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<title>compcert.lib.Ordered</title>
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<h1 class="libtitle">Library compcert.lib.Ordered</h1>
<div class="code">
<br/>
</div>
<div class="doc">
Constructions of ordered types, for use with the <span class="inlinecode"><span class="id" title="var">FSet</span></span> functors
for finite sets and the <span class="inlinecode"><span class="id" title="var">FMap</span></span> functors for finite maps.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.FSets.FSets.html#"><span class="id" title="library">FSets</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/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.lib.Maps.html#"><span class="id" title="library">Maps</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.lib.Integers.html#"><span class="id" title="library">Integers</span></a>.<br/>
<br/>
</div>
<div class="doc">
The ordered type of positive numbers
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="OrderedPositive"><span class="id" title="module">OrderedPositive</span></a> <: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPositive.t"><span class="id" title="definition">t</span></a> := <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>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPositive.eq"><span class="id" title="definition">eq</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Ordered.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.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPositive.lt"><span class="id" title="definition">lt</span></a> := <a class="idref" href="compcert.lib.Coqlib.html#Plt"><span class="id" title="definition">Plt</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.eq_refl"><span class="id" title="lemma">eq_refl</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.eq_sym"><span class="id" title="lemma">eq_sym</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="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_equal"><span class="id" title="abbreviation">sym_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.eq_trans"><span class="id" title="lemma">eq_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#trans_equal"><span class="id" title="abbreviation">trans_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPositive.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedPositive.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> <a class="idref" href="compcert.lib.Coqlib.html#Plt_trans"><span class="id" title="lemma">Plt_trans</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.lt_not_eq"><span class="id" title="lemma">lt_not_eq</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="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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="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.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span> <a class="idref" href="compcert.lib.Coqlib.html#Plt_ne"><span class="id" title="lemma">Plt_ne</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPositive.compare"><span class="id" title="lemma">compare</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="compcert.lib.Ordered.html#OrderedPositive.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#Compare"><span class="id" title="inductive">Compare</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</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.PArith.BinPos.html#Pos.compare"><span class="id" title="definition">Pos.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) <span class="id" title="keyword">as</span> [] <span class="id" title="var">eqn</span>:<span class="id" title="var">E</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#EQ"><span class="id" title="constructor">EQ</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.PArith.BinPos.html#Pos.compare_eq_iff"><span class="id" title="lemma">Pos.compare_eq_iff</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.PArith.BinPos.html#Pos.compare_gt_iff"><span class="id" title="lemma">Pos.compare_gt_iff</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPositive.eq_dec"><span class="id" title="definition">eq_dec</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.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="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.Ordered.html#OrderedPositive.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="compcert.lib.Coqlib.html#peq"><span class="id" title="definition">peq</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPositive"><span class="id" title="module">OrderedPositive</span></a>.<br/>
<br/>
</div>
<div class="doc">
The ordered type of integers
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="OrderedZ"><span class="id" title="module">OrderedZ</span></a> <: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedZ.t"><span class="id" title="definition">t</span></a> := <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>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedZ.eq"><span class="id" title="definition">eq</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Ordered.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.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedZ.lt"><span class="id" title="definition">lt</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zlt"><span class="id" title="abbreviation">Zlt</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.eq_refl"><span class="id" title="lemma">eq_refl</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.eq_sym"><span class="id" title="lemma">eq_sym</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="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_equal"><span class="id" title="abbreviation">sym_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.eq_trans"><span class="id" title="lemma">eq_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#trans_equal"><span class="id" title="abbreviation">trans_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_trans"><span class="id" title="abbreviation">Zlt_trans</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.lt_not_eq"><span class="id" title="lemma">lt_not_eq</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="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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="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.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">omega</span>. <span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedZ.compare"><span class="id" title="lemma">compare</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="compcert.lib.Ordered.html#OrderedZ.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#Compare"><span class="id" title="inductive">Compare</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</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.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) <span class="id" title="keyword">as</span> [] <span class="id" title="var">eqn</span>:<span class="id" title="var">E</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#EQ"><span class="id" title="constructor">EQ</span></a>. <span class="id" title="tactic">red</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.compare_eq_iff"><span class="id" title="lemma">Z.compare_eq_iff</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.compare_gt_iff"><span class="id" title="lemma">Z.compare_gt_iff</span></a>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedZ.eq_dec"><span class="id" title="definition">eq_dec</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.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="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.Ordered.html#OrderedZ.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="compcert.lib.Coqlib.html#zeq"><span class="id" title="definition">zeq</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedZ"><span class="id" title="module">OrderedZ</span></a>.<br/>
<br/>
</div>
<div class="doc">
The ordered type of machine integers
