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compcert.lib.Maps.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>compcert.lib.Maps</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library compcert.lib.Maps</h1>
<div class="code">
<br/>
</div>
<div class="doc">
Applicative finite maps are the main data structure used in this
project. A finite map associates data to keys. The two main operations
are <span class="inlinecode"><span class="id" title="tactic">set</span></span> <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a></span> <span class="inlinecode"><a class="idref" href="compcert.flocq.Appli.Fappli_rnd_odd.html#Odd_prop_aux.d"><span class="id" title="variable">d</span></a></span> <span class="inlinecode"><a class="idref" href="compcert.x86.SelectLongproof.html#CMCONSTR.m"><span class="id" title="variable">m</span></a></span>, which returns a map identical to <span class="inlinecode"><a class="idref" href="compcert.x86.SelectLongproof.html#CMCONSTR.m"><span class="id" title="variable">m</span></a></span> except that <span class="inlinecode"><a class="idref" href="compcert.flocq.Appli.Fappli_rnd_odd.html#Odd_prop_aux.d"><span class="id" title="variable">d</span></a></span>
is associated to <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a></span>, and <span class="inlinecode"><a class="idref" href="compcert.backend.Locations.html#Locmap.get"><span class="id" title="definition">get</span></a></span> <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a></span> <span class="inlinecode"><a class="idref" href="compcert.x86.SelectLongproof.html#CMCONSTR.m"><span class="id" title="variable">m</span></a></span> which returns the data associated
to key <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a></span> in map <span class="inlinecode"><a class="idref" href="compcert.x86.SelectLongproof.html#CMCONSTR.m"><span class="id" title="variable">m</span></a></span>. In this library, we distinguish two kinds of maps:
<ul class="doclist">
<li> Trees: the <span class="inlinecode"><a class="idref" href="compcert.backend.Locations.html#Locmap.get"><span class="id" title="definition">get</span></a></span> operation returns an option type, either <span class="inlinecode"><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a></span>
if no data is associated to the key, or <span class="inlinecode"><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a></span> <span class="inlinecode"><a class="idref" href="compcert.flocq.Appli.Fappli_rnd_odd.html#Odd_prop_aux.d"><span class="id" title="variable">d</span></a></span> otherwise.
</li>
<li> Maps: the <span class="inlinecode"><a class="idref" href="compcert.backend.Locations.html#Locmap.get"><span class="id" title="definition">get</span></a></span> operation always returns a data. If no data was explicitly
associated with the key, a default data provided at map initialization time
is returned.
<div class="paragraph"> </div>
In this library, we provide efficient implementations of trees and
maps whose keys range over the type <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> of binary positive
integers or any type that can be injected 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>. The
implementation is based on radix-2 search trees (uncompressed
Patricia trees) and guarantees logarithmic-time operations. An
inefficient implementation of maps as functions is also provided.
</li>
</ul>
</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.Classes.Equivalence.html#"><span class="id" title="library">Equivalence</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Classes.EquivDec.html#"><span class="id" title="library">EquivDec</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.lib.Coqlib.html#"><span class="id" title="library">Coqlib</span></a>.<br/>
<br/>
<span class="id" title="keyword">Local</span> <span class="id" title="keyword">Unset</span> <span class="id" title="var">Elimination</span> <span class="id" title="var">Schemes</span>.<br/>
<span class="id" title="keyword">Local</span> <span class="id" title="keyword">Unset</span> <span class="id" title="var">Case</span> <span class="id" title="var">Analysis</span> <span class="id" title="var">Schemes</span>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab8"></a><h1 class="section">The abstract signatures of trees</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="TREE"><span class="id" title="module">TREE</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.elt"><span class="id" title="axiom">elt</span></a>: <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.elt_eq"><span class="id" title="axiom">elt_eq</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.lib.Maps.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="compcert.lib.Maps.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#b"><span class="id" title="variable">b</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">Parameter</span> <a name="TREE.t"><span class="id" title="axiom">t</span></a>: <span class="id" title="keyword">Type</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.empty"><span class="id" title="axiom">empty</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.get"><span class="id" title="axiom">get</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.set"><span class="id" title="axiom">set</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="TREE.remove"><span class="id" title="axiom">remove</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<br/>
</div>
<div class="doc">
The ``good variables'' properties for trees, expressing
commutations between <span class="inlinecode"><a class="idref" href="compcert.backend.Locations.html#Locmap.get"><span class="id" title="definition">get</span></a></span>, <span class="inlinecode"><span class="id" title="tactic">set</span></span> and <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#ITree.remove"><span class="id" title="definition">remove</span></a></span>.
