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<h1 class="title">Aula 11 - Táticas Apply e Inversion</h1>
<div id="outline-container-orga8a3fff" class="outline-2">
<h2 id="orga8a3fff"><span class="section-number-2">1</span> A Tática <i>apply</i></h2>
<div class="outline-text-2" id="text-1">
</div>
<div id="outline-container-orgd5a34f8" class="outline-3">
<h3 id="orgd5a34f8"><span class="section-number-3">1.1</span> Uso de <i>apply</i> quando o objetivo é uma hipótese.</h3>
<div class="outline-text-3" id="text-1-1">
<ul class="org-ul">
<li>Algumas vezes o objetivo que queremos provar é exatamente algumas das hipóteses.</li>
<li>Na prova abaixo poderíamos apenas ter utilizado <i>rewrite -> eq2</i> após o primeiro <i>rewrite</i> e, então, /reflexivity.</li>
<li>Porém, podemos atingir o mesmo objetivo utilizando <i>apply eq2</i>.</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;">silly1</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m o p</span> : nat),
n = m ->
[n;o] = [n;p] ->
[n;o] = [m;p].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m o p eq1 eq2.
<span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> <- eq1.
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq2. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org3238ea9" class="outline-3">
<h3 id="org3238ea9"><span class="section-number-3">1.2</span> Uso de <i>apply</i> quando a conclusão de uma hipótese unifica com o objetivo</h3>
<div class="outline-text-3" id="text-1-2">
<ul class="org-ul">
<li>Na prova abaixo temos uma hipótese <i>eq2</i> quantificada com um <i>forall</i> sobre <i>q</i> e <i>r</i>.</li>
<li>A conclusão dessa hipótese <i>eq2</i> pode ser especializada/instanciada no objetivo.</li>
<li>A especialização irá trocar as variáveis (quantificadas) <i>q</i> por <i>n</i> e <i>r</i> por <i>m</i>.</li>
<li>A premissa da implicação <i>eq2</i> é adicionada como objetivo.</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;">silly2</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m o p</span> : nat),
n = m ->
(<span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">q r</span> : nat), q = r -> [q;o] = [r;p]) ->
[n;o] = [m;p].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m o p eq1 eq2.
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq2. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq1. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-orgdfd3996" class="outline-3">
<h3 id="orgdfd3996"><span class="section-number-3">1.3</span> Mais um exemplo!</h3>
<div class="outline-text-3" id="text-1-3">
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">silly2a</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m</span> : nat),
(n,n) = (m,m) ->
(<span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">q r</span> : nat), (q,q) = (r,r) -> [q] = [r]) ->
[n] = [m].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m eq1 eq2.
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq2. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq1. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-orge3596fd" class="outline-3">
<h3 id="orge3596fd"><span class="section-number-3">1.4</span> Exemplo em sala de aula</h3>
<div class="outline-text-3" id="text-1-4">
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">silly_ex</span> :
(<span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n</span>, evenb n = true -> oddb (S n) = true) ->
evenb 3 = true ->
oddb 4 = true.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<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: #9FC59F;">(** [] *)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org19415f0" class="outline-3">
<h3 id="org19415f0"><span class="section-number-3">1.5</span> A tática <i>symmetry</i></h3>
<div class="outline-text-3" id="text-1-5">
<ul class="org-ul">
<li>A tática <i>symmetry</i> inverte os lados da igualdade.</li>
<li>Aplicar <i>simpl</i> é opcional, pois <i>apply</i> iria realizar as simplificações se necessário.</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;">silly3_firsttry</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n</span> : nat),
true = beq_nat n 5 ->
beq_nat (S (S n)) 7 = true.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n H.
<span style="color: #BFEBBF; background-color: #3F3F3F;">symmetry</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</span>.
<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-orgf09f260" class="outline-3">
<h3 id="orgf09f260"><span class="section-number-3">1.6</span> Teorema: igualdade é transitiva</h3>
<div class="outline-text-3" id="text-1-6">
<ul class="org-ul">
<li>O teorema abaixo nos ajudará numa próxima prova.</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;">trans_eq</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">X</span>:<span style="color: #7CB8BB;">Type</span>) (<span style="color: #DFAF8F;">n m o</span> : X),
n = m -> m = o -> n = o.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> X n m o eq1 eq2. <span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> -> eq1. <span style="color: #BFEBBF; background-color: #3F3F3F;">rewrite</span> -> eq2.
