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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>Combi.Combi.std: Standard Words, i.e. Permutation as Words</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library Combi.Combi.std: Standard Words, i.e. Permutation as Words</h1>
<div class="code">
</div>
<div class="doc">
</div>
<div class="code">
</div>
<div class="doc">
<a id="lab76"></a><h1 class="section">Standard words</h1>
<div class="paragraph"> </div>
That is words wich are permutations of <span class="inlinecode">0,</span> <span class="inlinecode">1,</span> <span class="inlinecode">...,</span> <span class="inlinecode"><span class="id" title="var">n</span>-1</span>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">is_std</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> == <span class="inlinecode"><span class="id" title="var">s</span></span> is a standard word
</li>
<li> <span class="inlinecode"><span class="id" title="var">is_std_of_n</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> == <span class="inlinecode"><span class="id" title="var">s</span></span> is a standard word of size <span class="inlinecode"><span class="id" title="var">n</span></span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">wordperm</span></span> <span class="inlinecode"><span class="id" title="var">p</span></span> == the standard word associated with a permutation <span class="inlinecode"><span class="id" title="var">p</span></span> of <span class="inlinecode">'<span class="id" title="var">S_n</span></span>
</li>
</ul>
<div class="paragraph"> </div>
Sigma types for standard word
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">stdwordn</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> == a type for <span class="inlinecode"><span class="id" title="var">seq</span></span> <span class="inlinecode"><span class="id" title="var">nat</span></span> which standard words size <span class="inlinecode"><span class="id" title="var">n</span></span>;
it is canonically a <span class="inlinecode"><span class="id" title="var">finType</span></span>
</li>
</ul>
<div class="paragraph"> </div>
Inversions and standardisation of a word over a totally ordered alphabet
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">std</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> == given a word w over an <span class="inlinecode"><span class="id" title="var">ordType</span></span> returns the standard word with
the same set of inversion (lemmas <span class="inlinecode"><span class="id" title="var">stdP</span></span> and <span class="inlinecode"><span class="id" title="var">std_eq_invP</span></span>)
</li>
<li> <span class="inlinecode"><span class="id" title="var">versions</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode"><span class="id" title="var">j</span></span> == <span class="inlinecode"><span class="id" title="var">i</span></span> <span class="inlinecode"><span class="id" title="var">j</span></span> is an non-inversion of <span class="inlinecode"><span class="id" title="var">w</span></span> that is
<code>i <= j < size w and w_i <= w_j</code>
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode"><span class="id" title="var">eq_inv</span></span> <span class="inlinecode"><span class="id" title="var">w1</span></span> <span class="inlinecode"><span class="id" title="var">w2</span></span> == <span class="inlinecode"><span class="id" title="var">w1</span></span> and <span class="inlinecode"><span class="id" title="var">w2</span></span> have the same versions which is equivalent
to the same set of inversions; in particular <span class="inlinecode"><span class="id" title="var">w1</span></span> and <span class="inlinecode"><span class="id" title="var">w2</span></span>
have the same size.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode"><span class="id" title="var">std_spec</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">p</span></span> == <span class="inlinecode"><span class="id" title="var">p</span></span> is a standard word with the same versions as <span class="inlinecode"><span class="id" title="var">s</span></span>
</li>
</ul>
<div class="paragraph"> </div>
The main result here is <span class="inlinecode"><span class="id" title="var">std_eq_invP</span></span> which says that having the same inversions
is the same as having the same standardized.
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">linvseq</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> == <span class="inlinecode"><span class="id" title="var">t</span></span> is the left inverse of <span class="inlinecode"><span class="id" title="var">s</span></span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">invseq</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> == <span class="inlinecode"><span class="id" title="var">s</span></span> and <span class="inlinecode"><span class="id" title="var">t</span></span> are inverse on of each other
</li>
<li> <span class="inlinecode"><span class="id" title="var">invstd</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> == the inverse of the standardized of <span class="inlinecode"><span class="id" title="var">w</span></span> that is the permutation
which sorts <span class="inlinecode"><span class="id" title="var">w</span></span> in a stable way.
</li>
</ul>
<div class="paragraph"> </div>
</div>
<div class="code">
<span class="id" title="keyword">From</span> <span class="id" title="var">HB</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">structures</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">all_boot</span> <span class="id" title="library">order</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">perm</span> <span class="id" title="library">fingroup</span>.<br/>
<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Combi.SSRcomplements.tools.html#"><span class="id" title="library">tools</span></a> <a class="idref" href="Combi.Basic.combclass.html#"><span class="id" title="library">combclass</span></a> <a class="idref" href="Combi.Basic.ordtype.html#"><span class="id" title="library">ordtype</span></a> <a class="idref" href="Combi.SSRcomplements.permcomp.html#"><span class="id" title="library">permcomp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="module">Order.TTheory</span>.<br/>
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">nat_scope</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab77"></a><h1 class="section">Standard words and permutations</h1>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="StandardWords" class="idref" href="#StandardWords"><span class="id" title="section">StandardWords</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">n</span> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a id="is_std" class="idref" href="#is_std"><span class="id" title="definition">is_std</span></a> (<a id="s:1" class="idref" href="#s:1"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.std.html#s:1"><span class="id" title="variable">s</span></a> (<span class="id" title="definition">iota</span> 0 (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:1"><span class="id" title="variable">s</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="perm_std" class="idref" href="#perm_std"><span class="id" title="lemma">perm_std</span></a> <a id="u:2" class="idref" href="#u:2"><span class="id" title="binder">u</span></a> <a id="v:3" class="idref" href="#v:3"><span class="id" title="binder">v</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#u:2"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.std.html#v:3"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#u:2"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#v:3"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_perm" class="idref" href="#std_perm"><span class="id" title="lemma">std_perm</span></a> <a id="u:4" class="idref" href="#u:4"><span class="id" title="binder">u</span></a> <a id="v:5" class="idref" href="#v:5"><span class="id" title="binder">v</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#u:4"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:4"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.std.html#v:5"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#u:4"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="mem_std" class="idref" href="#mem_std"><span class="id" title="lemma">mem_std</span></a> <a id="p:6" class="idref" href="#p:6"><span class="id" title="binder">p</span></a> <a id="i:7" class="idref" href="#i:7"><span class="id" title="binder">i</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#p:6"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#i:7"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.std.html#p:6"><span class="id" title="variable">p</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#i:7"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#p:6"><span class="id" title="variable">p</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_uniq" class="idref" href="#std_uniq"><span class="id" title="lemma">std_uniq</span></a> <a id="u:8" class="idref" href="#u:8"><span class="id" title="binder">u</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#u:8"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.std.html#u:8"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_max" class="idref" href="#std_max"><span class="id" title="lemma">std_max</span></a> <a id="s0:9" class="idref" href="#s0:9"><span class="id" title="binder">s0</span></a> <a id="s:10" class="idref" href="#s:10"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> (<a class="idref" href="Combi.Combi.std.html#s0:9"><span class="id" title="variable">s0</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#s:10"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Basic.ordtype.html#maxL"><span class="id" title="definition">maxL</span></a> <a class="idref" href="Combi.Combi.std.html#s0:9"><span class="id" title="variable">s0</span></a> <a class="idref" href="Combi.Combi.std.html#s:10"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:10"><span class="id" title="variable">s</span></a>.<br/>
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<span class="id" title="keyword">Lemma</span> <a id="is_stdP" class="idref" href="#is_stdP"><span class="id" title="lemma">is_stdP</span></a> <a id="s:11" class="idref" href="#s:11"><span class="id" title="binder">s</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a> (<span class="id" title="keyword">forall</span> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</span></a>, <a class="idref" href="Combi.Combi.std.html#n:12"><span class="id" title="variable">n</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:11"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#n:12"><span class="id" title="variable">n</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.std.html#s:11"><span class="id" title="variable">s</span></a>) (<a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:11"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="wordperm" class="idref" href="#wordperm"><span class="id" title="definition">wordperm</span></a> <a id="n:13" class="idref" href="#n:13"><span class="id" title="binder">n</span></a> (<a id="p:14" class="idref" href="#p:14"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) := <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="notation">(</span><a class="idref" href="Combi.Combi.std.html#p:14"><span class="id" title="variable">p</span></a> <a class="idref" href="Combi.Combi.std.html#i:15"><span class="id" title="variable">i</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">|</span> <a id="i:15" class="idref" href="#i:15"><span class="id" title="binder"><span id="i:16" class="id"><span id="i:17" class="id"><span id="i:18" class="id">i</span></span></span></span></a> <a id="i:16" class="idref" href="#i:16"><span class="id" title="binder"><span id="i:18" class="id">:</span></span></a> <a id="i:16" class="idref" href="#i:16"><span class="id" title="binder"><span id="i:18" class="id">'</span></span></a><a id="i:16" class="idref" href="#i:16"><span class="id" title="binder"><span id="i:18" class="id">I_n</span></span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">group_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="wordperm_iota" class="idref" href="#wordperm_iota"><span class="id" title="lemma">wordperm_iota</span></a> <a id="n:19" class="idref" href="#n:19"><span class="id" title="binder">n</span></a> (<a id="p:20" class="idref" href="#p:20"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:20"><span class="id" title="variable">p</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <span class="id" title="definition">iota</span> 0 <a class="idref" href="Combi.Combi.std.html#n:19"><span class="id" title="variable">n</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="uniq_wordperm" class="idref" href="#uniq_wordperm"><span class="id" title="lemma">uniq_wordperm</span></a> <a id="n:21" class="idref" href="#n:21"><span class="id" title="binder">n</span></a> (<a id="p:22" class="idref" href="#p:22"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) : <span class="id" title="definition">uniq</span> (<a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:22"><span class="id" title="variable">p</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="wordperm_std" class="idref" href="#wordperm_std"><span class="id" title="lemma">wordperm_std</span></a> <a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> (<a id="p:24" class="idref" href="#p:24"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> (<a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:24"><span class="id" title="variable">p</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="perm_of_std" class="idref" href="#perm_of_std"><span class="id" title="lemma">perm_of_std</span></a> <a id="s:25" class="idref" href="#s:25"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a> <a id="p:26" class="idref" href="#p:26"><span class="id" title="binder">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <span class="id" title="notation">'</span><span class="id" title="notation">S_</span><span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:25"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="Combi.Combi.std.