</div>
<div class="code">
<span class="id" title="keyword">Axiom</span> <a name="TREE.gempty"><span class="id" title="axiom">gempty</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>), <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.empty"><span class="id" title="axiom">empty</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gss"><span class="id" title="axiom">gss</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gso"><span class="id" title="axiom">gso</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gsspec"><span class="id" title="axiom">gsspec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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> <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#TREE.elt_eq"><span class="id" title="axiom">elt_eq</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.grs"><span class="id" title="axiom">grs</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gro"><span class="id" title="axiom">gro</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.grspec"><span class="id" title="axiom">grspec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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> <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#TREE.elt_eq"><span class="id" title="axiom">elt_eq</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<br/>
</div>
<div class="doc">
Extensional equality between trees.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.beq"><span class="id" title="axiom">beq</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.beq_correct"><span class="id" title="axiom">beq_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">eqA</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">t1</span> <span class="id" title="var">t2</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.beq"><span class="id" title="axiom">beq</span></a> <a class="idref" href="compcert.lib.Maps.html#eqA"><span class="id" title="variable">eqA</span></a> <a class="idref" href="compcert.lib.Maps.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="compcert.lib.Maps.html#t2"><span class="id" title="variable">t2</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">↔</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>),<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#t1"><span class="id" title="variable">t1</span></a>, <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#t2"><span class="id" title="variable">t2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y1</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y2</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#eqA"><span class="id" title="variable">eqA</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
<br/>
</div>
<div class="doc">
Applying a function to all data of a tree.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.map"><span class="id" title="axiom">map</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gmap"><span class="id" title="axiom">gmap</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.map"><span class="id" title="axiom">map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> (<a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<br/>
</div>
<div class="doc">
Same as <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#EMap.map"><span class="id" title="definition">map</span></a></span>, but the function does not receive the <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#ITree.elt"><span class="id" title="definition">elt</span></a></span> argument.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.map1"><span class="id" title="axiom">map1</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gmap1"><span class="id" title="axiom">gmap1</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.map1"><span class="id" title="axiom">map1</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<br/>
</div>
<div class="doc">
Applying a function pairwise to all data of two trees.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.combine"><span class="id" title="axiom">combine</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span>: <span class="id" title="keyword">Type</span>), <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.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#C"><span class="id" title="variable">C</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#C"><span class="id" title="variable">C</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.gcombine"><span class="id" title="axiom">gcombine</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#C"><span class="id" title="variable">C</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">m1</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.combine"><span class="id" title="axiom">combine</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</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.Maps.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a>) (<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a>).<br/>
<br/>
</div>
<div class="doc">
Enumerating the bindings of a tree.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.elements"><span class="id" title="axiom">elements</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a>).<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.elements_correct"><span class="id" title="axiom">elements_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.elements_complete"><span class="id" title="axiom">elements_complete</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.elements_keys_norepet"><span class="id" title="axiom">elements_keys_norepet</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Coqlib.html#list_norepet"><span class="id" title="inductive">list_norepet</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.elt"><span class="id" title="axiom">elt</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>)).<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.elements_extensional"><span class="id" title="axiom">elements_extensional</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span> <span class="id" title="var">n</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.elements_remove"><span class="id" title="axiom">elements_remove</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) <span class="id" title="var">i</span> <span class="id" title="var">v</span> (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#l1"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="compcert.lib.Maps.html#l2"><span class="id" title="variable">l2</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.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> (<a class="idref" href="compcert.lib.Maps.html#TREE.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#l1"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="compcert.lib.Maps.html#l2"><span class="id" title="variable">l2</span></a>.<br/>
<br/>
</div>
<div class="doc">
Folding a function over all bindings of a tree.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.fold"><span class="id" title="axiom">fold</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#B"><span class="id" title="variable">B</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.fold_spec"><span class="id" title="axiom">fold_spec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#TREE.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">a</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Maps.html#p"><span class="id" title="variable">p</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.Maps.html#p"><span class="id" title="variable">p</span></a>)) (<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>.<br/>
</div>
<div class="doc">
Same as <span class="inlinecode"><span class="id" title="tactic">fold</span></span>, but the function does not receive the <span class="inlinecode"><a class="idref" href="compcert.lib.Maps.html#ITree.elt"><span class="id" title="definition">elt</span></a></span> argument.
</div>
<div class="code">
<span class="id" title="keyword">Parameter</span> <a name="TREE.fold1"><span class="id" title="axiom">fold1</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#B"><span class="id" title="variable">B</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="TREE.fold1_spec"><span class="id" title="axiom">fold1_spec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#TREE.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#TREE.fold1"><span class="id" title="axiom">fold1</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">a</span> <span class="id" title="var">p</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="compcert.lib.Maps.html#p"><span class="id" title="variable">p</span></a>)) (<a class="idref" href="compcert.lib.Maps.html#TREE.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>.<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Maps.html#TREE"><span class="id" title="module">TREE</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab9"></a><h1 class="section">The abstract signatures of maps</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="MAP"><span class="id" title="module">MAP</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.elt"><span class="id" title="axiom">elt</span></a>: <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.elt_eq"><span class="id" title="axiom">elt_eq</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</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.Maps.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.lib.Maps.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="compcert.lib.Maps.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#b"><span class="id" title="variable">b</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">Parameter</span> <a name="MAP.t"><span class="id" title="axiom">t</span></a>: <span class="id" title="keyword">Type</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.init"><span class="id" title="axiom">init</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.get"><span class="id" title="axiom">get</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</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.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.set"><span class="id" title="axiom">set</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>), <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gi"><span class="id" title="axiom">gi</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.init"><span class="id" title="axiom">init</span></a> <a class="idref" href="compcert.lib.Maps.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.Maps.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gss"><span class="id" title="axiom">gss</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gso"><span class="id" title="axiom">gso</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gsspec"><span class="id" title="axiom">gsspec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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> <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#MAP.elt_eq"><span class="id" title="axiom">elt_eq</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gsident"><span class="id" title="axiom">gsident</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.set"><span class="id" title="axiom">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Parameter</span> <a name="MAP.map"><span class="id" title="axiom">map</span></a>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>), <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.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>.<br/>
