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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=iso-8859-1"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>SfLib</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library SfLib</h1>
<div class="code">
</div>
<div class="doc">
<a name="lab1"></a><h1 class="section">SfLib: Modified From the Software Foundations Library </h1>
<div class="paragraph"> </div>
These are the basic notions that will go into our development.
<div class="paragraph"> </div>
We introduce identifiers and states, and discuss their properties.
<div class="paragraph"> </div>
This file in particular, as well as core part PrImp.v were based upon
Pierce et al's Software Foundations, available at:
http://www.cis.upenn.edu/~bcpierce/sf/current/index.html
<div class="paragraph"> </div>
Imports and basic notions
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.omega.Omega.html#"><span class="id" type="library">Omega</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="Style.html#"><span class="id" type="library">Style</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Bool.Bool.html#"><span class="id" type="library">Bool</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.Arith.html#"><span class="id" type="library">Arith</span></a>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#"><span class="id" type="library">Arith.EqNat</span></a>. <span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Logic.FunctionalExtensionality.html#"><span class="id" type="library">Coq.Logic.FunctionalExtensionality</span></a>.<br/>
<br/>
<span class="id" type="keyword">Require</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Strings.String.html#"><span class="id" type="library">String</span></a>. <span class="id" type="keyword">Open</span> <span class="id" type="keyword">Scope</span> <span class="id" type="var">string_scope</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="ble_nat"><span class="id" type="definition">ble_nat</span></a> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" type="inductive">bool</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">n'</span> =><br/>
<span class="id" type="keyword">match</span> <a class="idref" href="SfLib.html#m"><span class="id" type="variable">m</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">m'</span> => <a class="idref" href="SfLib.html#ble_nat"><span class="id" type="definition">ble_nat</span></a> <span class="id" type="var">n'</span> <span class="id" type="var">m'</span><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Arithmetic Identifiers
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="aid"><span class="id" type="inductive">aid</span></a> : <span class="id" type="keyword">Type</span> := <a name="Aid"><span class="id" type="constructor">Aid</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="SfLib.html#aid"><span class="id" type="inductive">aid</span></a>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="beq_aid"><span class="id" type="definition">beq_aid</span></a> (<span class="id" type="var">i1</span> <span class="id" type="var">i2</span> : <a class="idref" href="SfLib.html#aid"><span class="id" type="inductive">aid</span></a>) := <br/>
<span class="id" type="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#Aid"><span class="id" type="constructor">Aid</span></a> <span class="id" type="var">n1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="SfLib.html#Aid"><span class="id" type="constructor">Aid</span></a> <span class="id" type="var">n2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat"><span class="id" type="definition">beq_nat</span></a> <span class="id" type="var">n1</span> <span class="id" type="var">n2</span> <span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_aid_refl"><span class="id" type="lemma">beq_aid_refl</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i</span>, <a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#i"><span class="id" type="variable">i</span></a> <a class="idref" href="SfLib.html#i"><span class="id" type="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i</span>. <span class="id" type="tactic">symmetry</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_refl"><span class="id" type="lemma">beq_nat_refl</span></a>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_aid_eq"><span class="id" type="lemma">beq_aid_eq</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span>, <a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i1</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">i2</span>.<br/>
<span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_eq"><span class="id" type="definition">beq_nat_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_aid_false_iff"><span class="id" type="lemma">beq_aid_false_iff</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span>, <a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<->'_x"><span class="id" type="notation"><span style="font-family: arial;">↔</span></span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<>'_x"><span class="id" type="notation"><span style="font-family: serif;">≠</span></span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">split</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">eq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">rewrite</span> <a class="idref" href="SfLib.html#beq_aid_refl"><span class="id" type="lemma">beq_aid_refl</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i1</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">i2</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_false_iff"><span class="id" type="lemma">beq_nat_false_iff</span></a>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Boolean Identifiers
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="bid"><span class="id" type="inductive">bid</span></a> : <span class="id" type="keyword">Type</span> := <a name="Bid"><span class="id" type="constructor">Bid</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="SfLib.html#bid"><span class="id" type="inductive">bid</span></a>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="beq_bid"><span class="id" type="definition">beq_bid</span></a> (<span class="id" type="var">i1</span> <span class="id" type="var">i2</span> : <a class="idref" href="SfLib.html#bid"><span class="id" type="inductive">bid</span></a>) := <br/>