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:26"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="is_std_wordpermP" class="idref" href="#is_std_wordpermP"><span class="id" title="lemma">is_std_wordpermP</span></a> <a id="s:30" class="idref" href="#s:30"><span class="id" title="binder">s</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a> (<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">exists</span></a> <a id="p:31" class="idref" href="#p:31"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_</span><span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:30"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Combi.Combi.std.html#s:30"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:31"><span class="id" title="variable">p</span></a>) (<a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:30"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="wordperm_invP" class="idref" href="#wordperm_invP"><span class="id" title="lemma">wordperm_invP</span></a> <a id="n:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> (<a id="p:33" class="idref" href="#p:33"><span class="id" title="binder">p</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) <a id="i:34" class="idref" href="#i:34"><span class="id" title="binder">i</span></a> (<a id="j:35" class="idref" href="#j:35"><span class="id" title="binder">j</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">I_n</span>) :<br/>
<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Combi.std.html#n:32"><span class="id" title="variable">n</span></a> (<a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:33"><span class="id" title="variable">p</span></a>) <a class="idref" href="Combi.Combi.std.html#i:34"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#j:35"><span class="id" title="variable">j</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation"><-></span></a> <a class="idref" href="Combi.Combi.std.html#i:34"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">(</span><a class="idref" href="Combi.Combi.std.html#p:33"><span class="id" title="variable">p</span></a><span class="id" title="notation">^-1)</span> <span class="id" title="notation">j</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#StandardWords"><span class="id" title="section">StandardWords</span></a>.<br/>
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</div>
<div class="doc">
<a id="lab78"></a><h1 class="section">Standard words as a <span class="inlinecode"><span class="id" title="var">finType</span></span></h1>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="StdCombClass" class="idref" href="#StdCombClass"><span class="id" title="section">StdCombClass</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a id="StdCombClass.n" class="idref" href="#StdCombClass.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="is_std_of_n" class="idref" href="#is_std_of_n"><span class="id" title="definition">is_std_of_n</span></a> := <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#10dd1f0b03819efd5777bd139f7faec4"><span class="id" title="notation">[</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#10dd1f0b03819efd5777bd139f7faec4"><span class="id" title="notation">pred</span></a> <a id="w:38" class="idref" href="#w:38"><span class="id" title="binder">w</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#10dd1f0b03819efd5777bd139f7faec4"><span class="id" title="notation">|</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#w:38"><span class="id" title="variable">w</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">(</span></a><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w:38"><span class="id" title="variable">w</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.std.html#StdCombClass.n"><span class="id" title="variable">n</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#10dd1f0b03819efd5777bd139f7faec4"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a id="stdwordn" class="idref" href="#stdwordn"><span class="id" title="record">stdwordn</span></a> : <span class="id" title="keyword">Set</span> :=<br/>
<span class="id" title="var">StdWordN</span> {<a id="stdwordnval" class="idref" href="#stdwordnval"><span class="id" title="projection">stdwordnval</span></a> :> <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="Combi.Combi.std.html#is_std_of_n"><span class="id" title="definition">is_std_of_n</span></a> <a class="idref" href="Combi.Combi.std.html#stdwordnval:40"><span class="id" title="method">stdwordnval</span></a>}.<br/>
<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="stdwordnP" class="idref" href="#stdwordnP"><span class="id" title="lemma">stdwordnP</span></a> (<a id="s:41" class="idref" href="#s:41"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#stdwordn"><span class="id" title="record">stdwordn</span></a>) : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.std.html#s:41"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_sdtn" class="idref" href="#size_sdtn"><span class="id" title="lemma">size_sdtn</span></a> (<a id="s:42" class="idref" href="#s:42"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#stdwordn"><span class="id" title="record">stdwordn</span></a>) : <span class="id" title="definition">size</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.std.html#s:42"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#StdCombClass.n"><span class="id" title="variable">n</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="enum_stdwordn" class="idref" href="#enum_stdwordn"><span class="id" title="definition">enum_stdwordn</span></a> := <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#p:43"><span class="id" title="variable">p</span></a> <span class="id" title="notation">|</span> <a id="p:43" class="idref" href="#p:43"><span class="id" title="binder"><span id="p:44" class="id"><span id="p:45" class="id"><span id="p:46" class="id">p</span></span></span></span></a> <a id="p:44" class="idref" href="#p:44"><span class="id" title="binder"><span id="p:46" class="id">:</span></span></a> <a id="p:44" class="idref" href="#p:44"><span class="id" title="binder"><span id="p:46" class="id">'</span></span></a><a id="p:44" class="idref" href="#p:44"><span class="id" title="binder"><span id="p:46" class="id">S_n</span></span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="enum_stdwordnE" class="idref" href="#enum_stdwordnE"><span class="id" title="lemma">enum_stdwordnE</span></a> : <a class="idref" href="Combi.Combi.std.html#enum_stdwordn"><span class="id" title="definition">enum_stdwordn</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="Combi.Combi.std.html#is_std_of_n"><span class="id" title="definition">is_std_of_n</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="wordperm_inj" class="idref" href="#wordperm_inj"><span class="id" title="lemma">wordperm_inj</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (@<a class="idref" href="Combi.Combi.std.html#wordperm"><span class="id" title="definition">wordperm</span></a> <a class="idref" href="Combi.Combi.std.html#StdCombClass.n"><span class="id" title="variable">n</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="enum_stdwordn_uniq" class="idref" href="#enum_stdwordn_uniq"><span class="id" title="lemma">enum_stdwordn_uniq</span></a> : <span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.std.html#enum_stdwordn"><span class="id" title="definition">enum_stdwordn</span></a>.<br/>
<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="card_stdwordn" class="idref" href="#card_stdwordn"><span class="id" title="lemma">card_stdwordn</span></a> : <span class="id" title="notation">#|</span><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#fa3b33ae9d0a52de608b305a09f3b881"><span class="id" title="notation">{:</span></a> <a class="idref" href="Combi.Combi.std.html#stdwordn"><span class="id" title="record">stdwordn</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#fa3b33ae9d0a52de608b305a09f3b881"><span class="id" title="notation">}</span></a><span class="id" title="notation">|</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#StdCombClass.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#StdCombClass"><span class="id" title="section">StdCombClass</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab79"></a><h1 class="section">Standardisation of a word over a totally ordered alphabet</h1>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="Standardisation" class="idref" href="#Standardisation"><span class="id" title="section">Standardisation</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> {<a id="Standardisation.disp" class="idref" href="#Standardisation.disp"><span class="id" title="variable">disp</span></a>} {<a id="Standardisation.Alph" class="idref" href="#Standardisation.Alph"><span class="id" title="variable">Alph</span></a> : <span class="id" title="abbreviation">orderType</span> <a class="idref" href="Combi.Combi.std.html#disp:48"><span class="id" title="variable">disp</span></a>}.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> <span class="id" title="var">w</span> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Standardisation.Alph"><span class="id" title="variable">Alph</span></a>.<br/>
<br/>
<span class="id" title="keyword">Fixpoint</span> <a id="std_rec" class="idref" href="#std_rec"><span class="id" title="definition">std_rec</span></a> <a id="n:50" class="idref" href="#n:50"><span class="id" title="binder">n</span></a> <a id="s:51" class="idref" href="#s:51"><span class="id" title="binder">s</span></a> :=<br/>
<span class="id" title="keyword">if</span> <a class="idref" href="Combi.Combi.std.html#n:50"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><span class="id" title="notation">.+1</span> <span class="id" title="keyword">then</span><br/>
<span class="id" title="keyword">let</span> <a id="rec:54" class="idref" href="#rec:54"><span class="id" title="binder">rec</span></a> := <a class="idref" href="Combi.Combi.std.html#std_rec:52"><span class="id" title="definition">std_rec</span></a> <span class="id" title="var">n'</span> (<a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> <a class="idref" href="Combi.Combi.std.html#s:51"><span class="id" title="variable">s</span></a>) <span class="id" title="tactic">in</span><br/>
<span class="id" title="keyword">let</span> <a id="pos:55" class="idref" href="#pos:55"><span class="id" title="binder">pos</span></a> := <a class="idref" href="Combi.Basic.ordtype.html#posbig"><span class="id" title="definition">posbig</span></a> <a class="idref" href="Combi.Combi.std.html#s:51"><span class="id" title="variable">s</span></a> <span class="id" title="tactic">in</span><br/>