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="OrderedInt"><span class="id" title="module">OrderedInt</span></a> <: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedInt.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="compcert.lib.Integers.html#int"><span class="id" title="abbreviation">int</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedInt.eq"><span class="id" title="definition">eq</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Ordered.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.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedInt.lt"><span class="id" title="definition">lt</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</span></a> <a class="idref" href="compcert.lib.Ordered.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.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.eq_refl"><span class="id" title="lemma">eq_refl</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.eq_sym"><span class="id" title="lemma">eq_sym</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="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_equal"><span class="id" title="abbreviation">sym_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.eq_trans"><span class="id" title="lemma">eq_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#trans_equal"><span class="id" title="abbreviation">trans_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.lt_not_eq"><span class="id" title="lemma">lt_not_eq</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="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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="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.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a>,<a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedInt.compare"><span class="id" title="lemma">compare</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="compcert.lib.Ordered.html#OrderedInt.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#Compare"><span class="id" title="inductive">Compare</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Coqlib.html#zlt"><span class="id" title="definition">zlt</span></a> (<a class="idref" href="compcert.lib.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="compcert.lib.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</span></a> <span class="id" title="var">y</span>)).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Integers.html#Int.eq_dec"><span class="id" title="lemma">Int.eq_dec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#EQ"><span class="id" title="constructor">EQ</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<a class="idref" href="compcert.lib.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</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#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Integers.html#Int.unsigned"><span class="id" title="definition">Int.unsigned</span></a> <span class="id" title="var">y</span>).<br/>
<span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.lib.Integers.html#Int.repr_unsigned"><span class="id" title="lemma">Int.repr_unsigned</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.lib.Integers.html#Int.repr_unsigned"><span class="id" title="lemma">Int.repr_unsigned</span></a> <span class="id" title="var">y</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">red</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedInt.eq_dec"><span class="id" title="definition">eq_dec</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.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="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.Ordered.html#OrderedInt.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="compcert.lib.Integers.html#Int.eq_dec"><span class="id" title="lemma">Int.eq_dec</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedInt"><span class="id" title="module">OrderedInt</span></a>.<br/>
<br/>
</div>
<div class="doc">
Indexed types (those that inject into <span class="inlinecode"><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></span>) are ordered.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="OrderedIndexed"><span class="id" title="module">OrderedIndexed</span></a>(<span class="id" title="var">A</span>: <a class="idref" href="compcert.lib.Maps.html#INDEXED_TYPE"><span class="id" title="module">INDEXED_TYPE</span></a>) <: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedIndexed.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="compcert.lib.Ordered.html#A.t"><span class="id" title="axiom">A.t</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Ordered.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.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="compcert.lib.Coqlib.html#Plt"><span class="id" title="definition">Plt</span></a> (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.eq_refl"><span class="id" title="lemma">eq_refl</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.eq_sym"><span class="id" title="lemma">eq_sym</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="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_equal"><span class="id" title="abbreviation">sym_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>).<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.eq_trans"><span class="id" title="lemma">eq_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#trans_equal"><span class="id" title="abbreviation">trans_equal</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.lib.Coqlib.html#Plt_trans"><span class="id" title="lemma">Plt_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.lt_not_eq"><span class="id" title="lemma">lt_not_eq</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="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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="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.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a>; <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">red</span>; <span class="id" title="tactic">intro</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#Plt_strict"><span class="id" title="lemma">Plt_strict</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <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">Lemma</span> <a name="OrderedIndexed.compare"><span class="id" title="lemma">compare</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="compcert.lib.Ordered.html#OrderedIndexed.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#Compare"><span class="id" title="inductive">Compare</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#OrderedPositive.compare"><span class="id" title="lemma">OrderedPositive.compare</span></a> (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">exact</span> <span class="id" title="var">l</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#EQ"><span class="id" title="constructor">EQ</span></a>. <span class="id" title="tactic">red</span>; <span class="id" title="tactic">red</span> <span class="id" title="keyword">in</span> <span class="id" title="var">e</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.index_inj"><span class="id" title="axiom">A.index_inj</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>. <span class="id" title="tactic">exact</span> <span class="id" title="var">l</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedIndexed.eq_dec"><span class="id" title="lemma">eq_dec</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.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="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.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Coqlib.html#peq"><span class="id" title="definition">peq</span></a> (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="compcert.lib.Ordered.html#A.index"><span class="id" title="axiom">A.index</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">left</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.index_inj"><span class="id" title="axiom">A.index_inj</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">right</span>; <span class="id" title="tactic">red</span>; <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedIndexed"><span class="id" title="module">OrderedIndexed</span></a>.<br/>
<br/>
</div>
<div class="doc">
The product of two ordered types is ordered.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="OrderedPair"><span class="id" title="module">OrderedPair</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) <: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPair.t"><span class="id" title="definition">t</span></a> := (<a class="idref" href="compcert.lib.Ordered.html#A.t"><span class="id" title="axiom">A.t</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.Ordered.html#B.t"><span class="id" title="axiom">B.t</span></a>)%<span class="id" title="var">type</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPair.eq"><span class="id" title="definition">eq</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>) :=<br/>
