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<title>compcert.lib.Decidableplus</title>
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<h1 class="libtitle">Library compcert.lib.Decidableplus</h1>
<div class="code">
<br/>
</div>
<div class="doc">
This library provides type classes and tactics to decide logical
propositions by reflection into computable Boolean equalities.
It extends the <span class="inlinecode"><span class="id" title="library">DecidableClass</span></span> module from the standard library
of Coq 8.5 with more instances of decidable properties, including
universal and existential quantification over finite types.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#"><span class="id" title="library">DecidableClass</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.lib.Coqlib.html#"><span class="id" title="library">Coqlib</span></a>.<br/>
<br/>
<span class="id" title="keyword">Ltac</span> <span class="id" title="var">decide_goal</span> := <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">reflexivity</span>.<br/>
<br/>
</div>
<div class="doc">
Deciding logical connectives
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_not"><span class="id" title="instance">Decidable_not</span></a> (<span class="id" title="var">P</span>: <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">¬</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a>. <span class="id" title="tactic">split</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete_alt"><span class="id" title="lemma">Decidable_complete_alt</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound_alt"><span class="id" title="lemma">Decidable_sound_alt</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_equiv"><span class="id" title="instance">Decidable_equiv</span></a> (<span class="id" title="var">P</span> <span class="id" title="var">Q</span>: <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) (<span class="id" title="var">dQ</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">↔</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#eqb"><span class="id" title="definition">Bool.eqb</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span>) (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">Q</span> <span class="id" title="var">dQ</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#eqb_true_iff"><span class="id" title="lemma">eqb_true_iff</span></a>.<br/>
<span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; [<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">H</span> | <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>]; <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">destruct</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">Q</span> <span class="id" title="var">dQ</span>) <span class="id" title="var">eqn</span>:<span class="id" title="var">D</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound_alt"><span class="id" title="lemma">Decidable_sound_alt</span></a>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete_alt"><span class="id" title="lemma">Decidable_complete_alt</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_and"><span class="id" title="instance">Decidable_and</span></a> (<span class="id" title="var">P</span> <span class="id" title="var">Q</span>: <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) (<span class="id" title="var">dQ</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">Q</span> <span class="id" title="var">dQ</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a>. <span class="id" title="tactic">rewrite</span> ! <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_spec"><span class="id" title="method">Decidable_spec</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_or"><span class="id" title="instance">Decidable_or</span></a> (<span class="id" title="var">P</span> <span class="id" title="var">Q</span>: <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) (<span class="id" title="var">dQ</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">Q</span> <span class="id" title="var">dQ</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#orb_true_iff"><span class="id" title="lemma">orb_true_iff</span></a>. <span class="id" title="tactic">rewrite</span> ! <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_spec"><span class="id" title="method">Decidable_spec</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_implies"><span class="id" title="instance">Decidable_implies</span></a> (<span class="id" title="var">P</span> <span class="id" title="var">Q</span>: <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a>) (<span class="id" title="var">dQ</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Q"><span class="id" title="variable">Q</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <span class="id" title="keyword">if</span> @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span> <span class="id" title="keyword">then</span> @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">Q</span> <span class="id" title="var">dQ</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
- <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
- <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">destruct</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> <span class="id" title="var">P</span> <span class="id" title="var">dP</span>) <span class="id" title="var">eqn</span>:<span class="id" title="var">WP</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Deciding equalities.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Definition</span> <a name="Decidable_eq"><span class="id" title="definition">Decidable_eq</span></a> {<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>} (<span class="id" title="var">eqdec</span>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a><a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {|<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">Decidable_witness</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">:=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">proj_sumbool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">eqdec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Build_Decidable"><span class="id" title="constructor">)</span></a><br/>