<span class="id" title="definition">take</span> <a class="idref" href="Combi.Combi.std.html#pos:55"><span class="id" title="variable">pos</span></a> <a class="idref" href="Combi.Combi.std.html#rec:54"><span class="id" title="variable">rec</span></a> <span class="id" title="notation">++</span> <span class="id" title="var">n'</span> <span class="id" title="notation">::</span> <span class="id" title="definition">drop</span> <a class="idref" href="Combi.Combi.std.html#pos:55"><span class="id" title="variable">pos</span></a> <a class="idref" href="Combi.Combi.std.html#rec:54"><span class="id" title="variable">rec</span></a><br/>
<span class="id" title="keyword">else</span> <span class="id" title="notation">[::]</span>.<br/>
<span class="id" title="keyword">Definition</span> <a id="std" class="idref" href="#std"><span class="id" title="definition">std</span></a> <a id="s:56" class="idref" href="#s:56"><span class="id" title="binder">s</span></a> := <a class="idref" href="Combi.Combi.std.html#std_rec"><span class="id" title="definition">std_rec</span></a> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:56"><span class="id" title="variable">s</span></a>) <a class="idref" href="Combi.Combi.std.html#s:56"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_std_rec" class="idref" href="#size_std_rec"><span class="id" title="lemma">size_std_rec</span></a> <a id="n:57" class="idref" href="#n:57"><span class="id" title="binder">n</span></a> <a id="s:58" class="idref" href="#s:58"><span class="id" title="binder">s</span></a> : <span class="id" title="definition">size</span> (<a class="idref" href="Combi.Combi.std.html#std_rec"><span class="id" title="definition">std_rec</span></a> <a class="idref" href="Combi.Combi.std.html#n:57"><span class="id" title="variable">n</span></a> <a class="idref" href="Combi.Combi.std.html#s:58"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#n:57"><span class="id" title="variable">n</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_std" class="idref" href="#size_std"><span class="id" title="lemma">size_std</span></a> <a id="s:59" class="idref" href="#s:59"><span class="id" title="binder">s</span></a> : <span class="id" title="definition">size</span> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:59"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:59"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_is_std" class="idref" href="#std_is_std"><span class="id" title="lemma">std_is_std</span></a> <a id="s:60" class="idref" href="#s:60"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:60"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="in_std_ltn_size" class="idref" href="#in_std_ltn_size"><span class="id" title="lemma">in_std_ltn_size</span></a> <a id="s:61" class="idref" href="#s:61"><span class="id" title="binder">s</span></a> <a id="i:62" class="idref" href="#i:62"><span class="id" title="binder">i</span></a> : <a class="idref" href="Combi.Combi.std.html#i:62"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:61"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#i:62"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:61"><span class="id" title="variable">s</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="allLtn_std_rec" class="idref" href="#allLtn_std_rec"><span class="id" title="lemma">allLtn_std_rec</span></a> <a id="s:63" class="idref" href="#s:63"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#allLtn"><span class="id" title="definition">allLtn</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:63"><span class="id" title="variable">s</span></a>) (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:63"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="rembig_ins_std" class="idref" href="#rembig_ins_std"><span class="id" title="lemma">rembig_ins_std</span></a> <a id="s:64" class="idref" href="#s:64"><span class="id" title="binder">s</span></a> <a id="pos:65" class="idref" href="#pos:65"><span class="id" title="binder">pos</span></a> :<br/>
<a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> (<span class="id" title="definition">take</span> <a class="idref" href="Combi.Combi.std.html#pos:65"><span class="id" title="variable">pos</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:64"><span class="id" title="variable">s</span></a>) <span class="id" title="notation">++</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:64"><span class="id" title="variable">s</span></a> <span class="id" title="notation">::</span> <span class="id" title="definition">drop</span> <a class="idref" href="Combi.Combi.std.html#pos:65"><span class="id" title="variable">pos</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:64"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:64"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_rembig" class="idref" href="#std_rembig"><span class="id" title="lemma">std_rembig</span></a> <a id="s:66" class="idref" href="#s:66"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> <a class="idref" href="Combi.Combi.std.html#s:66"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:66"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_posbig" class="idref" href="#std_posbig"><span class="id" title="lemma">std_posbig</span></a> <a id="s:67" class="idref" href="#s:67"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#posbig"><span class="id" title="definition">posbig</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:67"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Basic.ordtype.html#posbig"><span class="id" title="definition">posbig</span></a> <a class="idref" href="Combi.Combi.std.html#s:67"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#Standardisation"><span class="id" title="section">Standardisation</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_std" class="idref" href="#std_std"><span class="id" title="lemma">std_std</span></a> <a id="s:68" class="idref" href="#s:68"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:68"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:68"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#s:68"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_stdE" class="idref" href="#std_stdE"><span class="id" title="lemma">std_stdE</span></a> <a id="disp:69" class="idref" href="#disp:69"><span class="id" title="binder">disp</span></a> (<a id="T:70" class="idref" href="#T:70"><span class="id" title="binder">T</span></a> : <span class="id" title="abbreviation">orderType</span> <a class="idref" href="Combi.Combi.std.html#disp:69"><span class="id" title="variable">disp</span></a>) (<a id="s:71" class="idref" href="#s:71"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:70"><span class="id" title="variable">T</span></a>) : <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:71"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:71"><span class="id" title="variable">s</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab80"></a><h2 class="section">Inversion sets and standardization</h2>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="EqInvDef" class="idref" href="#EqInvDef"><span class="id" title="section">EqInvDef</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="versions" class="idref" href="#versions"><span class="id" title="definition">versions</span></a> <a id="d:72" class="idref" href="#d:72"><span class="id" title="binder">d</span></a> (<a id="T:73" class="idref" href="#T:73"><span class="id" title="binder">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d:72"><span class="id" title="variable">d</span></a>) (<a id="w:74" class="idref" href="#w:74"><span class="id" title="binder">w</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:73"><span class="id" title="variable">T</span></a>) : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> :=<br/>
<span class="id" title="keyword">fun</span> <a id="i:75" class="idref" href="#i:75"><span class="id" title="binder">i</span></a> <a id="j:76" class="idref" href="#j:76"><span class="id" title="binder">j</span></a> => <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#i:75"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Combi.std.html#j:76"><span class="id" title="variable">j</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w:74"><span class="id" title="variable">w</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">&&</span></a> (<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w:74"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.std.html#i:75"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w:74"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.std.html#j:76"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">O</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="eq_inv" class="idref" href="#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a id="d1:77" class="idref" href="#d1:77"><span class="id" title="binder">d1</span></a> <a id="d2:78" class="idref" href="#d2:78"><span class="id" title="binder">d2</span></a> (<a id="T1:79" class="idref" href="#T1:79"><span class="id" title="binder">T1</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d1:77"><span class="id" title="variable">d1</span></a>) (<a id="T2:80" class="idref" href="#T2:80"><span class="id" title="binder">T2</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d2:78"><span class="id" title="variable">d2</span></a>)<br/>
(<a id="w1:81" class="idref" href="#w1:81"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T1:79"><span class="id" title="variable">T1</span></a>) (<a id="w2:82" class="idref" href="#w2:82"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T2:80"><span class="id" title="variable">T2</span></a>) :=<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#fe2a1a98339f5fcdd989ce8af348292e"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#versions"><span class="id" title="definition">versions</span></a> <a class="idref" href="Combi.Combi.std.html#w1:81"><span class="id" title="variable">w1</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#fe2a1a98339f5fcdd989ce8af348292e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#fe2a1a98339f5fcdd989ce8af348292e"><span class="id" title="notation">=2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#fe2a1a98339f5fcdd989ce8af348292e"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#versions"><span class="id" title="definition">versions</span></a> <a class="idref" href="Combi.Combi.std.html#w2:82"><span class="id" title="variable">w2</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#fe2a1a98339f5fcdd989ce8af348292e"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="EqInvDef.disp1" class="idref" href="#EqInvDef.disp1"><span class="id" title="variable">disp1</span></a> <a id="EqInvDef.disp2" class="idref" href="#EqInvDef.disp2"><span class="id" title="variable">disp2</span></a> <a id="EqInvDef.disp3" class="idref" href="#EqInvDef.disp3"><span class="id" title="variable">disp3</span></a><br/>
(<a id="EqInvDef.S" class="idref" href="#EqInvDef.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp1:83"><span class="id" title="variable">disp1</span></a>)<br/>
(<a id="EqInvDef.T" class="idref" href="#EqInvDef.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp2:84"><span class="id" title="variable">disp2</span></a>)<br/>