<span class="id" title="keyword">Axiom</span> <a name="MAP.gmap"><span class="id" title="axiom">gmap</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.elt"><span class="id" title="axiom">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#MAP.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#MAP.map"><span class="id" title="axiom">map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#f"><span class="id" title="variable">f</span></a>(<a class="idref" href="compcert.lib.Maps.html#MAP.get"><span class="id" title="axiom">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Maps.html#MAP"><span class="id" title="module">MAP</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab10"></a><h1 class="section">An implementation of trees over type <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></h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Module</span> <a name="PTree"><span class="id" title="module">PTree</span></a> <: <a class="idref" href="compcert.lib.Maps.html#TREE"><span class="id" title="module">TREE</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.elt"><span class="id" title="definition">elt</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="PTree.elt_eq"><span class="id" title="definition">elt_eq</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">Inductive</span> <a name="PTree.tree"><span class="id" title="inductive">tree</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
| <a name="PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> : <a class="idref" href="compcert.lib.Maps.html#tree"><span class="id" title="inductive">tree</span></a> <span class="id" title="var">A</span><br/>
| <a name="PTree.Node"><span class="id" title="constructor">Node</span></a> : <a class="idref" href="compcert.lib.Maps.html#tree"><span class="id" title="inductive">tree</span></a> <span class="id" title="var">A</span> <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.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <span class="id" title="var">A</span> <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.Maps.html#tree"><span class="id" title="inductive">tree</span></a> <span class="id" title="var">A</span> <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.Maps.html#tree"><span class="id" title="inductive">tree</span></a> <span class="id" title="var">A</span>.<br/>
<br/>
<span class="id" title="var">Arguments</span> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> [<span class="id" title="var">A</span>].<br/>
<span class="id" title="var">Arguments</span> <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> [<span class="id" title="var">A</span>].<br/>
<span class="id" title="keyword">Scheme</span> <a name="PTree.tree_ind"><span class="id" title="definition">tree_ind</span></a> := <span class="id" title="keyword">Induction</span> <span class="id" title="keyword">for</span> <a class="idref" href="compcert.lib.Maps.html#PTree.tree"><span class="id" title="inductive">tree</span></a> <span class="id" title="keyword">Sort</span> <span class="id" title="keyword">Prop</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="compcert.lib.Maps.html#PTree.tree"><span class="id" title="inductive">tree</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.empty"><span class="id" title="definition">empty</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) := (<a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>).<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.get"><span class="id" title="definition">get</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">i</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <span class="id" title="var">o</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#get"><span class="id" title="definition">get</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">l</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#get"><span class="id" title="definition">get</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">r</span><br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.set"><span class="id" title="definition">set</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">v</span> : <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">i</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>) <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> (<a class="idref" href="compcert.lib.Maps.html#set"><span class="id" title="definition">set</span></a> <span class="id" title="var">ii</span> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> (<a class="idref" href="compcert.lib.Maps.html#set"><span class="id" title="definition">set</span></a> <span class="id" title="var">ii</span> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a>)<br/>
<span class="id" title="keyword">end</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>) <span class="id" title="var">r</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> (<a class="idref" href="compcert.lib.Maps.html#set"><span class="id" title="definition">set</span></a> <span class="id" title="var">ii</span> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a> <span class="id" title="var">l</span>) <span class="id" title="var">o</span> <span class="id" title="var">r</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">ii</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> (<a class="idref" href="compcert.lib.Maps.html#set"><span class="id" title="definition">set</span></a> <span class="id" title="var">ii</span> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.remove"><span class="id" title="definition">remove</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">i</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="var">o</span> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">r</span><br/>
<span class="id" title="keyword">end</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">ii</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <span class="id" title="var">mm</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">mm</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
<span class="id" title="keyword">end</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> (<a class="idref" href="compcert.lib.Maps.html#remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">l</span>) <span class="id" title="var">o</span> <span class="id" title="var">r</span><br/>
<span class="id" title="keyword">end</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">ii</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">r</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">r</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <span class="id" title="var">mm</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">mm</span><br/>
<span class="id" title="keyword">end</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> (<a class="idref" href="compcert.lib.Maps.html#remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">ii</span> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gempty"><span class="id" title="lemma">gempty</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>), <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.empty"><span class="id" title="definition">empty</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gss"><span class="id" title="lemma">gss</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</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="PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>), <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gempty"><span class="id" title="lemma">gempty</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gso"><span class="id" title="lemma">gso</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">j</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>; <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gsspec"><span class="id" title="lemma">gsspec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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> <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Coqlib.html#peq"><span class="id" title="definition">peq</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Coqlib.html#peq"><span class="id" title="definition">peq</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span>); [ <span class="id" title="tactic">rewrite</span> <span class="id" title="var">e</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gss"><span class="id" title="lemma">gss</span></a> | <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gso"><span class="id" title="lemma">gso</span></a>; <span class="id" title="tactic">auto</span> ].<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gsident"><span class="id" title="lemma">gsident</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHi</span> <span class="id" title="var">m2</span> <span class="id" title="var">v</span> <span class="id" title="var">H</span>); <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHi</span> <span class="id" title="var">m1</span> <span class="id" title="var">v</span> <span class="id" title="var">H</span>); <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.set2"><span class="id" title="lemma">set2</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">v1</span> <span class="id" title="var">v2</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#v2"><span class="id" title="variable">v2</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#v1"><span class="id" title="variable">v1</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.set"><span class="id" title="definition">set</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#v2"><span class="id" title="variable">v2</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">try</span> (<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHi</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="PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> : <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>), <a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.grs"><span class="id" title="lemma">grs</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>.<br/>
<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span> <span class="id" title="keyword">as</span> [ | <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>]; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>)) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>)); <span class="id" title="tactic">auto</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>.<br/>
<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">m1</span> <span class="id" title="keyword">as</span> [ | <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>]; <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>)) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">ll</span> <span class="id" title="var">oo</span> <span class="id" title="var">rr</span>)); <span class="id" title="tactic">auto</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>.<br/>