|}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="var">InvBooleans</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">subst</span> <span class="id" title="var">y</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Coqlib.html#dec_eq_true"><span class="id" title="lemma">dec_eq_true</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_eq_bool"><span class="id" title="instance">Decidable_eq_bool</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#eqb"><span class="id" title="definition">Bool.eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#eqb_true_iff"><span class="id" title="lemma">eqb_true_iff</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_eq_nat"><span class="id" title="instance">Decidable_eq_nat</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Arith.EqNat.html#beq_nat"><span class="id" title="abbreviation">beq_nat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Arith.EqNat.html#beq_nat_true_iff"><span class="id" title="abbreviation">beq_nat_true_iff</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_eq_positive"><span class="id" title="instance">Decidable_eq_positive</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.PArith.BinPos.html#Pos.eqb"><span class="id" title="definition">Pos.eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.PArith.BinPos.html#Pos.eqb_eq"><span class="id" title="lemma">Pos.eqb_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_eq_Z"><span class="id" title="instance">Decidable_eq_Z</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eqb_eq"><span class="id" title="lemma">Z.eqb_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Deciding order on Z
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_le_Z"><span class="id" title="instance">Decidable_le_Z</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.leb"><span class="id" title="definition">Z.leb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.leb_le"><span class="id" title="lemma">Z.leb_le</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_lt_Z"><span class="id" title="instance">Decidable_lt_Z</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_ge_Z"><span class="id" title="instance">Decidable_ge_Z</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#162be8364124d2d389b2e8a3fc881ab2"><span class="id" title="notation">≥</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.geb"><span class="id" title="definition">Z.geb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.geb_le"><span class="id" title="lemma">Z.geb_le</span></a>. <span class="id" title="tactic">intuition</span> <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_gt_Z"><span class="id" title="instance">Decidable_gt_Z</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a1c07c706e79df7cc9eee5b1e369014a"><span class="id" title="notation">></span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.gtb"><span class="id" title="definition">Z.gtb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.gtb_lt"><span class="id" title="lemma">Z.gtb_lt</span></a>. <span class="id" title="tactic">intuition</span> <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_divides"><span class="id" title="instance">Decidable_divides</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#22aee23095fa75a48e17ac5311892ed3"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#22aee23095fa75a48e17ac5311892ed3"><span class="id" title="notation">|</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#22aee23095fa75a48e17ac5311892ed3"><span class="id" title="notation">)</span></a> := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <span class="id" title="var">x</span>)%<span class="id" title="var">Z</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eqb_eq"><span class="id" title="lemma">Z.eqb_eq</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">∃</span> (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <span class="id" title="var">x</span>). <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">intros</span> [<span class="id" title="var">k</span> <span class="id" title="var">EQ</span>]. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eqb_eq"><span class="id" title="lemma">Z.eqb_eq</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.eq_dec"><span class="id" title="definition">Z.eq_dec</span></a> <span class="id" title="var">x</span> 0).<br/>
<span class="id" title="tactic">subst</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.mul_0_r"><span class="id" title="lemma">Z.mul_0_r</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">subst</span> <span class="id" title="var">y</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">k</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <span class="id" title="var">x</span>).<br/>
{ <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_unique_full"><span class="id" title="lemma">Zdiv_unique_full</span></a> <span class="id" title="keyword">with</span> 0. <span class="id" title="tactic">red</span>; <span class="id" title="tactic">omega</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">EQ</span>; <span class="id" title="tactic">ring</span>. }<br/>
<span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Deciding properties over lists
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_forall_in_list"><span class="id" title="instance">Decidable_forall_in_list</span></a> : <br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">l</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">P</span>: <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> (<span class="id" title="var">P</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) (<span class="id" title="var">dP</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)) <span class="id" title="var">l</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#forallb_forall"><span class="id" title="lemma">List.forallb_forall</span></a>. <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
- <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_exists_in_list"><span class="id" title="instance">Decidable_exists_in_list</span></a> : <br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">l</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">P</span>: <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#existsb"><span class="id" title="definition">List.existsb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> (<span class="id" title="var">P</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) (<span class="id" title="var">dP</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)) <span class="id" title="var">l</span><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#existsb_exists"><span class="id" title="lemma">List.existsb_exists</span></a>. <span class="id" title="tactic">split</span>.<br/>
- <span class="id" title="tactic">intros</span> (<span class="id" title="var">x</span> & <span class="id" title="var">U</span> & <span class="id" title="var">V</span>). <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">split</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