(<a id="EqInvDef.U" class="idref" href="#EqInvDef.U"><span class="id" title="variable">U</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp3:85"><span class="id" title="variable">disp3</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_refl" class="idref" href="#eq_inv_refl"><span class="id" title="lemma">eq_inv_refl</span></a> (<a id="w:89" class="idref" href="#w:89"><span class="id" title="binder">w</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w:89"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.std.html#w:89"><span class="id" title="variable">w</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_nil" class="idref" href="#eq_inv_nil"><span class="id" title="lemma">eq_inv_nil</span></a> : <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (@<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="Combi.Combi.std.html#EqInvDef.S"><span class="id" title="variable">S</span></a>) (@<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="Combi.Combi.std.html#EqInvDef.T"><span class="id" title="variable">T</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_sym" class="idref" href="#eq_inv_sym"><span class="id" title="lemma">eq_inv_sym</span></a> (<a id="w1:90" class="idref" href="#w1:90"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.S"><span class="id" title="variable">S</span></a>) (<a id="w2:91" class="idref" href="#w2:91"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:90"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w2:91"><span class="id" title="variable">w2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w2:91"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#w1:90"><span class="id" title="variable">w1</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_trans" class="idref" href="#eq_inv_trans"><span class="id" title="lemma">eq_inv_trans</span></a> (<a id="w1:92" class="idref" href="#w1:92"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.S"><span class="id" title="variable">S</span></a>) (<a id="w2:93" class="idref" href="#w2:93"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.T"><span class="id" title="variable">T</span></a>) (<a id="w3:94" class="idref" href="#w3:94"><span class="id" title="binder">w3</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef.U"><span class="id" title="variable">U</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:92"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w2:93"><span class="id" title="variable">w2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w2:93"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#w3:94"><span class="id" title="variable">w3</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:92"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w3:94"><span class="id" title="variable">w3</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#EqInvDef"><span class="id" title="section">EqInvDef</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_size" class="idref" href="#eq_inv_size"><span class="id" title="lemma">eq_inv_size</span></a> <a id="d1:95" class="idref" href="#d1:95"><span class="id" title="binder">d1</span></a> <a id="d2:96" class="idref" href="#d2:96"><span class="id" title="binder">d2</span></a> (<a id="T1:97" class="idref" href="#T1:97"><span class="id" title="binder">T1</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d1:95"><span class="id" title="variable">d1</span></a>) (<a id="T2:98" class="idref" href="#T2:98"><span class="id" title="binder">T2</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d2:96"><span class="id" title="variable">d2</span></a>)<br/>
(<a id="w1:99" class="idref" href="#w1:99"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T1:97"><span class="id" title="variable">T1</span></a>) (<a id="w2:100" class="idref" href="#w2:100"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T2:98"><span class="id" title="variable">T2</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:99"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w2:100"><span class="id" title="variable">w2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w1:99"><span class="id" title="variable">w1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w2:100"><span class="id" title="variable">w2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="EqInvAltDef" class="idref" href="#EqInvAltDef"><span class="id" title="section">EqInvAltDef</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="EqInvAltDef.disp1" class="idref" href="#EqInvAltDef.disp1"><span class="id" title="variable">disp1</span></a> <a id="EqInvAltDef.disp2" class="idref" href="#EqInvAltDef.disp2"><span class="id" title="variable">disp2</span></a> <a id="EqInvAltDef.disp3" class="idref" href="#EqInvAltDef.disp3"><span class="id" title="variable">disp3</span></a><br/>
(<a id="EqInvAltDef.S" class="idref" href="#EqInvAltDef.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp1:101"><span class="id" title="variable">disp1</span></a>)<br/>
(<a id="EqInvAltDef.T" class="idref" href="#EqInvAltDef.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp2:102"><span class="id" title="variable">disp2</span></a>)<br/>
(<a id="EqInvAltDef.U" class="idref" href="#EqInvAltDef.U"><span class="id" title="variable">U</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp3:103"><span class="id" title="variable">disp3</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_invP" class="idref" href="#eq_invP"><span class="id" title="lemma">eq_invP</span></a> (<a id="w1:107" class="idref" href="#w1:107"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.S"><span class="id" title="variable">S</span></a>) (<a id="w2:108" class="idref" href="#w2:108"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">(</span></a><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w1:107"><span class="id" title="variable">w1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w2:108"><span class="id" title="variable">w2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\</span></a><br/>
<span class="id" title="keyword">forall</span> <a id="i:109" class="idref" href="#i:109"><span class="id" title="binder">i</span></a> <a id="j:110" class="idref" href="#j:110"><span class="id" title="binder">j</span></a>, <a class="idref" href="Combi.Combi.std.html#i:109"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Combi.std.html#j:110"><span class="id" title="variable">j</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w1:107"><span class="id" title="variable">w1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
(<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w1:107"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#i:109"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w1:107"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#j:110"><span class="id" title="variable">j</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w2:108"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#i:109"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w2:108"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#j:110"><span class="id" title="variable">j</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">O</span><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">)</span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation"><-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:107"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w2:108"><span class="id" title="variable">w2</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_inversionP" class="idref" href="#eq_inv_inversionP"><span class="id" title="lemma">eq_inv_inversionP</span></a> (<a id="w1:111" class="idref" href="#w1:111"><span class="id" title="binder">w1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.S"><span class="id" title="variable">S</span></a>) (<a id="w2:112" class="idref" href="#w2:112"><span class="id" title="binder">w2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.T"><span class="id" title="variable">T</span></a>) :<br/>
(<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w1:111"><span class="id" title="variable">w1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w2:112"><span class="id" title="variable">w2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\</span></a><br/>
<span class="id" title="keyword">forall</span> <a id="i:113" class="idref" href="#i:113"><span class="id" title="binder">i</span></a> <a id="j:114" class="idref" href="#j:114"><span class="id" title="binder">j</span></a>, <a class="idref" href="Combi.Combi.std.html#i:113"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Combi.std.html#j:114"><span class="id" title="variable">j</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#w1:111"><span class="id" title="variable">w1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w1:111"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#i:113"><span class="id" title="variable">i</span></a> <span class="id" title="notation">></span> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w1:111"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#j:114"><span class="id" title="variable">j</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w2:112"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#i:113"><span class="id" title="variable">i</span></a> <span class="id" title="notation">></span> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> <a class="idref" href="Combi.Combi.std.html#w2:112"><span class="id" title="variable">w2</span></a> <a class="idref" href="Combi.Combi.std.html#j:114"><span class="id" title="variable">j</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">O</span><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation"><-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#w1:111"><span class="id" title="variable">w1</span></a> <a class="idref" href="Combi.Combi.std.html#w2:112"><span class="id" title="variable">w2</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_consK" class="idref" href="#eq_inv_consK"><span class="id" title="lemma">eq_inv_consK</span></a> (<a id="a:115" class="idref" href="#a:115"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.S"><span class="id" title="variable">S</span></a>) <a id="u:116" class="idref" href="#u:116"><span class="id" title="binder">u</span></a> (<a id="b:117" class="idref" href="#b:117"><span class="id" title="binder">b</span></a> : <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.T"><span class="id" title="variable">T</span></a>) <a id="v:118" class="idref" href="#v:118"><span class="id" title="binder">v</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#a:115"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#u:116"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Combi.std.html#b:117"><span class="id" title="variable">b</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#v:118"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:116"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:118"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_rconsK" class="idref" href="#eq_inv_rconsK"><span class="id" title="lemma">eq_inv_rconsK</span></a> (<a id="a:119" class="idref" href="#a:119"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.S"><span class="id" title="variable">S</span></a>) <a id="u:120" class="idref" href="#u:120"><span class="id" title="binder">u</span></a> (<a id="b:121" class="idref" href="#b:121"><span class="id" title="binder">b</span></a> : <a class="idref" href="Combi.Combi.std.html#EqInvAltDef.T"><span class="id" title="variable">T</span></a>) <a id="v:122" class="idref" href="#v:122"><span class="id" title="binder">v</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.std.html#u:120"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#a:119"><span class="id" title="variable">a</span></a>) (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.std.html#v:122"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#b:121"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:120"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:122"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#EqInvAltDef"><span class="id" title="section">EqInvAltDef</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_allgt_imply" class="idref" href="#eq_inv_allgt_imply"><span class="id" title="lemma">eq_inv_allgt_imply</span></a><br/>
<a id="d1:123" class="idref" href="#d1:123"><span class="id" title="binder">d1</span></a> <a id="d2:124" class="idref" href="#d2:124"><span class="id" title="binder">d2</span></a> (<a id="S:125" class="idref" href="#S:125"><span class="id" title="binder">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d1:123"><span class="id" title="variable">d1</span></a>) (<a id="T:126" class="idref" href="#T:126"><span class="id" title="binder">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d2:124"><span class="id" title="variable">d2</span></a>) (<a id="a:127" class="idref" href="#a:127"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.std.html#S:125"><span class="id" title="variable">S</span></a>) <a id="u:128" class="idref" href="#u:128"><span class="id" title="binder">u</span></a> (<a id="b:129" class="idref" href="#b:129"><span class="id" title="binder">b</span></a> : <a class="idref" href="Combi.Combi.std.html#T:126"><span class="id" title="variable">T</span></a>) <a id="v:130" class="idref" href="#v:130"><span class="id" title="binder">v</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#a:127"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#u:128"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Combi.std.html#b:129"><span class="id" title="variable">b</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#v:130"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Basic.ordtype.html#allLtn"><span class="id" title="definition">allLtn</span></a> <a class="idref" href="Combi.Combi.std.html#u:128"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#a:127"><span class="id" title="variable">a</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Basic.ordtype.html#allLtn"><span class="id" title="definition">allLtn</span></a> <a class="idref" href="Combi.Combi.std.html#v:130"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#b:129"><span class="id" title="variable">b</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_allgtnX" class="idref" href="#eq_inv_allgtnX"><span class="id" title="lemma">eq_inv_allgtnX</span></a><br/>