<span class="id" type="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a> <span class="id" type="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#Bid"><span class="id" type="constructor">Bid</span></a> <span class="id" type="var">n1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="SfLib.html#Bid"><span class="id" type="constructor">Bid</span></a> <span class="id" type="var">n2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat"><span class="id" type="definition">beq_nat</span></a> <span class="id" type="var">n1</span> <span class="id" type="var">n2</span> <span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_bid_refl"><span class="id" type="lemma">beq_bid_refl</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i</span>, <a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <a class="idref" href="SfLib.html#i"><span class="id" type="variable">i</span></a> <a class="idref" href="SfLib.html#i"><span class="id" type="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i</span>. <span class="id" type="tactic">symmetry</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_refl"><span class="id" type="lemma">beq_nat_refl</span></a>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_bid_eq"><span class="id" type="lemma">beq_bid_eq</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span>, <a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i1</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">i2</span>.<br/>
<span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_eq"><span class="id" type="definition">beq_nat_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_bid_false_iff"><span class="id" type="lemma">beq_bid_false_iff</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">i1</span> <span class="id" type="var">i2</span>, <a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<->'_x"><span class="id" type="notation"><span style="font-family: arial;">↔</span></span></a> <a class="idref" href="SfLib.html#i1"><span class="id" type="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'<>'_x"><span class="id" type="notation"><span style="font-family: serif;">≠</span></span></a> <a class="idref" href="SfLib.html#i2"><span class="id" type="variable">i2</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">split</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span> <span class="id" type="var">eq</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">eq</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">rewrite</span> <a class="idref" href="SfLib.html#beq_bid_refl"><span class="id" type="lemma">beq_bid_refl</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">i1</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">i2</span>. <span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Arith.EqNat.html#beq_nat_false_iff"><span class="id" type="lemma">beq_nat_false_iff</span></a>. <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab2"></a><h2 class="section">States</h2>
<div class="paragraph"> </div>
A <i>state</i> represents the current values of all the variables at
some point in the execution of a program. For simplicity (to avoid dealing with partial functions), we
let the state be defined for <i>all</i> variables, even though any
given program is only going to mention a finite number of them.
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Definition</span> <a name="state"><span class="id" type="definition">state</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:type_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#aid"><span class="id" type="inductive">aid</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:type_scope:x_'*'_x"><span class="id" type="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:type_scope:x_'*'_x"><span class="id" type="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:type_scope:x_'*'_x"><span class="id" type="notation">(</span></a><a class="idref" href="SfLib.html#bid"><span class="id" type="inductive">bid</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" type="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:type_scope:x_'*'_x"><span class="id" type="notation">)</span></a>)%<span class="id" type="var">type</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="empty_state"><span class="id" type="definition">empty_state</span></a> : <a class="idref" href="SfLib.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> => 0<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <span class="id" type="keyword">fun</span> <span class="id" type="var">_</span> => <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="update"><span class="id" type="definition">update</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="SfLib.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">x</span> : <a class="idref" href="SfLib.html#aid"><span class="id" type="inductive">aid</span></a>) (<span class="id" type="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <a class="idref" href="SfLib.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="keyword">fun</span> <span class="id" type="var">x'</span> => <span class="id" type="keyword">if</span> <a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="SfLib.html#x'"><span class="id" type="variable">x'</span></a> <span class="id" type="keyword">then</span> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">else</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a>) <a class="idref" href="SfLib.html#x'"><span class="id" type="variable">x'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="update_b"><span class="id" type="definition">update_b</span></a> (<span class="id" type="var">st</span> : <a class="idref" href="SfLib.html#state"><span class="id" type="definition">state</span></a>) (<span class="id" type="var">y</span> : <a class="idref" href="SfLib.html#bid"><span class="id" type="inductive">bid</span></a>) (<span class="id" type="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" type="inductive">bool</span></a>) : <a class="idref" href="SfLib.html#state"><span class="id" type="definition">state</span></a> :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <span class="id" type="keyword">fun</span> <span class="id" type="var">y'</span> => <span class="id" type="keyword">if</span> <a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <a class="idref" href="SfLib.html#y"><span class="id" type="variable">y</span></a> <a class="idref" href="SfLib.html#y'"><span class="id" type="variable">y'</span></a> <span class="id" type="keyword">then</span> <a class="idref" href="SfLib.html#b"><span class="id" type="variable">b</span></a> <span class="id" type="keyword">else</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a>) <a class="idref" href="SfLib.html#y'"><span class="id" type="variable">y'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>.<br/>
<br/>
</div>
<div class="doc">
Properties of <span class="inlinecode"><span class="id" type="var">update</span></span>.