<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gro"><span class="id" title="lemma">gro</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">j</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">j</span>));<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">j</span>));<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> 1); <span class="id" title="tactic">auto</span>;<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>;<br/>
<span class="id" title="tactic">simpl</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">congruence</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.rleaf"><span class="id" title="lemma">rleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">j</span>); <span class="id" title="tactic">auto</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m2_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2_2</span>)) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m2_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2_2</span>));<br/>
[ <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m2_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2_2</span>)); <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span><br/>
| <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>; <span class="id" title="tactic">congruence</span> ].<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o0</span> <span class="id" title="var">m1_2</span>)); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>);<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m2_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2_2</span>)); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>);<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o0</span> <span class="id" title="var">m1_2</span>)) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o0</span> <span class="id" title="var">m1_2</span>));<br/>
[ <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o0</span> <span class="id" title="var">m1_2</span>)); <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>); <span class="id" title="tactic">auto</span><br/>
| <span class="id" title="tactic">apply</span> <span class="id" title="var">IHi</span>; <span class="id" title="tactic">congruence</span> ].<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m2_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2_2</span>)); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>);<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">j</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o0</span> <span class="id" title="var">m1_2</span>)); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>);<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.grspec"><span class="id" title="lemma">grspec</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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> <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#PTree.elt_eq"><span class="id" title="definition">elt_eq</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.elt_eq"><span class="id" title="definition">elt_eq</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span>). <span class="id" title="tactic">subst</span> <span class="id" title="var">j</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.grs"><span class="id" title="lemma">grs</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gro"><span class="id" title="lemma">gro</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="PTree.BOOLEAN_EQUALITY"><span class="id" title="section">BOOLEAN_EQUALITY</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="PTree.BOOLEAN_EQUALITY.A"><span class="id" title="variable">A</span></a>: <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Variable</span> <a name="PTree.BOOLEAN_EQUALITY.beqA"><span class="id" title="variable">beqA</span></a>: <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.bempty"><span class="id" title="definition">bempty</span></a> (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#bempty"><span class="id" title="definition">bempty</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="compcert.lib.Maps.html#bempty"><span class="id" title="definition">bempty</span></a> <span class="id" title="var">r</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">r</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.beq"><span class="id" title="definition">beq</span></a> (<span class="id" title="var">m1</span> <span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m1</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a>, <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.bempty"><span class="id" title="definition">bempty</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a><br/>
| <span class="id" title="var">_</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.bempty"><span class="id" title="definition">bempty</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">o1</span> <span class="id" title="var">r1</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l2</span> <span class="id" title="var">o2</span> <span class="id" title="var">r2</span> ⇒<br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">o1</span>, <span class="id" title="var">o2</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y1</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y2</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.beqA"><span class="id" title="variable">beqA</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
<span class="id" title="keyword">end</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="compcert.lib.Maps.html#beq"><span class="id" title="definition">beq</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="compcert.lib.Maps.html#beq"><span class="id" title="definition">beq</span></a> <span class="id" title="var">r1</span> <span class="id" title="var">r2</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.bempty_correct"><span class="id" title="lemma">bempty_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.bempty"><span class="id" title="definition">bempty</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H</span>). <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H0</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm2</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">simpl</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.Init.Datatypes.html#andb_true_intro"><span class="id" title="lemma">andb_true_intro</span></a>; <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm1</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">x</span>)).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm2</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">x</span>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.beq_correct"><span class="id" title="lemma">beq_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span>,<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.beq"><span class="id" title="definition">beq</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">↔</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a>),<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y1</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y2</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY.beqA"><span class="id" title="variable">beqA</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.bempty_correct"><span class="id" title="lemma">bempty_correct</span></a>. <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <span class="id" title="var">x</span>). <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">x</span> <span class="id" title="var">m2</span>); <span class="id" title="tactic">tauto</span>.<br/>
- <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>.<br/>
+ <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.beq"><span class="id" title="definition">beq</span></a>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.bempty_correct"><span class="id" title="lemma">bempty_correct</span></a>. <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <span class="id" title="var">x</span>). <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">x</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1_1</span> <span class="id" title="var">o</span> <span class="id" title="var">m1_2</span>)); <span class="id" title="tactic">tauto</span>.<br/>
+ <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">split</span>; <span class="id" title="tactic">intros</span>.<br/>
× <span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H</span>). <span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H0</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1_1</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H3</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1_2</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H1</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H3</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">o0</span>; <span class="id" title="tactic">auto</span> || <span class="id" title="tactic">congruence</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_true_intro"><span class="id" title="lemma">andb_true_intro</span></a>. <span class="id" title="tactic">split</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_true_intro"><span class="id" title="lemma">andb_true_intro</span></a>. <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>); <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">o0</span>; <span class="id" title="tactic">tauto</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm1_1</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">x</span>)).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm1_2</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">x</span>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Maps.html#PTree.BOOLEAN_EQUALITY"><span class="id" title="section">BOOLEAN_EQUALITY</span></a>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.prev_append"><span class="id" title="definition">prev_append</span></a> (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">i</span>} : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">i'</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#prev_append"><span class="id" title="definition">prev_append</span></a> <span class="id" title="var">i'</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a>)<br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">i'</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#prev_append"><span class="id" title="definition">prev_append</span></a> <span class="id" title="var">i'</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.prev"><span class="id" title="definition">prev</span></a> (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> :=<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span>:<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">j</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">i</span> <span class="id" title="var">IH</span>|<span class="id" title="var">i</span> <span class="id" title="var">IH</span>|]. 3: <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">j</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">j</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.prev_involutive"><span class="id" title="lemma">prev_involutive</span></a> <span class="id" title="var">i</span> :<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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.Maps.html#i"><span class="id" title="variable">i</span></a>.<br/>