<a id="d1:131" class="idref" href="#d1:131"><span class="id" title="binder">d1</span></a> <a id="d2:132" class="idref" href="#d2:132"><span class="id" title="binder">d2</span></a> (<a id="S:133" class="idref" href="#S:133"><span class="id" title="binder">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d1:131"><span class="id" title="variable">d1</span></a>) (<a id="T:134" class="idref" href="#T:134"><span class="id" title="binder">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d2:132"><span class="id" title="variable">d2</span></a>) (<a id="a:135" class="idref" href="#a:135"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.std.html#S:133"><span class="id" title="variable">S</span></a>) <a id="u:136" class="idref" href="#u:136"><span class="id" title="binder">u</span></a> (<a id="b:137" class="idref" href="#b:137"><span class="id" title="binder">b</span></a> : <a class="idref" href="Combi.Combi.std.html#T:134"><span class="id" title="variable">T</span></a>) <a id="v:138" class="idref" href="#v:138"><span class="id" title="binder">v</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#a:135"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#u:136"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Combi.std.html#b:137"><span class="id" title="variable">b</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.std.html#v:138"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Basic.ordtype.html#allLtn"><span class="id" title="definition">allLtn</span></a> <a class="idref" href="Combi.Combi.std.html#u:136"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#a:135"><span class="id" title="variable">a</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Basic.ordtype.html#allLtn"><span class="id" title="definition">allLtn</span></a> <a class="idref" href="Combi.Combi.std.html#v:138"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#b:137"><span class="id" title="variable">b</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="EqInvPosRemBig" class="idref" href="#EqInvPosRemBig"><span class="id" title="section">EqInvPosRemBig</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="EqInvPosRemBig.disp1" class="idref" href="#EqInvPosRemBig.disp1"><span class="id" title="variable">disp1</span></a> <a id="EqInvPosRemBig.disp2" class="idref" href="#EqInvPosRemBig.disp2"><span class="id" title="variable">disp2</span></a> (<a id="EqInvPosRemBig.S" class="idref" href="#EqInvPosRemBig.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp1:139"><span class="id" title="variable">disp1</span></a>) (<a id="EqInvPosRemBig.T" class="idref" href="#EqInvPosRemBig.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp2:140"><span class="id" title="variable">disp2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_posbig" class="idref" href="#eq_inv_posbig"><span class="id" title="lemma">eq_inv_posbig</span></a> (<a id="u:143" class="idref" href="#u:143"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.S"><span class="id" title="variable">S</span></a>) (<a id="v:144" class="idref" href="#v:144"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:143"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:144"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Basic.ordtype.html#posbig"><span class="id" title="definition">posbig</span></a> <a class="idref" href="Combi.Combi.std.html#u:143"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Basic.ordtype.html#posbig"><span class="id" title="definition">posbig</span></a> <a class="idref" href="Combi.Combi.std.html#v:144"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_rembig" class="idref" href="#eq_inv_rembig"><span class="id" title="lemma">eq_inv_rembig</span></a> (<a id="u:145" class="idref" href="#u:145"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.S"><span class="id" title="variable">S</span></a>) (<a id="v:146" class="idref" href="#v:146"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:145"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:146"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> <a class="idref" href="Combi.Combi.std.html#u:145"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Basic.ordtype.html#rembig"><span class="id" title="definition">rembig</span></a> <a class="idref" href="Combi.Combi.std.html#v:146"><span class="id" title="variable">v</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_eq_inv" class="idref" href="#std_eq_inv"><span class="id" title="lemma">std_eq_inv</span></a> (<a id="u:147" class="idref" href="#u:147"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.S"><span class="id" title="variable">S</span></a>) (<a id="v:148" class="idref" href="#v:148"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:147"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:148"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:147"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#v:148"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_std" class="idref" href="#eq_inv_std"><span class="id" title="lemma">eq_inv_std</span></a> (<a id="u:149" class="idref" href="#u:149"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig.T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:149"><span class="id" title="variable">u</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:149"><span class="id" title="variable">u</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#EqInvPosRemBig"><span class="id" title="section">EqInvPosRemBig</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="Spec" class="idref" href="#Spec"><span class="id" title="section">Spec</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="Spec.disp1" class="idref" href="#Spec.disp1"><span class="id" title="variable">disp1</span></a> <a id="Spec.disp2" class="idref" href="#Spec.disp2"><span class="id" title="variable">disp2</span></a> (<a id="Spec.S" class="idref" href="#Spec.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp1:152"><span class="id" title="variable">disp1</span></a>) (<a id="Spec.T" class="idref" href="#Spec.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp2:153"><span class="id" title="variable">disp2</span></a>).<br/>
<br/>
<span class="id" title="keyword">Variant</span> <a id="std_spec" class="idref" href="#std_spec"><span class="id" title="inductive">std_spec</span></a> (<a id="s:156" class="idref" href="#s:156"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) (<a id="p:157" class="idref" href="#p:157"><span class="id" title="binder">p</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="StdSpec" class="idref" href="#StdSpec"><span class="id" title="constructor">StdSpec</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#p:157"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#s:156"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#p:157"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std_spec:158"><span class="id" title="inductive">std_spec</span></a> <a class="idref" href="Combi.Combi.std.html#s:156"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#p:157"><span class="id" title="variable">p</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_spec_uniq" class="idref" href="#std_spec_uniq"><span class="id" title="lemma">std_spec_uniq</span></a> (<a id="u:160" class="idref" href="#u:160"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) <a id="p:161" class="idref" href="#p:161"><span class="id" title="binder">p</span></a> <a id="q:162" class="idref" href="#q:162"><span class="id" title="binder">q</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#std_spec"><span class="id" title="inductive">std_spec</span></a> <a class="idref" href="Combi.Combi.std.html#u:160"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#p:161"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std_spec"><span class="id" title="inductive">std_spec</span></a> <a class="idref" href="Combi.Combi.std.html#u:160"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#q:162"><span class="id" title="variable">q</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#p:161"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#q:162"><span class="id" title="variable">q</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_specP" class="idref" href="#std_specP"><span class="id" title="lemma">std_specP</span></a> (<a id="s:163" class="idref" href="#s:163"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="Combi.Combi.std.html#std_spec"><span class="id" title="inductive">std_spec</span></a> <a class="idref" href="Combi.Combi.std.html#s:163"><span class="id" title="variable">s</span></a> (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:163"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="stdP" class="idref" href="#stdP"><span class="id" title="lemma">stdP</span></a> (<a id="s:164" class="idref" href="#s:164"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) <a id="p:165" class="idref" href="#p:165"><span class="id" title="binder">p</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a> (<a class="idref" href="Combi.Combi.std.html#std_spec"><span class="id" title="inductive">std_spec</span></a> <a class="idref" href="Combi.Combi.std.html#s:164"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#p:165"><span class="id" title="variable">p</span></a>) (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#s:164"><span class="id" title="variable">s</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.std.html#p:165"><span class="id" title="variable">p</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_eq_invP" class="idref" href="#std_eq_invP"><span class="id" title="lemma">std_eq_invP</span></a> (<a id="u:166" class="idref" href="#u:166"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.S"><span class="id" title="variable">S</span></a>) (<a id="v:167" class="idref" href="#v:167"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a> (<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u:166"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:167"><span class="id" title="variable">v</span></a>) (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:166"><span class="id" title="variable">u</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#v:167"><span class="id" title="variable">v</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_rconsK" class="idref" href="#std_rconsK"><span class="id" title="lemma">std_rconsK</span></a> (<a id="u:168" class="idref" href="#u:168"><span class="id" title="binder">u</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.S"><span class="id" title="variable">S</span></a>) (<a id="v:169" class="idref" href="#v:169"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) <a id="a:170" class="idref" href="#a:170"><span class="id" title="binder">a</span></a> <a id="b:171" class="idref" href="#b:171"><span class="id" title="binder">b</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.std.html#u:168"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#a:170"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.std.html#v:169"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Combi.std.html#b:171"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:168"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#v:169"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_catl" class="idref" href="#eq_inv_catl"><span class="id" title="lemma">eq_inv_catl</span></a> (<a id="u1:172" class="idref" href="#u1:172"><span