipo verdade em Coq</h3>
<div class="outline-text-3" id="text-3-7">
<ul class="org-ul">
<li>O tipo verdade (True) é a constante verdadeira de tipo <i>Prop</i>.</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Lemma</span> <span style="color: #93E0E3;">True_is_true</span> : <span style="color: #7CB8BB;">True</span>.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> I. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Print</span> <span style="color: #7CB8BB;">True</span>.
</pre>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="date">Date: 30/05/2018</p>
<p class="author">Author: Rafael Castro - rafaelcgs10.github.io/coq</p>
<p class="date">Created: 2018-06-01 sex 14:18</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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