<span class="id" title="keyword">Proof</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.prev_append_inj"><span class="id" title="lemma">prev_append_inj</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="var">j'</span> :<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j'"><span class="id" title="variable">j'</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.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#j'"><span class="id" title="variable">j'</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">j</span> <span class="id" title="var">j'</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">i</span> <span class="id" title="var">Hi</span>|<span class="id" title="var">i</span> <span class="id" title="var">Hi</span>|]; <span class="id" title="tactic">intros</span> <span class="id" title="var">j</span> <span class="id" title="var">j'</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">auto</span>;<br/>
<span class="id" title="tactic">specialize</span> (<span class="id" title="var">Hi</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H</span>); <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.xmap"><span class="id" title="definition">xmap</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)<br/>
{<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> (<a class="idref" href="compcert.lib.Maps.html#xmap"><span class="id" title="definition">xmap</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>))<br/>
(<span class="id" title="keyword">match</span> <span class="id" title="var">o</span> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</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#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) <span class="id" title="var">x</span>) <span class="id" title="keyword">end</span>)<br/>
(<a class="idref" href="compcert.lib.Maps.html#xmap"><span class="id" title="definition">xmap</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">r</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>))<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.map"><span class="id" title="definition">map</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">m</span> := <a class="idref" href="compcert.lib.Maps.html#PTree.xmap"><span class="id" title="definition">xmap</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.xgmap"><span class="id" title="lemma">xgmap</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xmap"><span class="id" title="definition">xmap</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> (<a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a>))) (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gmap"><span class="id" title="lemma">gmap</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.map"><span class="id" title="definition">map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> (<a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">f</span> <span class="id" title="var">i</span> <span class="id" title="var">m</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.map"><span class="id" title="definition">map</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xgmap"><span class="id" title="lemma">xgmap</span></a>. <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">f_equal</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_involutive"><span class="id" title="lemma">prev_involutive</span></a> <span class="id" title="var">i</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.map1"><span class="id" title="definition">map1</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> (<a class="idref" href="compcert.lib.Maps.html#map1"><span class="id" title="definition">map1</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span>) (<a class="idref" href="compcert.lib.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">o</span>) (<a class="idref" href="compcert.lib.Maps.html#map1"><span class="id" title="definition">map1</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gmap1"><span class="id" title="lemma">gmap1</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.map1"><span class="id" title="definition">map1</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Coqlib.html#option_map"><span class="id" title="definition">option_map</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.Node'"><span class="id" title="definition">Node'</span></a> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">l</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</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#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">r</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>): <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#l"><span class="id" title="variable">l</span></a>, <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a>, <a class="idref" href="compcert.lib.Maps.html#r"><span class="id" title="variable">r</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>, <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#r"><span class="id" title="variable">r</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.gnode'"><span class="id" title="lemma">gnode'</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">l</span> <span class="id" title="var">r</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</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#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a> <a class="idref" href="compcert.lib.Maps.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#r"><span class="id" title="variable">r</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#r"><span class="id" title="variable">r</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">l</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">r</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.filter1"><span class="id" title="definition">filter1</span></a> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">pred</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">o'</span> := <span class="id" title="keyword">match</span> <span class="id" title="var">o</span> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> ⇒ <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#pred"><span class="id" title="variable">pred</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> <span class="id" title="var">o</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="keyword">end</span> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a> (<a class="idref" href="compcert.lib.Maps.html#filter1"><span class="id" title="definition">filter1</span></a> <a class="idref" href="compcert.lib.Maps.html#pred"><span class="id" title="variable">pred</span></a> <span class="id" title="var">l</span>) <a class="idref" href="compcert.lib.Maps.html#o'"><span class="id" title="variable">o'</span></a> (<a class="idref" href="compcert.lib.Maps.html#filter1"><span class="id" title="definition">filter1</span></a> <a class="idref" href="compcert.lib.Maps.html#pred"><span class="id" title="variable">pred</span></a> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gfilter1"><span class="id" title="lemma">gfilter1</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">pred</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.filter1"><span class="id" title="definition">filter1</span></a> <a class="idref" href="compcert.lib.Maps.html#pred"><span class="id" title="variable">pred</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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><br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> ⇒ <span class="id" title="keyword">if</span> <a class="idref" href="compcert.lib.Maps.html#pred"><span class="id" title="variable">pred</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="tactic">until</span> <span class="id" title="var">m</span>. <span class="id" title="var">revert</span> <span class="id" title="var">m</span> <span class="id" title="var">i</span>. <span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gnode'"><span class="id" title="lemma">gnode'</span></a>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="PTree.COMBINE"><span class="id" title="section">COMBINE</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> <a name="PTree.COMBINE.A"><span class="id" title="variable">A</span></a> <a name="PTree.COMBINE.B"><span class="id" title="variable">B</span></a> <a name="PTree.COMBINE.C"><span class="id" title="variable">C</span></a>: <span class="id" title="keyword">Type</span>.<br/>
<span class="id" title="keyword">Variable</span> <a name="PTree.COMBINE.f"><span class="id" title="variable">f</span></a>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.B"><span class="id" title="variable">B</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.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.C"><span class="id" title="variable">C</span></a>.<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="PTree.COMBINE.f_none_none"><span class="id" title="variable">f_none_none</span></a>: <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.xcombine_l"><span class="id" title="definition">xcombine_l</span></a> (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.A"><span class="id" title="variable">A</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.C"><span class="id" title="variable">C</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a> (<a class="idref" href="compcert.lib.Maps.html#xcombine_l"><span class="id" title="definition">xcombine_l</span></a> <span class="id" title="var">l</span>) (<a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> <span class="id" title="var">o</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>) (<a class="idref" href="compcert.lib.Maps.html#xcombine_l"><span class="id" title="definition">xcombine_l</span></a> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.xgcombine_l"><span class="id" title="lemma">xgcombine_l</span></a> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_l"><span class="id" title="definition">xcombine_l</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gnode'"><span class="id" title="lemma">gnode'</span></a>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.xcombine_r"><span class="id" title="definition">xcombine_r</span></a> (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.B"><span class="id" title="variable">B</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.C"><span class="id" title="variable">C</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <span class="id" title="var">o</span> <span class="id" title="var">r</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a> (<a class="idref" href="compcert.lib.Maps.html#xcombine_r"><span class="id" title="definition">xcombine_r</span></a> <span class="id" title="var">l</span>) (<a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">o</span>) (<a class="idref" href="compcert.lib.Maps.html#xcombine_r"><span class="id" title="definition">xcombine_r</span></a> <span class="id" title="var">r</span>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.xgcombine_r"><span class="id" title="lemma">xgcombine_r</span></a> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_r"><span class="id" title="definition">xcombine_r</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gnode'"><span