class="id" title="binder">u1</span></a> <a id="v1:173" class="idref" href="#v1:173"><span class="id" title="binder">v1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.S"><span class="id" title="variable">S</span></a>) (<a id="u2:174" class="idref" href="#u2:174"><span class="id" title="binder">u2</span></a> <a id="v2:175" class="idref" href="#v2:175"><span class="id" title="binder">v2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#u1:172"><span class="id" title="variable">u1</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v1:173"><span class="id" title="variable">v1</span></a>) (<a class="idref" href="Combi.Combi.std.html#u2:174"><span class="id" title="variable">u2</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v2:175"><span class="id" title="variable">v2</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u1:172"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u2:174"><span class="id" title="variable">u2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#u1:172"><span class="id" title="variable">u1</span></a> <a class="idref" href="Combi.Combi.std.html#u2:174"><span class="id" title="variable">u2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_catr" class="idref" href="#eq_inv_catr"><span class="id" title="lemma">eq_inv_catr</span></a> (<a id="u1:176" class="idref" href="#u1:176"><span class="id" title="binder">u1</span></a> <a id="v1:177" class="idref" href="#v1:177"><span class="id" title="binder">v1</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.S"><span class="id" title="variable">S</span></a>) (<a id="u2:178" class="idref" href="#u2:178"><span class="id" title="binder">u2</span></a> <a id="v2:179" class="idref" href="#v2:179"><span class="id" title="binder">v2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Spec.T"><span class="id" title="variable">T</span></a>) :<br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#u1:176"><span class="id" title="variable">u1</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v1:177"><span class="id" title="variable">v1</span></a>) (<a class="idref" href="Combi.Combi.std.html#u2:178"><span class="id" title="variable">u2</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v2:179"><span class="id" title="variable">v2</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u1:176"><span class="id" title="variable">u1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u2:178"><span class="id" title="variable">u2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> <a class="idref" href="Combi.Combi.std.html#v1:177"><span class="id" title="variable">v1</span></a> <a class="idref" href="Combi.Combi.std.html#v2:179"><span class="id" title="variable">v2</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#Spec"><span class="id" title="section">Spec</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="StdTakeDrop" class="idref" href="#StdTakeDrop"><span class="id" title="section">StdTakeDrop</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="StdTakeDrop.disp1" class="idref" href="#StdTakeDrop.disp1"><span class="id" title="variable">disp1</span></a> <a id="StdTakeDrop.disp2" class="idref" href="#StdTakeDrop.disp2"><span class="id" title="variable">disp2</span></a> (<a id="StdTakeDrop.S" class="idref" href="#StdTakeDrop.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp1:180"><span class="id" title="variable">disp1</span></a>) (<a id="StdTakeDrop.T" class="idref" href="#StdTakeDrop.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp2:181"><span class="id" title="variable">disp2</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#StdTakeDrop.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_take_std" class="idref" href="#std_take_std"><span class="id" title="lemma">std_take_std</span></a> <a id="u:184" class="idref" href="#u:184"><span class="id" title="binder">u</span></a> <a id="v:185" class="idref" href="#v:185"><span class="id" title="binder">v</span></a> : <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<span class="id" title="definition">take</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:184"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#u:184"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:185"><span class="id" title="variable">v</span></a>))) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:184"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_drop_std" class="idref" href="#std_drop_std"><span class="id" title="lemma">std_drop_std</span></a> <a id="u:186" class="idref" href="#u:186"><span class="id" title="binder">u</span></a> <a id="v:187" class="idref" href="#v:187"><span class="id" title="binder">v</span></a> : <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<span class="id" title="definition">drop</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:186"><span class="id" title="variable">u</span></a>) (<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#u:186"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:187"><span class="id" title="variable">v</span></a>))) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#v:187"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#StdTakeDrop"><span class="id" title="section">StdTakeDrop</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="PermEq" class="idref" href="#PermEq"><span class="id" title="section">PermEq</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="PermEq.disp" class="idref" href="#PermEq.disp"><span class="id" title="variable">disp</span></a> (<a id="PermEq.Alph" class="idref" href="#PermEq.Alph"><span class="id" title="variable">Alph</span></a> : <span class="id" title="abbreviation">orderType</span> <a class="idref" href="Combi.Combi.std.html#disp:188"><span class="id" title="variable">disp</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#PermEq.Alph"><span class="id" title="variable">Alph</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a id="perm_stdE" class="idref" href="#perm_stdE"><span class="id" title="lemma">perm_stdE</span></a> <a id="u:190" class="idref" href="#u:190"><span class="id" title="binder">u</span></a> <a id="v:191" class="idref" href="#v:191"><span class="id" title="binder">v</span></a> : <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.std.html#u:190"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Combi.std.html#v:191"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#u:190"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> <a class="idref" href="Combi.Combi.std.html#v:191"><span class="id" title="variable">v</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#u:190"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#v:191"><span class="id" title="variable">v</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#PermEq"><span class="id" title="section">PermEq</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab81"></a><h2 class="section">Standardization and elementary transpositions of a word</h2>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="Transp" class="idref" href="#Transp"><span class="id" title="section">Transp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> <a id="Transp.disp" class="idref" href="#Transp.disp"><span class="id" title="variable">disp</span></a> (<a id="Transp.Alph" class="idref" href="#Transp.Alph"><span class="id" title="variable">Alph</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#disp:192"><span class="id" title="variable">disp</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#Transp.Alph"><span class="id" title="variable">Alph</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="nth_transp" class="idref" href="#nth_transp"><span class="id" title="lemma">nth_transp</span></a> <a id="u:194" class="idref" href="#u:194"><span class="id" title="binder">u</span></a> <a id="v:195" class="idref" href="#v:195"><span class="id" title="binder">v</span></a> <a id="a:196" class="idref" href="#a:196"><span class="id" title="binder">a</span></a> <a id="b:197" class="idref" href="#b:197"><span class="id" title="binder">b</span></a> <a id="i:198" class="idref" href="#i:198"><span class="id" title="binder">i</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#i:198"><span class="id" title="variable">i</span></a> <span class="id" title="notation">!=</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:194"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#i:198"><span class="id" title="variable">i</span></a> <span class="id" title="notation">!=</span> <span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:194"><span class="id" title="variable">u</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> (<a class="idref" href="Combi.Combi.std.html#u:194"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:196"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:197"><span class="id" title="variable">b</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:195"><span class="id" title="variable">v</span></a>) <a class="idref" href="Combi.Combi.std.html#i:198"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> (<a class="idref" href="Combi.Combi.std.html#u:194"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#b:197"><span class="id" title="variable">b</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#a:196"><span class="id" title="variable">a</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:195"><span class="id" title="variable">v</span></a>) <a class="idref" href="Combi.Combi.std.html#i:198"><span class="id" title="variable">i</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="nth_sizeu" class="idref" href="#nth_sizeu"><span class="id" title="lemma">nth_sizeu</span></a> <a id="u:199" class="idref" href="#u:199"><span class="id" title="binder">u</span></a> <a id="v:200" class="idref" href="#v:200"><span class="id" title="binder">v</span></a> <a id="a:201" class="idref" href="#a:201"><span class="id" title="binder">a</span></a> <a id="b:202" class="idref" href="#b:202"><span class="id" title="binder">b</span></a> :<br/>
<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> (<a class="idref" href="Combi.Combi.std.html#u:199"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:201"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:202"><span class="id" title="variable">b</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:200"><span class="id" title="variable">v</span></a>) (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:199"><span class="id" title="variable">u</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#a:201"><span class="id" title="variable">a</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="nth_sizeu1" class="idref" href="#nth_sizeu1"><span class="id" title="lemma">nth_sizeu1</span></a> <a id="u:203" class="idref" href="#u:203"><span class="id" title="binder">u</span></a> <a id="v:204" class="idref" href="#v:204"><span class="id" title="binder">v</span></a> <a id="a:205" class="idref" href="#a:205"><span class="id" title="binder">a</span></a> <a id="b:206" class="idref" href="#b:206"><span class="id" title="binder">b</span></a> :<br/>
<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> (<a class="idref" href="Combi.Combi.std.html#u:203"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:205"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:206"><span class="id" title="variable">b</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:204"><span class="id" title="variable">v</span></a>) <span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:203"><span class="id" title="variable">u</span></a><span class="id" title="notation">).+1</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#b:206"><span class="id" title="variable">b</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="nth_sizeu2" class="idref" href="#nth_sizeu2"><span class="id" title="lemma">nth_sizeu2</span></a> <a id="u:207" class="idref" href="#u:207"><span class="id" title="binder">u</span></a> <a id="v:208" class="idref" href="#v:208"><span class="id" title="binder">v</span></a> <a id="a:209" class="idref" href="#a:209"><span class="id" title="binder">a</span></a> <a id="b:210" class="idref" href="#b:210"><span class="id" title="binder">b</span></a> <a id="c:211" class="idref" href="#c:211"><span class="id" title="binder">c</span></a> :<br/>