<span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org60525f3" class="outline-3">
<h3 id="org60525f3"><span class="section-number-3">1.7</span> A Tática <i>apply … with …</i></h3>
<div class="outline-text-3" id="text-1-7">
<ul class="org-ul">
<li>Podemos aplicar (<i>apply</i>) o teorema <i>trans _eq</i> para prova o teorema abaixo.</li>
<li>A tática <i>apply</i> pode receber, além de hipóteses, teoremas anteriores.</li>
<li><i>apply</i> gera a substituição de <i>n</i> para <i>[a;b]</i> e <i>o</i> para <i>[e;f]</i>.</li>
<li>Nota-se que coq não seria capaz de adivinhar como a variável <i>m</i> do teorema <i>trans _eq</i> devem ser especializada.</li>
<li>A cláusula <i>with</i> permite especificar como uma variável quantificada deve ser especializada.</li>
<li>A atribuição <i>(m:=[c;d])</i> diz que a variável <i>m</i> do teorema <i>trans _eq</i> deve ser trocada pela lista <i>[c;d]</i>.</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;">trans_eq_example</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">a b c d e f</span> : nat),
[a;b] = [c;d] ->
[c;d] = [e;f] ->
[a;b] = [e;f].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> a b c d e f eq1 eq2.
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> trans_eq <span style="color: #7CB8BB;">with</span> (m:=[c;d]).
<span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq1. <span style="color: #BFEBBF; background-color: #3F3F3F;">apply</span> eq2. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org5cf3af9" class="outline-2">
<h2 id="org5cf3af9"><span class="section-number-2">2</span> A Tática <i>inversion</i></h2>
<div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-orga96796c" class="outline-3">
<h3 id="orga96796c"><span class="section-number-3">2.1</span> Construtores de tipos indutivos</h3>
<div class="outline-text-3" id="text-2-1">
<ul class="org-ul">
<li>O construtor do tipo indutivo <i>nat</i> define que ou um número natural é zero ou é um sucessor de um número natural.</li>
<li>Há fatos sobre essa definição que estão implícitos:
<ol class="org-ol">
<li>O construtor <i>S</i> é injetivo: Se <i>S n = S m</i>, então <i>n = m</i>.</li>
<li>Os construtores <i>0</i> e <i>S</i> são disjuntos: <i>0</i> não é igual a <i>S n</i> para qualquer <i>n</i>.</li>
</ol></li>
<li>Ser injetivo significa que uma função mapea diferentes argumentos em diferentes resultados!</li>
<li>Na realidade, isso se aplica para todos os tipos definidos indutivamente!</li>
</ul>
<div class="org-src-container">
<pre class="src src-coq"><span style="color: #9FC59F;">(** </span>
<span style="color: #9FC59F;"> Inductive nat : Type :=</span>
<span style="color: #9FC59F;"> | O : nat</span>
<span style="color: #9FC59F;"> | S : nat -> nat.</span>
<span style="color: #9FC59F;">*)</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgaf91556" class="outline-3">
<h3 id="orgaf91556"><span class="section-number-3">2.2</span> A Tática <i>inversion</i></h3>
<div class="outline-text-3" id="text-2-2">
<ul class="org-ul">
<li>A tática <i>inversion</i> permite explorar esses dois fatos sobre os tipos indutivos.</li>
<li>Para msotrar como <i>inversion</i> funciona vamos demonstrar que <i>S</i> é injetivo.</li>
<li>A utilizar <i>inversion H</i> estamos pedindo para o Coq gerar todas as equações que podem ser inferidas de <i>H</i> e reescrevendo as mesmas no objetivo ao longo disso.</li>
<li>No caso abaixo, Coq inferiu <i>n = m</i> e fez a reescrita no objetivo.</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;">S_injective</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m</span> : nat),
S n = S m ->
n = m.
<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;">inversion</span> H. <span style="color: #BFEBBF;">reflexivity</span>.