- <span class="id" title="tactic">intros</span> (<span class="id" title="var">x</span> & <span class="id" title="var">U</span> & <span class="id" title="var">V</span>). <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">split</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Types with finitely many elements.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Class</span> <a name="Finite"><span class="id" title="record">Finite</span></a> (<span class="id" title="var">T</span>: <span class="id" title="keyword">Type</span>) := {<br/>
<a name="Finite_elements"><span class="id" title="projection">Finite_elements</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>;<br/>
<a name="Finite_elements_spec"><span class="id" title="projection">Finite_elements_spec</span></a>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a><br/>
}.<br/>
<br/>
</div>
<div class="doc">
Deciding forall and exists quantification over finite types.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_forall"><span class="id" title="instance">Decidable_forall</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">T</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">fT</span>: <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">P</span>: <a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> (<span class="id" title="var">P</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) (<span class="id" title="var">dP</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)) (@<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> <span class="id" title="var">T</span> <span class="id" title="var">fT</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#forallb_forall"><span class="id" title="lemma">List.forallb_forall</span></a>. <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements_spec"><span class="id" title="method">Finite_elements_spec</span></a>.<br/>
- <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Decidable_exists"><span class="id" title="instance">Decidable_exists</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">T</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">fT</span>: <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">P</span>: <a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">dP</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>:<a class="idref" href="compcert.lib.Decidableplus.html#T"><span class="id" title="variable">T</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)), <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable"><span class="id" title="class">Decidable</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) := {<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#existsb"><span class="id" title="definition">List.existsb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ @<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_witness"><span class="id" title="method">Decidable_witness</span></a> (<span class="id" title="var">P</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>) (<span class="id" title="var">dP</span> <a class="idref" href="compcert.lib.Decidableplus.html#x"><span class="id" title="variable">x</span></a>)) (@<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> <span class="id" title="var">T</span> <span class="id" title="var">fT</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#existsb_exists"><span class="id" title="lemma">List.existsb_exists</span></a>. <span class="id" title="tactic">split</span>.<br/>
- <span class="id" title="tactic">intros</span> (<span class="id" title="var">x</span> & <span class="id" title="var">A</span> & <span class="id" title="var">B</span>). <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_sound"><span class="id" title="lemma">Decidable_sound</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
- <span class="id" title="tactic">intros</span> (<span class="id" title="var">x</span> & <span class="id" title="var">A</span>). <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">split</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements_spec"><span class="id" title="method">Finite_elements_spec</span></a>. <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.DecidableClass.html#Decidable_complete"><span class="id" title="lemma">Decidable_complete</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Some examples of finite types.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Finite_bool"><span class="id" title="instance">Finite_bool</span></a> : <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := {<br/>
<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Finite_pair"><span class="id" title="instance">Finite_pair</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">fA</span>: <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">fB</span>: <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#B"><span class="id" title="variable">B</span></a>), <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> (<a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#B"><span class="id" title="variable">B</span></a>) := {<br/>
<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#list_prod"><span class="id" title="definition">list_prod</span></a> (@<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> <span class="id" title="var">A</span> <span class="id" title="var">fA</span>) (@<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> <span class="id" title="var">B</span> <span class="id" title="var">fB</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#in_prod"><span class="id" title="lemma">List.in_prod</span></a>; <span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements_spec"><span class="id" title="method">Finite_elements_spec</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Program Instance</span> <a name="Finite_option"><span class="id" title="instance">Finite_option</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">fA</span>: <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Decidableplus.html#Finite"><span class="id" title="class">Finite</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Decidableplus.html#A"><span class="id" title="variable">A</span></a>) := {<br/>
<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">A</span>) (@<a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements"><span class="id" title="method">Finite_elements</span></a> <span class="id" title="var">A</span> <span class="id" title="var">fA</span>)<br/>
}.<br/>
<span class="id" title="keyword">Next</span> <span class="id" title="keyword">Obligation</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">right</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#in_map"><span class="id" title="lemma">List.in_map</span></a>; <span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.lib.Decidableplus.html#Finite_elements_spec"><span class="id" title="method">Finite_elements_spec</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
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