class="id" title="lemma">gnode'</span></a>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.combine"><span class="id" title="definition">combine</span></a> (<span class="id" title="var">m1</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.B"><span class="id" title="variable">B</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m1</span>} : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.C"><span class="id" title="variable">C</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_r"><span class="id" title="definition">xcombine_r</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">o1</span> <span class="id" title="var">r1</span> ⇒<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_l"><span class="id" title="definition">xcombine_l</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l2</span> <span class="id" title="var">o2</span> <span class="id" title="var">r2</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.Node'"><span class="id" title="definition">Node'</span></a> (<a class="idref" href="compcert.lib.Maps.html#combine"><span class="id" title="definition">combine</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span>) (<a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> <span class="id" title="var">o1</span> <span class="id" title="var">o2</span>) (<a class="idref" href="compcert.lib.Maps.html#combine"><span class="id" title="definition">combine</span></a> <span class="id" title="var">r1</span> <span class="id" title="var">r2</span>)<br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.gcombine"><span class="id" title="lemma">gcombine</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">m1</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE.B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.combine"><span class="id" title="definition">combine</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</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.Maps.html#PTree.COMBINE.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xgcombine_r"><span class="id" title="lemma">xgcombine_r</span></a>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a>. <span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.lib.Maps.html#PTree.xgcombine_l"><span class="id" title="lemma">xgcombine_l</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gnode'"><span class="id" title="lemma">gnode'</span></a>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.lib.Maps.html#PTree.COMBINE"><span class="id" title="section">COMBINE</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.xcombine_lr"><span class="id" title="lemma">xcombine_lr</span></a> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span> <span class="id" title="var">g</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_l"><span class="id" title="definition">xcombine_l</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.xcombine_r"><span class="id" title="definition">xcombine_r</span></a> <a class="idref" href="compcert.lib.Maps.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm2</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.combine_commut"><span class="id" title="lemma">combine_commut</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">f</span> <span class="id" title="var">g</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>), <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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.Maps.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">m1</span> <span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.combine"><span class="id" title="definition">combine</span></a> <a class="idref" href="compcert.lib.Maps.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</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.Maps.html#PTree.combine"><span class="id" title="definition">combine</span></a> <a class="idref" href="compcert.lib.Maps.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">f</span> <span class="id" title="var">g</span> <span class="id" title="var">EQ1</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">EQ2</span>: <span class="id" title="keyword">∀</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <span class="id" title="var">A</span>), <span class="id" title="var">g</span> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</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> <span class="id" title="var">f</span> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>).<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m1</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">m2</span>; <span class="id" title="tactic">simpl</span>;<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">EQ1</span>;<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_lr"><span class="id" title="lemma">xcombine_lr</span></a> <span class="id" title="var">f</span> <span class="id" title="var">g</span>);<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.xcombine_lr"><span class="id" title="lemma">xcombine_lr</span></a> <span class="id" title="var">g</span> <span class="id" title="var">f</span>);<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1_1</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1_2</span>.<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="PTree.xelements"><span class="id" title="definition">xelements</span></a> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>)<br/>
(<span class="id" title="var">k</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>)) {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>}<br/>
: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> ⇒ <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a><br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">r</span> ⇒<br/>
<a class="idref" href="compcert.lib.Maps.html#xelements"><span class="id" title="definition">xelements</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="compcert.lib.Maps.html#xelements"><span class="id" title="definition">xelements</span></a> <span class="id" title="var">r</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a>)<br/>
| <a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">r</span> ⇒<br/>
<a class="idref" href="compcert.lib.Maps.html#xelements"><span class="id" title="definition">xelements</span></a> <span class="id" title="var">l</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>)<br/>
(<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="compcert.lib.Maps.html#xelements"><span class="id" title="definition">xelements</span></a> <span class="id" title="var">r</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a>)<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.elements"><span class="id" title="definition">elements</span></a> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span> : <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xelements_append"><span class="id" title="lemma">xelements_append</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">i</span> <span class="id" title="var">k1</span> <span class="id" title="var">k2</span>,<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="compcert.lib.Maps.html#k1"><span class="id" title="variable">k1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="compcert.lib.Maps.html#k2"><span class="id" title="variable">k2</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.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#k1"><span class="id" title="variable">k1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="compcert.lib.Maps.html#k2"><span class="id" title="variable">k2</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>.<br/>
- <span class="id" title="tactic">auto</span>.<br/>
- <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm2</span>; <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">IHm1</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xelements_leaf"><span class="id" title="lemma">xelements_leaf</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> (@<a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xelements_node"><span class="id" title="lemma">xelements_node</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> (<span class="id" title="var">m1</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">o</span> (<span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">i</span>,<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#o"><span class="id" title="variable">o</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a>) <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#o"><span class="id" title="variable">o</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#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">v</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="keyword">end</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_append"><span class="id" title="lemma">xelements_append</span></a>; <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="PTree.xelements_incl"><span class="id" title="lemma">xelements_incl</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) <span class="id" title="var">k</span> <span class="id" title="var">x</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</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.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm1</span>. <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">right</span>; <span class="id" title="tactic">auto</span>.<br/>
<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="PTree.xelements_correct"><span class="id" title="lemma">xelements_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">k</span>,<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.gleaf"><span class="id" title="lemma">gleaf</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">destruct</span> <span class="id" title="var">i</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_incl"><span class="id" title="lemma">xelements_incl</span></a>. <span class="id" title="tactic">right</span>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="var">inv</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_incl"><span class="id" title="lemma">xelements_incl</span></a>. <span class="id" title="tactic">left</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_incl"><span class="id" title="lemma">xelements_incl</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="var">inv</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_correct"><span class="id" title="lemma">elements_correct</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</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.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">m</span> <span class="id" title="var">i</span> <span class="id" title="var">v</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements_correct"><span class="id" title="lemma">xelements_correct</span></a> <span class="id" title="var">m</span> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">H</span>). <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a>. <span class="id" title="tactic">exact</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#id"><span class="id" title="definition">id</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.in_xelements"><span class="id" title="lemma">in_xelements</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> <span class="id" title="var">k</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) ,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</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.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_leaf"><span class="id" title="lemma">xelements_leaf</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="var">contradiction</span>.<br/>