<a class="idref" href="compcert.lib.Ordered.html#A.eq"><span class="id" title="axiom">A.eq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#B.eq"><span class="id" title="axiom">B.eq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.eq_refl"><span class="id" title="lemma">eq_refl</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">split</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.eq_sym"><span class="id" title="lemma">eq_sym</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="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">intuition</span> <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.eq_trans"><span class="id" title="lemma">eq_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">intuition</span> <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="OrderedPair.lt"><span class="id" title="definition">lt</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>) :=<br/>
<a class="idref" href="compcert.lib.Ordered.html#A.lt"><span class="id" title="axiom">A.lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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><br/>
<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.Ordered.html#A.eq"><span class="id" title="axiom">A.eq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#B.lt"><span class="id" title="axiom">B.lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.lt_trans"><span class="id" title="lemma">lt_trans</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</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.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">elim</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">intros</span>.<br/>
<br/>
<span class="id" title="tactic">left</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.lt_trans"><span class="id" title="axiom">A.lt_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>); <span class="id" title="tactic">auto</span>.<br/>
<br/>
<span class="id" title="tactic">left</span>. <span class="id" title="tactic">elim</span> <span class="id" title="var">H1</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#A.compare"><span class="id" title="axiom">A.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">z</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.lib.Ordered.html#A.lt_not_eq"><span class="id" title="axiom">A.lt_not_eq</span></a> <span class="id" title="var">H2</span>); <span class="id" title="tactic">intro</span>. <span class="id" title="tactic">elim</span> <span class="id" title="var">H5</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_trans"><span class="id" title="axiom">A.eq_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">z</span>). <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">generalize</span> (@<a class="idref" href="compcert.lib.Ordered.html#A.lt_not_eq"><span class="id" title="axiom">A.lt_not_eq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">z</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H5</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.lt_trans"><span class="id" title="axiom">A.lt_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_sym"><span class="id" title="axiom">A.eq_sym</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<br/>
<span class="id" title="tactic">left</span>. <span class="id" title="tactic">elim</span> <span class="id" title="var">H2</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#A.compare"><span class="id" title="axiom">A.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">z</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.lib.Ordered.html#A.lt_not_eq"><span class="id" title="axiom">A.lt_not_eq</span></a> <span class="id" title="var">H1</span>); <span class="id" title="tactic">intro</span>. <span class="id" title="tactic">elim</span> <span class="id" title="var">H5</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_trans"><span class="id" title="axiom">A.eq_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_sym"><span class="id" title="axiom">A.eq_sym</span></a>. <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">generalize</span> (@<a class="idref" href="compcert.lib.Ordered.html#A.lt_not_eq"><span class="id" title="axiom">A.lt_not_eq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H5</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.lt_trans"><span class="id" title="axiom">A.lt_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">z</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_sym"><span class="id" title="axiom">A.eq_sym</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<br/>
<span class="id" title="tactic">right</span>. <span class="id" title="tactic">elim</span> <span class="id" title="var">H1</span>; <span class="id" title="tactic">elim</span> <span class="id" title="var">H2</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">split</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_trans"><span class="id" title="axiom">A.eq_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#B.lt_trans"><span class="id" title="axiom">B.lt_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <span class="id" title="var">y</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.lt_not_eq"><span class="id" title="lemma">lt_not_eq</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="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</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="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.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a>, <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#not"><span class="id" title="definition">not</span></a>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">apply</span> (@<a class="idref" href="compcert.lib.Ordered.html#A.lt_not_eq"><span class="id" title="axiom">A.lt_not_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H3</span> <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">H3</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">apply</span> (@<a class="idref" href="compcert.lib.Ordered.html#B.lt_not_eq"><span class="id" title="axiom">B.lt_not_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H5</span> <span class="id" title="var">H2</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.compare"><span class="id" title="lemma">compare</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="compcert.lib.Ordered.html#OrderedPair.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#Compare"><span class="id" title="inductive">Compare</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#A.compare"><span class="id" title="axiom">A.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">left</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#B.compare"><span class="id" title="axiom">B.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intro</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#LT"><span class="id" title="constructor">LT</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">right</span>. <span class="id" title="tactic">tauto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#EQ"><span class="id" title="constructor">EQ</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">tauto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">right</span>. <span class="id" title="tactic">split</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Ordered.html#A.eq_sym"><span class="id" title="axiom">A.eq_sym</span></a>. <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Structures.OrderedType.html#GT"><span class="id" title="constructor">GT</span></a>. <span class="id" title="tactic">red</span>. <span class="id" title="tactic">left</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="OrderedPair.eq_dec"><span class="id" title="lemma">eq_dec</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.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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="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.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="compcert.lib.Ordered.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Ordered.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>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair.eq"><span class="id" title="definition">eq</span></a>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#A.eq_dec"><span class="id" title="axiom">A.eq_dec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">case</span> (<a class="idref" href="compcert.lib.Ordered.html#B.eq_dec"><span class="id" title="axiom">B.eq_dec</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <span class="id" title="var">y</span>)); <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">left</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">right</span>; <span class="id" title="tactic">intuition</span>.<br/>
<span class="id" title="tactic">right</span>; <span class="id" title="tactic">intuition</span>.<br/>
<span class="id" title="keyword">Defined</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Ordered.html#OrderedPair"><span class="id" title="module">OrderedPair</span></a>.<br/>
<br/>
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