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_eq"><span class="id" type="lemma">update_eq</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">n</span> <span class="id" type="var">x</span> <span class="id" type="var">st</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> (<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="SfLib.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">unfold</span> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a>; <span class="id" type="tactic">simpl</span>; <span class="id" type="tactic">rewrite</span> <a class="idref" href="SfLib.html#beq_aid_refl"><span class="id" type="lemma">beq_aid_refl</span></a>; <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_neq"><span class="id" type="lemma">update_neq</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">x2</span> <span class="id" type="var">x1</span> <span class="id" type="var">n</span> <span class="id" type="var">st</span>,<br/>
<a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#x2"><span class="id" type="variable">x2</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> (<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x2"><span class="id" type="variable">x2</span></a> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">)</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">x2</span> <span class="id" type="var">x1</span> <span class="id" type="var">n</span> <span class="id" type="var">st</span> <span class="id" type="var">Hneq</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Hneq</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_same"><span class="id" type="lemma">update_same</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">n1</span> <span class="id" type="var">x1</span> <span class="id" type="var">st</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#n1"><span class="id" type="variable">n1</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="SfLib.html#n1"><span class="id" type="variable">n1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n1</span> <span class="id" type="var">x1</span> <span class="id" type="var">st</span> <span class="id" type="var">Heq</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#injective_projections"><span class="id" type="lemma">injective_projections</span></a>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Logic.FunctionalExtensionality.html#functional_extensionality"><span class="id" type="lemma">functional_extensionality</span></a>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <span class="id" type="var">x1</span> <span class="id" type="var">x</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eq</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_aid_eq"><span class="id" type="lemma">beq_aid_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Eq</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_permute"><span class="id" type="lemma">update_permute</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">n1</span> <span class="id" type="var">n2</span> <span class="id" type="var">x1</span> <span class="id" type="var">x2</span> <span class="id" type="var">st</span>,<br/>
<a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <a class="idref" href="SfLib.html#x2"><span class="id" type="variable">x2</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> (<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x2"><span class="id" type="variable">x2</span></a> <a class="idref" href="SfLib.html#n1"><span class="id" type="variable">n1</span></a>) <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="SfLib.html#n2"><span class="id" type="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> (<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x1"><span class="id" type="variable">x1</span></a> <a class="idref" href="SfLib.html#n2"><span class="id" type="variable">n2</span></a>) <a class="idref" href="SfLib.html#x2"><span class="id" type="variable">x2</span></a> <a class="idref" href="SfLib.html#n1"><span class="id" type="variable">n1</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n1</span> <span class="id" type="var">n2</span> <span class="id" type="var">x1</span> <span class="id" type="var">x2</span> <span class="id" type="var">st</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#injective_projections"><span class="id" type="lemma">injective_projections</span></a>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Logic.FunctionalExtensionality.html#functional_extensionality"><span class="id" type="lemma">functional_extensionality</span></a>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <span class="id" type="var">x1</span> <span class="id" type="var">x</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eq1</span>, (<a class="idref" href="SfLib.html#beq_aid"><span class="id" type="definition">beq_aid</span></a> <span class="id" type="var">x2</span> <span class="id" type="var">x</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eq2</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_aid_eq"><span class="id" type="lemma">beq_aid_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Eq1</span>; <span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_aid_eq"><span class="id" type="lemma">beq_aid_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Eq2</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_aid_false_iff"><span class="id" type="lemma">beq_aid_false_iff</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">contradict</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_permute_b"><span class="id" type="lemma">update_permute_b</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span class="id" type="var">y1</span> <span class="id" type="var">y2</span> <span class="id" type="var">st</span>,<br/>