<span class="id" title="definition">nth</span> <a class="idref" href="Combi.Basic.ordtype.html#inh"><span class="id" title="definition">inh</span></a> (<a class="idref" href="Combi.Combi.std.html#u:207"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:209"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:210"><span class="id" title="variable">b</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#c:211"><span class="id" title="variable">c</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:208"><span class="id" title="variable">v</span></a>) <span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:207"><span class="id" title="variable">u</span></a><span class="id" title="notation">).+2</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#c:211"><span class="id" title="variable">c</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#Transp"><span class="id" title="section">Transp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_inv_transp" class="idref" href="#eq_inv_transp"><span class="id" title="lemma">eq_inv_transp</span></a> <a id="d1:212" class="idref" href="#d1:212"><span class="id" title="binder">d1</span></a> <a id="d2:213" class="idref" href="#d2:213"><span class="id" title="binder">d2</span></a> (<a id="S:214" class="idref" href="#S:214"><span class="id" title="binder">S</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d1:212"><span class="id" title="variable">d1</span></a>) (<a id="T:215" class="idref" href="#T:215"><span class="id" title="binder">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d2:213"><span class="id" title="variable">d2</span></a>)<br/>
(<a id="u:216" class="idref" href="#u:216"><span class="id" title="binder">u</span></a> <a id="v:217" class="idref" href="#v:217"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#S:214"><span class="id" title="variable">S</span></a>) <a id="a:218" class="idref" href="#a:218"><span class="id" title="binder">a</span></a> <a id="b:219" class="idref" href="#b:219"><span class="id" title="binder">b</span></a> (<a id="U:220" class="idref" href="#U:220"><span class="id" title="binder">U</span></a> <a id="V:221" class="idref" href="#V:221"><span class="id" title="binder">V</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:215"><span class="id" title="variable">T</span></a>) <a id="A:222" class="idref" href="#A:222"><span class="id" title="binder">A</span></a> <a id="B:223" class="idref" href="#B:223"><span class="id" title="binder">B</span></a> :<br/>
(<a class="idref" href="Combi.Combi.std.html#a:218"><span class="id" title="variable">a</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.std.html#b:219"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> (<a class="idref" href="Combi.Combi.std.html#A:222"><span class="id" title="variable">A</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.std.html#B:223"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:216"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#U:220"><span class="id" title="variable">U</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#u:216"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:218"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:219"><span class="id" title="variable">b</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:217"><span class="id" title="variable">v</span></a>) (<a class="idref" href="Combi.Combi.std.html#U:220"><span class="id" title="variable">U</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#A:222"><span class="id" title="variable">A</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#B:223"><span class="id" title="variable">B</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#V:221"><span class="id" title="variable">V</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="Combi.Combi.std.html#eq_inv"><span class="id" title="definition">eq_inv</span></a> (<a class="idref" href="Combi.Combi.std.html#u:216"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#b:219"><span class="id" title="variable">b</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#a:218"><span class="id" title="variable">a</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:217"><span class="id" title="variable">v</span></a>) (<a class="idref" href="Combi.Combi.std.html#U:220"><span class="id" title="variable">U</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#B:223"><span class="id" title="variable">B</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#A:222"><span class="id" title="variable">A</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#V:221"><span class="id" title="variable">V</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_transp" class="idref" href="#std_transp"><span class="id" title="lemma">std_transp</span></a> <a id="d:229" class="idref" href="#d:229"><span class="id" title="binder">d</span></a> (<a id="T:230" class="idref" href="#T:230"><span class="id" title="binder">T</span></a> : <a class="idref" href="Combi.Basic.ordtype.html#InhOrder.Exports.inhOrderType"><span class="id" title="abbreviation">inhOrderType</span></a> <a class="idref" href="Combi.Combi.std.html#d:229"><span class="id" title="variable">d</span></a>)<br/>
(<a id="u:231" class="idref" href="#u:231"><span class="id" title="binder">u</span></a> <a id="v:232" class="idref" href="#v:232"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:230"><span class="id" title="variable">T</span></a>) <a id="a:233" class="idref" href="#a:233"><span class="id" title="binder">a</span></a> <a id="b:234" class="idref" href="#b:234"><span class="id" title="binder">b</span></a><br/>
(<a id="U:235" class="idref" href="#U:235"><span class="id" title="binder">U</span></a> <a id="V:236" class="idref" href="#V:236"><span class="id" title="binder">V</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a id="A:237" class="idref" href="#A:237"><span class="id" title="binder">A</span></a> <a id="B:238" class="idref" href="#B:238"><span class="id" title="binder">B</span></a> :<br/>
(<a class="idref" href="Combi.Combi.std.html#a:233"><span class="id" title="variable">a</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.std.html#b:234"><span class="id" title="variable">b</span></a>)%<span class="id" title="var">O</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:231"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#U:235"><span class="id" title="variable">U</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#u:231"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:233"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:234"><span class="id" title="variable">b</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:232"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#U:235"><span class="id" title="variable">U</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#A:237"><span class="id" title="variable">A</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#B:238"><span class="id" title="variable">B</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#V:236"><span class="id" title="variable">V</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#u:231"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#b:234"><span class="id" title="variable">b</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#a:233"><span class="id" title="variable">a</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:232"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#U:235"><span class="id" title="variable">U</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#B:238"><span class="id" title="variable">B</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#A:237"><span class="id" title="variable">A</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#V:236"><span class="id" title="variable">V</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="std_cutabc" class="idref" href="#std_cutabc"><span class="id" title="lemma">std_cutabc</span></a> <a id="d:240" class="idref" href="#d:240"><span class="id" title="binder">d</span></a> (<a id="T:241" class="idref" href="#T:241"><span class="id" title="binder">T</span></a> : <span class="id" title="abbreviation">orderType</span> <a class="idref" href="Combi.Combi.std.html#d:240"><span class="id" title="variable">d</span></a>) (<a id="u:242" class="idref" href="#u:242"><span class="id" title="binder">u</span></a> <a id="v:243" class="idref" href="#v:243"><span class="id" title="binder">v</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:241"><span class="id" title="variable">T</span></a>) <a id="a:244" class="idref" href="#a:244"><span class="id" title="binder">a</span></a> <a id="b:245" class="idref" href="#b:245"><span class="id" title="binder">b</span></a> <a id="c:246" class="idref" href="#c:246"><span class="id" title="binder">c</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">exists</span></a> <a id="U:247" class="idref" href="#U:247"><span class="id" title="binder">U</span></a> <a id="V:248" class="idref" href="#V:248"><span class="id" title="binder">V</span></a> <a id="A:249" class="idref" href="#A:249"><span class="id" title="binder">A</span></a> <a id="B:250" class="idref" href="#B:250"><span class="id" title="binder">B</span></a> <a id="C:251" class="idref" href="#C:251"><span class="id" title="binder">C</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#u:242"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#U:247"><span class="id" title="variable">U</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\</span></a><br/>
<a class="idref" href="Combi.Combi.std.html#std"><span class="id" title="definition">std</span></a> (<a class="idref" href="Combi.Combi.std.html#u:242"><span class="id" title="variable">u</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#a:244"><span class="id" title="variable">a</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#b:245"><span class="id" title="variable">b</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#c:246"><span class="id" title="variable">c</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#v:243"><span class="id" title="variable">v</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#U:247"><span class="id" title="variable">U</span></a> <span class="id" title="notation">++</span> <span class="id" title="notation">[::</span> <a class="idref" href="Combi.Combi.std.html#A:249"><span class="id" title="variable">A</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#B:250"><span class="id" title="variable">B</span></a><span class="id" title="notation">;</span> <a class="idref" href="Combi.Combi.std.html#C:251"><span class="id" title="variable">C</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">++</span> <a class="idref" href="Combi.Combi.std.html#V:248"><span class="id" title="variable">V</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab82"></a><h1 class="section">Inverse of a standard word</h1>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="perm_size_uniq" class="idref" href="#perm_size_uniq"><span class="id" title="lemma">perm_size_uniq</span></a> (<a id="T:252" class="idref" href="#T:252"><span class="id" title="binder">T</span></a> : <span class="id" title="abbreviation">eqType</span>) (<a id="s1:253" class="idref" href="#s1:253"><span class="id" title="binder">s1</span></a> <a id="s2:254" class="idref" href="#s2:254"><span class="id" title="binder">s2</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.std.html#T:252"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.std.html#s1:253"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="Combi.Combi.std.html#s1:253"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation"><=</span></a> <a class="idref" href="Combi.Combi.std.html#s2:254"><span class="id" title="variable">s2</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s2:254"><span class="id" title="variable">s2</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s1:253"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.std.html#s1:253"><span class="id" title="variable">s1</span></a> <a class="idref" href="Combi.Combi.std.html#s2:254"><span class="id" title="variable">s2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="InvSeq" class="idref" href="#InvSeq"><span class="id" title="section">InvSeq</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">n</span> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="linvseq" class="idref" href="#linvseq"><span class="id" title="definition">linvseq</span></a> <a id="s:255" class="idref" href="#s:255"><span class="id" title="binder">s</span></a> :=<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#8f75c029d3960a4603acd8dc22047b3c"><span class="id" title="notation">[</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#8f75c029d3960a4603acd8dc22047b3c"><span class="id" title="notation">fun</span></a> <a id="t:256" class="idref" href="#t:256"><span class="id" title="binder">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#8f75c029d3960a4603acd8dc22047b3c"><span class="id" title="notation">=></span></a> <span class="id" title="definition">all</span> (<span class="id" title="keyword">fun</span> <a id="i:257" class="idref" href="#i:257"><span class="id" title="binder">i</span></a> => <span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:255"><span class="id" title="variable">s</span></a>) <a class="idref" href="Combi.Combi.std.html#t:256"><span class="id" title="variable">t</span></a> (<span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:256"><span class="id" title="variable">t</span></a>) <a class="idref" href="Combi.Combi.std.html#s:255"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#i:257"><span class="id" title="variable">i</span></a>) <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.std.html#i:257"><span class="id" title="variable">i</span></a>)<br/>
