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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<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>more_list</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">more_list</h1>
<div class="code">
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#"><span class="id" type="library">List</span></a>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="length_invnil"><span class="id" type="lemma">length_invnil</span></a> : <span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> (<span class="id" type="var">l</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> 0 -> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">l</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">auto</span>;<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="length_invcons"><span class="id" type="lemma">length_invcons</span></a> : <span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> -> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:'exists'_x_'..'_x_','_x"><span class="id" type="notation">exists</span></a> <span class="id" type="var">x</span> <span class="id" type="var">l'</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:'exists'_x_'..'_x_','_x"><span class="id" type="notation">,</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#x"><span class="id" type="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">auto</span>;<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">exists</span> <span class="id" type="var">a</span>.<br/>
<span class="id" type="tactic">exists</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab70"></a><h1 class="section">List Split Operations</h1>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Function</span> <span class="id" type="var">split1</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>) {<span class="id" type="keyword">struct</span> <span class="id" type="var">n</span>} :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a>,<a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">_</span>,<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a><br/>
| <span class="id" type="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><span class="id" type="var">l'</span> , <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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="more_list.html#split1"><span class="id" type="definition">split1</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a> <span class="id" type="var">n'</span> <span class="id" type="var">l'</span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><span class="id" type="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><span class="id" type="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><span class="id" type="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">_</span> => <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Function</span> <span class="id" type="var">split2</span> {<span class="id" type="var">A</span>} <span class="id" type="var">n</span> <span class="id" type="var">m</span> (<span class="id" type="var">l</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>) :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><span class="id" type="var">l2'</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a> =><br/>
<span class="id" type="keyword">match</span> <a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <span class="id" type="var">l2'</span> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a> => <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><span class="id" type="var">l3</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><span class="id" type="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,(</span></a><span class="id" type="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><span class="id" type="var">l3</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">))</span></a><br/>
<span class="id" type="keyword">end</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab71"></a><h1 class="section">List Split Lemmas</h1>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_equation1"><span class="id" type="lemma">split2_equation1</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> (<span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,(</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">))</span></a><br/>
-> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> (<a class="idref" href="more_list.html#e"><span class="id" type="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#e"><span class="id" type="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_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">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> <span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="var">functional</span> <span class="id" type="tactic">induction</span> (<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">l</span>);<span class="id" type="tactic">try</span> <span class="id" type="tactic">discriminate</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">e0</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">e1</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_equation2"><span class="id" type="lemma">split2_equation2</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> (<span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> 0 <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,(</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">))</span></a><br/>
-> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> 0 (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a>) (<a class="idref" href="more_list.html#e"><span class="id" type="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#e"><span class="id" type="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a> <a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_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">A</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> <span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">l</span>.<br/>
- <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a>).<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
- <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> (<span class="id" type="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a> <span class="id" type="var">l</span>)).<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
+ <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_equation3"><span class="id" type="lemma">split2_equation3</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> (<span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a><br/>
-> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> (<a class="idref" href="more_list.html#e"><span class="id" type="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">e</span> <span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>).<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <span class="id" type="var">l4</span>).<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
+ <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split1_correct"><span class="id" type="lemma">split1_correct</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
(<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">_</span> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>)<br/>
-> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="var">functional</span> <span class="id" type="tactic">induction</span> <a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">simpl</span>;<span class="id" type="tactic">intros</span>.<br/>
- <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">split</span>; <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">IHo</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">e2</span>).<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHo</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">IHo1</span> <span class="id" type="var">IHo2</span>].<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">split</span>;<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
- <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split1_correct_eq"><span class="id" type="lemma">split1_correct_eq</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
(<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">_</span> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>)<br/>
-> (<a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> (<a class="idref" href="more_list.html#split1_correct"><span class="id" type="lemma">split1_correct</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H</span>;<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split1_correct_length"><span class="id" type="lemma">split1_correct_length</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
(<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">_</span> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">)</span></a>)<br/>
-> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.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> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> (<a class="idref" href="more_list.html#split1_correct"><span class="id" type="lemma">split1_correct</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>) <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">H</span>;<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split1_complete"><span class="id" type="lemma">split1_complete</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> (<span class="id" type="var">l</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a><br/>
-> (<a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a>)<br/>
-> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">_</span> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_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">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="var">functional</span> <span class="id" type="tactic">induction</span> <a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">simpl</span>;<span class="id" type="tactic">intros</span>.<br/>
- <span class="id" type="tactic">rewrite</span> (<a class="idref" href="more_list.html#length_invnil"><span class="id" type="lemma">length_invnil</span></a> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H</span>) <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#length_invcons"><span class="id" type="lemma">length_invcons</span></a> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H</span>) <span class="id" type="keyword">as</span> [<span class="id" type="var">e'</span> <span class="id" type="var">l''</span>].<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">l''</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>; <span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">IHo</span> <span class="id" type="var">l0</span> <span class="id" type="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" type="abbreviation">refl_equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" type="abbreviation">refl_equal</span></a>).<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">e2</span> <span class="id" type="keyword">in</span> <span class="id" type="var">IHo</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">IHo</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">l1</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">discriminate</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">IHo</span> <span class="id" type="var">l2</span> <span class="id" type="var">l1</span>).<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">IHo</span> <span class="id" type="keyword">in</span> <span class="id" type="var">e2</span>;<span class="id" type="tactic">try</span> <span class="id" type="tactic">discriminate</span>.<br/>
* <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
* <span class="id" type="tactic">assumption</span>.<br/>
- <span class="id" type="tactic">apply</span> <a class="idref" href="more_list.html#length_invcons"><span class="id" type="lemma">length_invcons</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">decompose</span> [<span class="id" type="var">ex</span>] <span class="id" type="var">H</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_correct"><span class="id" type="lemma">split2_correct</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> (<span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,(</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">))</span></a><br/>
-> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/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/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">heq</span>.<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="more_list.html#split1_correct"><span class="id" type="lemma">split1_correct</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">heq</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">heq</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">heq1</span> <span class="id" type="var">heq2</span>].<br/>
<span class="id" type="tactic">subst</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <span class="id" type="var">l4</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">heq'</span>.<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>;<span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>;<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="more_list.html#split1_correct"><span class="id" type="lemma">split1_correct</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">heq'</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">heq'</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">heq'1</span> <span class="id" type="var">heq'2</span>].<br/>
<span class="id" type="tactic">subst</span>;<span class="id" type="tactic">auto</span>.<br/>
+ <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
- <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.omega.Omega.html#"><span class="id" type="library">Omega</span></a>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_length_ok"><span class="id" type="lemma">split2_length_ok</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> (<span class="id" type="var">l</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>) <span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'>='_x"><span class="id" type="notation">>=</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'+'_x"><span class="id" type="notation">+</span></a><a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a><br/>
-> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#None"><span class="id" type="constructor">None</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">simpl</span>;<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">h</span>.<br/>
- <span class="id" type="tactic">assert</span> (<span class="id" type="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a>0) <span class="id" type="tactic">by</span> <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a>0) <span class="id" type="tactic">by</span> <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">discriminate</span>.<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>.<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">m</span>.<br/>
* <span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">discriminate</span>.<br/>
* <span class="id" type="tactic">intro</span> <span class="id" type="var">abs</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">h'</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <span class="id" type="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'>='_x"><span class="id" type="notation">>=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <span class="id" type="var">m</span>) <span class="id" type="tactic">by</span> <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">generalize</span> (<span class="id" type="var">IHl</span> 0 <span class="id" type="var">m</span> <span class="id" type="var">h'</span>).<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">IHl'</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">abs</span>,<span class="id" type="var">IHl'</span>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
{ <span class="id" type="tactic">destruct</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
-<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" type="constructor">nil</span></a>);<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">abs</span>.<br/>
+ <span class="id" type="var">contradiction</span>.<br/>
- <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> (<span class="id" type="var">a0</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'::'_x"><span class="id" type="notation">::</span></a> <span class="id" type="var">l</span>));<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
+ <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">abs</span>.<br/>
+ <span class="id" type="var">contradiction</span>. }<br/>
+ <span class="id" type="tactic">assert</span> (<span class="id" type="var">h'</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <span class="id" type="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'>='_x"><span class="id" type="notation">>=</span></a> <span class="id" type="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Peano.html#:nat_scope:x_'+'_x"><span class="id" type="notation">+</span></a> <span class="id" type="var">m</span>) <span class="id" type="tactic">by</span> <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">generalize</span> (<span class="id" type="var">IHl</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">h'</span>).<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">IHl'</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">IHl'</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a>.<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">l</span>);<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
* <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