<a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <a class="idref" href="SfLib.html#y2"><span class="id" type="variable">y2</span></a> <a class="idref" href="SfLib.html#y1"><span class="id" type="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#false"><span class="id" type="constructor">false</span></a> <span style="font-family: arial;">→</span><br/>
<a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> (<a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#y2"><span class="id" type="variable">y2</span></a> <a class="idref" href="SfLib.html#b1"><span class="id" type="variable">b1</span></a>) <a class="idref" href="SfLib.html#y1"><span class="id" type="variable">y1</span></a> <a class="idref" href="SfLib.html#b2"><span class="id" type="variable">b2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> (<a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#y1"><span class="id" type="variable">y1</span></a> <a class="idref" href="SfLib.html#b2"><span class="id" type="variable">b2</span></a>) <a class="idref" href="SfLib.html#y2"><span class="id" type="variable">y2</span></a> <a class="idref" href="SfLib.html#b1"><span class="id" type="variable">b1</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b1</span> <span class="id" type="var">b2</span> <span class="id" type="var">y1</span> <span class="id" type="var">y2</span> <span class="id" type="var">st</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#injective_projections"><span class="id" type="lemma">injective_projections</span></a>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Logic.FunctionalExtensionality.html#functional_extensionality"><span class="id" type="lemma">functional_extensionality</span></a>; <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <span class="id" type="var">y1</span> <span class="id" type="var">x</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eq1</span>, (<a class="idref" href="SfLib.html#beq_bid"><span class="id" type="definition">beq_bid</span></a> <span class="id" type="var">y2</span> <span class="id" type="var">x</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">Eq2</span>; <span class="id" type="tactic">trivial</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_bid_eq"><span class="id" type="lemma">beq_bid_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Eq1</span>; <span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_bid_eq"><span class="id" type="lemma">beq_bid_eq</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">Eq2</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="SfLib.html#beq_bid_false_iff"><span class="id" type="lemma">beq_bid_false_iff</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="var">contradict</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The following is trivial due to the non-conflicting nature of our updates,
but worth expressing explicitly. (In fact, reflexivity alone can prove this.)
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="update_interpermute"><span class="id" type="lemma">update_interpermute</span></a> : <span class="id" type="keyword"><span style="font-family: arial;">∀</span></span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">n</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span>,<br/>
<a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> (<a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="SfLib.html#y"><span class="id" type="variable">y</span></a> <a class="idref" href="SfLib.html#b"><span class="id" type="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="SfLib.html#update"><span class="id" type="definition">update</span></a> (<a class="idref" href="SfLib.html#update_b"><span class="id" type="definition">update_b</span></a> <a class="idref" href="SfLib.html#st"><span class="id" type="variable">st</span></a> <a class="idref" href="SfLib.html#y"><span class="id" type="variable">y</span></a> <a class="idref" href="SfLib.html#b"><span class="id" type="variable">b</span></a>) <a class="idref" href="SfLib.html#x"><span class="id" type="variable">x</span></a> <a class="idref" href="SfLib.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4/stdlib/Coq.Init.Datatypes.html#injective_projections"><span class="id" type="lemma">injective_projections</span></a>; <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
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