- <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_node"><span class="id" title="lemma">xelements_node</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">rewrite</span> ! <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">P</span> | [<span class="id" title="var">P</span> | <span class="id" title="var">P</span>]].<br/>
+ <span class="id" title="var">exploit</span> <span class="id" title="var">IHm1</span>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j</span> & <span class="id" title="var">Q</span> & <span class="id" title="var">R</span>). <span class="id" title="tactic">∃</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">j</span>); <span class="id" title="tactic">auto</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">P</span>; <span class="id" title="tactic">intuition</span> <span class="id" title="tactic">auto</span>. <span class="id" title="var">inv</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">∃</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>; <span class="id" title="tactic">auto</span>.<br/>
+ <span class="id" title="var">exploit</span> <span class="id" title="var">IHm2</span>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j</span> & <span class="id" title="var">Q</span> & <span class="id" title="var">R</span>). <span class="id" title="tactic">∃</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">j</span>); <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_complete"><span class="id" title="lemma">elements_complete</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) (<span class="id" title="var">v</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.html#v"><span class="id" title="variable">v</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a>. <span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">m</span> <span class="id" title="var">i</span> <span class="id" title="var">v</span> <span class="id" title="var">H</span>. <span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xelements"><span class="id" title="lemma">in_xelements</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j</span> & <span class="id" title="var">P</span> & <span class="id" title="var">Q</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">P</span>. <span class="id" title="tactic">change</span> <span class="id" title="var">i</span> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> 1 <span class="id" title="var">i</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">P</span>.<br/>
<span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_inj"><span class="id" title="lemma">prev_append_inj</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="PTree.xkeys"><span class="id" title="definition">xkeys</span></a> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>) :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</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> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xkeys_leaf"><span class="id" title="lemma">xkeys_leaf</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> (@<a class="idref" href="compcert.lib.Maps.html#PTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xkeys_node"><span class="id" title="lemma">xkeys_node</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">A</span> (<span class="id" title="var">m1</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">o</span> (<span class="id" title="var">m2</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">i</span>,<br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="compcert.lib.Maps.html#o"><span class="id" title="variable">o</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a>) <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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><br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <a class="idref" href="compcert.lib.Maps.html#m1"><span class="id" title="variable">m1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>)<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="compcert.lib.Maps.html#o"><span class="id" title="variable">o</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#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">v</span> ⇒ <a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="keyword">end</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <a class="idref" href="compcert.lib.Maps.html#m2"><span class="id" title="variable">m2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_node"><span class="id" title="lemma">xelements_node</span></a>. <span class="id" title="tactic">rewrite</span> ! <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#map_app"><span class="id" title="lemma">map_app</span></a>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</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="PTree.in_xkeys"><span class="id" title="lemma">in_xkeys</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span> <span class="id" title="var">k</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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><br/>
<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#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#k"><span class="id" title="variable">k</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.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev_append"><span class="id" title="definition">prev_append</span></a> <a class="idref" href="compcert.lib.Maps.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</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>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.lib.Coqlib.html#list_in_map_inv"><span class="id" title="lemma">list_in_map_inv</span></a>) <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span> <span class="id" title="keyword">as</span> ((<span class="id" title="var">j</span>, <span class="id" title="var">v</span>) & → & <span class="id" title="var">H</span>).<br/>
<span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xelements"><span class="id" title="lemma">in_xelements</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">k</span> & <span class="id" title="var">P</span> & <span class="id" title="var">Q</span>). <span class="id" title="tactic">∃</span> <span class="id" title="var">k</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="PTree.xelements_keys_norepet"><span class="id" title="lemma">xelements_keys_norepet</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">i</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#positive"><span class="id" title="inductive">positive</span></a>),<br/>
<a class="idref" href="compcert.lib.Coqlib.html#list_norepet"><span class="id" title="inductive">list_norepet</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys_leaf"><span class="id" title="lemma">xkeys_leaf</span></a>; <span class="id" title="tactic">constructor</span>.<br/>
- <span class="id" title="tactic">assert</span> (<span class="id" title="var">NOTIN1</span>: <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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <span class="id" title="var">i</span>) (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <span class="id" title="var">m1</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">i</span>))).<br/>
{ <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xkeys"><span class="id" title="lemma">in_xkeys</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j</span> & <span class="id" title="var">EQ</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_inj"><span class="id" title="lemma">prev_append_inj</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">discriminate</span>. }<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">NOTIN2</span>: <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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.prev"><span class="id" title="definition">prev</span></a> <span class="id" title="var">i</span>) (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <span class="id" title="var">m2</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">i</span>))).<br/>
{ <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>. <span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xkeys"><span class="id" title="lemma">in_xkeys</span></a>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j</span> & <span class="id" title="var">EQ</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_inj"><span class="id" title="lemma">prev_append_inj</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ</span>. <span class="id" title="tactic">discriminate</span>. }<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">DISJ</span>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <span class="id" title="var">m1</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">i</span>)) <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.Lists.List.html#In"><span class="id" title="definition">In</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.xkeys"><span class="id" title="definition">xkeys</span></a> <span class="id" title="var">m2</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">i</span>)) <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#False"><span class="id" title="inductive">False</span></a>).<br/>
{ <span class="id" title="tactic">intros</span>. <span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xkeys"><span class="id" title="lemma">in_xkeys</span></a>. <span class="id" title="tactic">eexact</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j1</span> & <span class="id" title="var">EQ1</span>).<br/>