{ <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split1"><span class="id" type="definition">split1</span></a> <span class="id" type="var">A</span> <span class="id" type="var">m</span> <span class="id" type="var">l1</span>).<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">discriminate</span>.<br/>
- <span class="id" type="tactic">assumption</span>. }<br/>
* <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="app_same_length_eq"><span class="id" type="lemma">app_same_length_eq</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> (<span class="id" type="var">l</span> <span class="id" type="var">l'</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a><br/>
-> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a><a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a><a class="idref" href="more_list.html#m'"><span class="id" type="variable">m'</span></a> -> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation">/\</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a><a class="idref" href="more_list.html#m'"><span class="id" type="variable">m'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">A</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">l</span>;<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *;<span class="id" type="tactic">intros</span>.<br/>
- <span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="more_list.html#length_invnil"><span class="id" type="lemma">length_invnil</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="tactic">split</span>;<span class="id" type="tactic">auto</span>.<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">l'</span>.<br/>
+ <span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
+ <span class="id" type="tactic">specialize</span> (<span class="id" type="var">IHl</span> <span class="id" type="var">l'</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span>).<br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> *.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">specialize</span> (<span class="id" type="var">IHl</span> <span class="id" type="var">H4</span> <span class="id" type="var">H3</span>).<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHl</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">split</span>;<span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="app_same_length_eq2"><span class="id" type="lemma">app_same_length_eq2</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> (<span class="id" type="var">l</span> <span class="id" type="var">l'</span> <span class="id" type="var">m</span> <span class="id" type="var">m'</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <a class="idref" href="more_list.html#m'"><span class="id" type="variable">m'</span></a><br/>
-> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a><a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a><a class="idref" href="more_list.html#m'"><span class="id" type="variable">m'</span></a> -> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l'"><span class="id" type="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'/\'_x"><span class="id" type="notation">/\</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a><a class="idref" href="more_list.html#m'"><span class="id" type="variable">m'</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H1</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> (<span class="id" type="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <span class="id" type="var">m</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> (<span class="id" type="var">l'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <span class="id" type="var">m'</span>)).<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H0</span>; <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H2</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <span class="id" type="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#length"><span class="id" type="definition">length</span></a> <span class="id" type="var">l'</span>).<br/>
<span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#app_length"><span class="id" type="lemma">app_length</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H1</span>. <span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#app_length"><span class="id" type="lemma">app_length</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H1</span>; <span class="id" type="tactic">omega</span>.<br/>
<span class="id" type="tactic">specialize</span> (<a class="idref" href="more_list.html#app_same_length_eq"><span class="id" type="lemma">app_same_length_eq</span></a> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H2</span> <span class="id" type="var">H0</span>); <span class="id" type="tactic">auto</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="split2_complete"><span class="id" type="lemma">split2_complete</span></a> :<br/>
<span class="id" type="keyword">forall</span> <span class="id" type="var">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> (<span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#list"><span class="id" type="inductive">list</span></a> <a class="idref" href="more_list.html#A"><span class="id" type="variable">A</span></a>),<br/>
<a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:list_scope:x_'++'_x"><span class="id" type="notation">++</span></a> <a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><br/>
-> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a><br/>
-> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#length"><span class="id" type="abbreviation">List.length</span></a> <a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a><br/>
-> <a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <a class="idref" href="more_list.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="more_list.html#m"><span class="id" type="variable">m</span></a> <a class="idref" href="more_list.html#l"><span class="id" type="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/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/8.4pl4/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" type="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">(</span></a><a class="idref" href="more_list.html#l1"><span class="id" type="variable">l1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,(</span></a><a class="idref" href="more_list.html#l2"><span class="id" type="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_x_','_'..'_','_x_')'"><span class="id" type="notation">,</span></a><a class="idref" href="more_list.html#l3"><span class="id" type="variable">l3</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Datatypes.html#:core_scope:'('_x_','_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">A</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">l</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span> <span class="id" type="var">H</span> <span class="id" type="var">H0</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split2"><span class="id" type="definition">split2</span></a> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">l</span>) <span class="id" type="var">eqn</span>:<span class="id" type="var">heq</span>.<br/>
- <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="more_list.html#split2_correct"><span class="id" type="lemma">split2_correct</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">heq</span>.<br/>
<span class="id" type="var">decompose</span> [<span class="id" type="var">and</span>] <span class="id" type="var">heq</span>.<br/>
<span class="id" type="tactic">clear</span> <span class="id" type="var">heq</span>.<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H4</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#app_same_length_eq"><span class="id" type="lemma">app_same_length_eq</span></a> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H4</span> <span class="id" type="var">H2</span>).<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">symmetry</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#app_same_length_eq"><span class="id" type="lemma">app_same_length_eq</span></a> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">_</span> <span class="id" type="var">H5</span> <span class="id" type="var">H0</span>).<br/>
<span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="id" type="tactic">destruct</span> (<a class="idref" href="more_list.html#split2_length_ok"><span class="id" type="lemma">split2_length_ok</span></a> <span class="id" type="var">_</span> <span class="id" type="var">l</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>).<br/>
+ <span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">repeat</span> <span class="id" type="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Lists.List.html#app_length"><span class="id" type="lemma">app_length</span></a>.<br/>
<span class="id" type="tactic">omega</span>.<br/>
+ <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
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