<span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org27ee17d" class="outline-3">
<h3 id="org27ee17d"><span class="section-number-3">2.3</span> Mais um exemplo de <i>inversion</i></h3>
<div class="outline-text-3" id="text-2-3">
<ul class="org-ul">
<li>O exemplo abaixo mostra como o <i>inversion</i> consegue derivar novas equações analisando a definição do tipo indutivo de listas.</li>
<li><i>inversion</i> cria equações que satisfazem a igualdade de <i>H</i>.</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;">inversion_ex1</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m o</span> : nat),
[n; m] = [o; o] ->
[n] = [m].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m o H. <span style="color: #BFEBBF; background-color: #3F3F3F;">inversion</span> H. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org21e3bbb" class="outline-3">
<h3 id="org21e3bbb"><span class="section-number-3">2.4</span> Cláusula <i>as</i></h3>
<div class="outline-text-3" id="text-2-4">
<ul class="org-ul">
<li>Podemos especificar os nomes das equações com a cláusula <i>as</i>.</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;">inversion_ex2</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m</span> : nat),
[n] = [m] ->
n = m.
<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;">inversion</span> H <span style="color: #7CB8BB;">as</span> [Hnm]. <span style="color: #BFEBBF;">reflexivity</span>. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
</pre>
</div>
</div>
</div>
<div id="outline-container-org2b79c31" class="outline-3">
<h3 id="org2b79c31"><span class="section-number-3">2.5</span> Hipóteses falsas</h3>
<div class="outline-text-3" id="text-2-5">
<ul class="org-ul">
<li>Quando há uma hipótese com uma igualade entre dos construtores diferentes, por exemplo <i>S n = 0</i>, <i>inversion</i> resolve o objetivo imediatamente.</li>
<li>No exemplo abaixo temos uma clara contradição ou falsidade na hipótese <i>H</i> do segundo caso.</li>
<li>O principio da explosão diz que do falso tudo se segue. Então quando tem-se uma hipótese falsa, qualquer objetivo é imediatamente provado. Ex: Se 0 = 1, então P = NP.</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;">beq_nat_0_l</span> : <span style="color: #7CB8BB;">forall</span> <span style="color: #DFAF8F;">n</span>,
beq_nat 0 n = true -> n = 0.
<span style="color: #F0DFAF; font-weight: bold;">Proof</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: #5F7F5F;">(* </span><span style="color: #7F9F7F;">n = 0 </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> H. <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;">n = S n' </span><span style="color: #5F7F5F;">*)</span> <span style="color: #BFEBBF; background-color: #3F3F3F;">simpl</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-org78bbae7" class="outline-3">
<h3 id="org78bbae7"><span class="section-number-3">2.6</span> Mais exemplos de explosões</h3>
<div class="outline-text-3" id="text-2-6">
<ul class="org-ul">
<li>Supondo premissas falsas, tudo é verdade!</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;">inversion_ex4</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n</span> : nat),
S n = O ->
2 + 2 = 5.
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n contra. <span style="color: #BFEBBF; background-color: #3F3F3F;">inversion</span> contra. <span style="color: #F0DFAF; font-weight: bold;">Qed</span>.
<span style="color: #F0DFAF; font-weight: bold;">Theorem</span> <span style="color: #93E0E3;">inversion_ex5</span> : <span style="color: #7CB8BB;">forall</span> (<span style="color: #DFAF8F;">n m</span> : nat),
false = true ->
[n] = [m].
<span style="color: #F0DFAF; font-weight: bold;">Proof</span>.
<span style="color: #BFEBBF; background-color: #3F3F3F;">intros</span> n m 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-org07d56ba" class="outline-3">
<h3 id="org07d56ba"><span class="section-number-3">2.7</span> Como <i>inversion</i> funciona de maneira geral (para igualdades)</h3>
<div class="outline-text-3" id="text-2-7">
<ul class="org-ul">
<li>Numa hipótese <i>H</i> da forma <i>c a1 a2 … an = d b1 b2 b3 … bm</i>:
<ol class="org-ol">
<li>Se <i>c = b</i>, então por injetividade sabemos que <i>a1 = b1, a2 = b2…</i>. <i>inversion</i> adicona esses fatos ao contexto.</li>
<li>Se <i>c</i> e <i>b</i> são diferentes, então a hipótese <i>H</i> é uma contradição. Neste caso o objetivo é imediatamente provado por <i>inversion</i>.</li>
</ol></li>
</ul>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="date">Date: 28/05/2018</p>
<p class="author">Author: Rafael Castro - rafaelcgs10.github.io/coq</p>
<p class="date">Created: 2018-05-25 sex 20:06</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
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