<span class="id" title="var">exploit</span> <a class="idref" href="compcert.lib.Maps.html#PTree.in_xkeys"><span class="id" title="lemma">in_xkeys</span></a>. <span class="id" title="tactic">eexact</span> <span class="id" title="var">H0</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">j2</span> & <span class="id" title="var">EQ2</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_prev"><span class="id" title="lemma">prev_append_prev</span></a> <span class="id" title="keyword">in</span> ×. <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> ×. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">EQ2</span> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ1</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.prev_append_inj"><span class="id" title="lemma">prev_append_inj</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">EQ1</span>. <span class="id" title="tactic">discriminate</span>. }<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xkeys_node"><span class="id" title="lemma">xkeys_node</span></a>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Coqlib.html#list_norepet_append"><span class="id" title="lemma">list_norepet_append</span></a>. <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">constructor</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">red</span>; <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="var">y</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H0</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H0</span>. <span class="id" title="tactic">subst</span> <span class="id" title="var">x</span>. <span class="id" title="tactic">tauto</span>. <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_keys_norepet"><span class="id" title="lemma">elements_keys_norepet</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="compcert.lib.Coqlib.html#list_norepet"><span class="id" title="inductive">list_norepet</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>)).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements_keys_norepet"><span class="id" title="lemma">xelements_keys_norepet</span></a> <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Remark</span> <a name="PTree.xelements_empty"><span class="id" title="lemma">xelements_empty</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">i</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements"><span class="id" title="definition">xelements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_node"><span class="id" title="lemma">xelements_node</span></a>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHm1</span>, <span class="id" title="var">IHm2</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">o</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">i0</span>)).<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">i0</span>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_canonical_order'"><span class="id" title="lemma">elements_canonical_order'</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">R</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">n</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Coqlib.html#option_rel"><span class="id" title="inductive">option_rel</span></a> <a class="idref" href="compcert.lib.Maps.html#R"><span class="id" title="variable">R</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.lib.Coqlib.html#list_forall2"><span class="id" title="inductive">list_forall2</span></a><br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">i_x</span> <span class="id" title="var">i_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> <a class="idref" href="compcert.lib.Maps.html#i_x"><span class="id" title="variable">i_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="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.Maps.html#i_y"><span class="id" title="variable">i_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.Maps.html#R"><span class="id" title="variable">R</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.Maps.html#i_x"><span class="id" title="variable">i_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.Maps.html#i_y"><span class="id" title="variable">i_y</span></a>))<br/>
(<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="tactic">until</span> <span class="id" title="var">n</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a>. <span class="id" title="tactic">generalize</span> 1%<span class="id" title="var">positive</span>. <span class="id" title="var">revert</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span>; <span class="id" title="tactic">intros</span>.<br/>
- <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_empty"><span class="id" title="lemma">xelements_empty</span></a>. <span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> <span class="id" title="var">i</span>). <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gempty"><span class="id" title="lemma">gempty</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="var">inv</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">auto</span>.<br/>
- <span class="id" title="tactic">destruct</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> [ | <span class="id" title="var">n1</span> <span class="id" title="var">o'</span> <span class="id" title="var">n2</span> ].<br/>
+ <span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.xelements_empty"><span class="id" title="lemma">xelements_empty</span></a> (<a class="idref" href="compcert.lib.Maps.html#PTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">o</span> <span class="id" title="var">m2</span>)). <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> <span class="id" title="var">i</span>). <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.lib.Maps.html#PTree.gempty"><span class="id" title="lemma">gempty</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>. <span class="id" title="var">inv</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">auto</span>.<br/>
+ <span class="id" title="tactic">rewrite</span> ! <a class="idref" href="compcert.lib.Maps.html#PTree.xelements_node"><span class="id" title="lemma">xelements_node</span></a>. <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Coqlib.html#list_forall2_app"><span class="id" title="lemma">list_forall2_app</span></a>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm1</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> <span class="id" title="var">i</span>)).<br/>
<span class="id" title="tactic">generalize</span> (<span class="id" title="var">H</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>); <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">OR</span>; <span class="id" title="var">inv</span> <span class="id" title="var">OR</span>.<br/>
<span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="tactic">constructor</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">IHm2</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xI"><span class="id" title="constructor">xI</span></a> <span class="id" title="var">i</span>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_canonical_order"><span class="id" title="lemma">elements_canonical_order</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">R</span>: <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">m</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">n</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#B"><span class="id" title="variable">B</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">x</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.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.Maps.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">y</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.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#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="compcert.lib.Maps.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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="compcert.lib.Maps.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="compcert.lib.Maps.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.lib.Maps.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.lib.Coqlib.html#list_forall2"><span class="id" title="inductive">list_forall2</span></a><br/>
(<span class="id" title="keyword">fun</span> <span class="id" title="var">i_x</span> <span class="id" title="var">i_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> <a class="idref" href="compcert.lib.Maps.html#i_x"><span class="id" title="variable">i_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="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.Maps.html#i_y"><span class="id" title="variable">i_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.Maps.html#R"><span class="id" title="variable">R</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.Maps.html#i_x"><span class="id" title="variable">i_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.Maps.html#i_y"><span class="id" title="variable">i_y</span></a>))<br/>
(<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.lib.Maps.html#PTree.elements_canonical_order'"><span class="id" title="lemma">elements_canonical_order'</span></a>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> <span class="id" title="var">m</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">x</span>|] <span class="id" title="var">eqn</span>:<span class="id" title="var">GM</span>.<br/>
<span class="id" title="var">exploit</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">y</span> & <span class="id" title="var">P</span> & <span class="id" title="var">Q</span>). <span class="id" title="tactic">rewrite</span> <span class="id" title="var">P</span>; <span class="id" title="tactic">constructor</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span>|] <span class="id" title="var">eqn</span>:<span class="id" title="var">GN</span>.<br/>
<span class="id" title="var">exploit</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">eauto</span>. <span class="id" title="tactic">intros</span> (<span class="id" title="var">x</span> & <span class="id" title="var">P</span> & <span class="id" title="var">Q</span>). <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="PTree.elements_extensional"><span class="id" title="lemma">elements_extensional</span></a>:<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span>: <span class="id" title="keyword">Type</span>) (<span class="id" title="var">m</span> <span class="id" title="var">n</span>: <a class="idref" href="compcert.lib.Maps.html#PTree.t"><span class="id" title="definition">t</span></a> <a class="idref" href="compcert.lib.Maps.html#A"><span class="id" title="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <a class="idref" href="compcert.lib.Maps.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.lib.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#m"><span class="id" title="variable">m</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.Maps.html#PTree.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="compcert.lib.Maps.html#n"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>.<br/>
<span class="id" title="var">exploit</span> (@<a class="idref" href="compcert.lib.Maps.html#PTree.elements_canonical_order'"><span class="id" title="lemma">elements_canonical_order'</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<span class="id" title="keyword">fun</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span>: <span class="id" title="var">A</span>) ⇒ <a class="idref" href="compcert.lib.Maps.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.Maps.html#y"><span class="id" title="variable">y</span></a>) <span class="id" title="var">m</span> <span class="id" title="var">n</span>).<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.lib.Maps.html#PTree.get"><span class="id" title="definition">get</span></a> <span class="id" title="var">i</span> <span class="id" title="var">n</span>); <span class="id" title="tactic">constructor</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">induction</span> 1. <span class="id" title="tactic">auto</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">a1</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">a2</span> <span class="id" title="var">a3</span>]; <span class="id" title="tactic">destruct</span> <span class="id" title="var">b1</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">b2</span> <span class="id" title="var">b3</span>]; <span class="id" title="tactic">simpl</span> <span class="id" title="keyword">in</span> ×.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H0</span>. <span class="id" title="tactic">congruence</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="PTree.xelements_remove"><span class="id" title="lemma">xelements_remove</span></a>:<br/>