(<span class="id" title="definition">iota</span> 0 (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:255"><span class="id" title="variable">s</span></a>))<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#8f75c029d3960a4603acd8dc22047b3c"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a id="invseq" class="idref" href="#invseq"><span class="id" title="definition">invseq</span></a> <a id="s:258" class="idref" href="#s:258"><span class="id" title="binder">s</span></a> <a id="t:259" class="idref" href="#t:259"><span class="id" title="binder">t</span></a> := <a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:258"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:259"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">&&</span></a> <a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#t:259"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Combi.std.html#s:258"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="linvseqP" class="idref" href="#linvseqP"><span class="id" title="lemma">linvseqP</span></a> <a id="s:260" class="idref" href="#s:260"><span class="id" title="binder">s</span></a> <a id="t:261" class="idref" href="#t:261"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a><br/>
(<span class="id" title="keyword">forall</span> <a id="i:262" class="idref" href="#i:262"><span class="id" title="binder">i</span></a>, <a class="idref" href="Combi.Combi.std.html#i:262"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:260"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:260"><span class="id" title="variable">s</span></a>) <a class="idref" href="Combi.Combi.std.html#t:261"><span class="id" title="variable">t</span></a> (<span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:261"><span class="id" title="variable">t</span></a>) <a class="idref" href="Combi.Combi.std.html#s:260"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#i:262"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#i:262"><span class="id" title="variable">i</span></a>)<br/>
(<a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:260"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:261"><span class="id" title="variable">t</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invseq_sym" class="idref" href="#invseq_sym"><span class="id" title="lemma">invseq_sym</span></a> <a id="s:263" class="idref" href="#s:263"><span class="id" title="binder">s</span></a> <a id="t:264" class="idref" href="#t:264"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:263"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:264"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#t:264"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Combi.std.html#s:263"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_all_leq" class="idref" href="#size_all_leq"><span class="id" title="lemma">size_all_leq</span></a> <a id="n:265" class="idref" href="#n:265"><span class="id" title="binder">n</span></a> <a id="t:266" class="idref" href="#t:266"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="i:267" class="idref" href="#i:267"><span class="id" title="binder">i</span></a>, <a class="idref" href="Combi.Combi.std.html#i:267"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.std.html#n:265"><span class="id" title="variable">n</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#i:267"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.std.html#t:266"><span class="id" title="variable">t</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#n:265"><span class="id" title="variable">n</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:266"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="linvseq_ltn_szt" class="idref" href="#linvseq_ltn_szt"><span class="id" title="lemma">linvseq_ltn_szt</span></a> <a id="s:268" class="idref" href="#s:268"><span class="id" title="binder">s</span></a> <a id="t:269" class="idref" href="#t:269"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:268"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:269"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="keyword">forall</span> <a id="i:270" class="idref" href="#i:270"><span class="id" title="binder">i</span></a>, <a class="idref" href="Combi.Combi.std.html#i:270"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:268"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:269"><span class="id" title="variable">t</span></a>) <a class="idref" href="Combi.Combi.std.html#s:268"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#i:270"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:269"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_leq_invseq" class="idref" href="#size_leq_invseq"><span class="id" title="lemma">size_leq_invseq</span></a> <a id="s:271" class="idref" href="#s:271"><span class="id" title="binder">s</span></a> <a id="t:272" class="idref" href="#t:272"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:271"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:272"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:271"><span class="id" title="variable">s</span></a> <span class="id" title="notation"><=</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:272"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_invseq" class="idref" href="#size_invseq"><span class="id" title="lemma">size_invseq</span></a> <a id="s:273" class="idref" href="#s:273"><span class="id" title="binder">s</span></a> <a id="t:274" class="idref" href="#t:274"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:273"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:274"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:273"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:274"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="linvseq_subset_iota" class="idref" href="#linvseq_subset_iota"><span class="id" title="lemma">linvseq_subset_iota</span></a> <a id="s:275" class="idref" href="#s:275"><span class="id" title="binder">s</span></a> <a id="t:276" class="idref" href="#t:276"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:275"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:276"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <span class="id" title="definition">iota</span> 0 (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:275"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation"><=</span></a> <a class="idref" href="Combi.Combi.std.html#t:276"><span class="id" title="variable">t</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invseq_is_std" class="idref" href="#invseq_is_std"><span class="id" title="lemma">invseq_is_std</span></a> <a id="s:277" class="idref" href="#s:277"><span class="id" title="binder">s</span></a> <a id="t:278" class="idref" href="#t:278"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:277"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:278"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:277"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="linvseq_sizeP" class="idref" href="#linvseq_sizeP"><span class="id" title="lemma">linvseq_sizeP</span></a> <a id="s:279" class="idref" href="#s:279"><span class="id" title="binder">s</span></a> <a id="t:280" class="idref" href="#t:280"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#linvseq"><span class="id" title="definition">linvseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:279"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:280"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:279"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:280"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:279"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:280"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invseq_nthE" class="idref" href="#invseq_nthE"><span class="id" title="lemma">invseq_nthE</span></a> <a id="s:281" class="idref" href="#s:281"><span class="id" title="binder">s</span></a> <a id="t:282" class="idref" href="#t:282"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:281"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t:282"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<span class="id" title="keyword">forall</span> <a id="i:283" class="idref" href="#i:283"><span class="id" title="binder">i</span></a> <a id="j:284" class="idref" href="#j:284"><span class="id" title="binder">j</span></a>, <a class="idref" href="Combi.Combi.std.html#i:283"><span class="id" title="variable">i</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:281"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#j:284"><span class="id" title="variable">j</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:282"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#i:283"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:281"><span class="id" title="variable">s</span></a>) <a class="idref" href="Combi.Combi.std.html#t:282"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Combi.std.html#j:284"><span class="id" title="variable">j</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'<->'_x"><span class="id" title="notation"><-></span></a> <span class="id" title="definition">nth</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#t:282"><span class="id" title="variable">t</span></a>) <a class="idref" href="Combi.Combi.std.html#s:281"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#i:283"><span class="id" title="variable">i</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#j:284"><span class="id" title="variable">j</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab83"></a><h1 class="section">inverse of the standardized of w</h1>
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a id="invstd" class="idref" href="#invstd"><span class="id" title="definition">invstd</span></a> <a id="s:285" class="idref" href="#s:285"><span class="id" title="binder">s</span></a> := <span class="id" title="definition">mkseq</span> (<span class="id" title="keyword">fun</span> <a id="i:286" class="idref" href="#i:286"><span class="id" title="binder">i</span></a> => <span class="id" title="definition">index</span> <a class="idref" href="Combi.Combi.std.html#i:286"><span class="id" title="variable">i</span></a> <a class="idref" href="Combi.Combi.std.html#s:285"><span class="id" title="variable">s</span></a>) (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#s:285"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invseq_invstd" class="idref" href="#invseq_invstd"><span class="id" title="lemma">invseq_invstd</span></a> <a id="s:287" class="idref" href="#s:287"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:287"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:287"><span class="id" title="variable">s</span></a> (<a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#s:287"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_invstd" class="idref" href="#size_invstd"><span class="id" title="lemma">size_invstd</span></a> <a id="p:288" class="idref" href="#p:288"><span class="id" title="binder">p</span></a> : <span class="id" title="definition">size</span> (<a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#p:288"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.std.html#p:288"><span class="id" title="variable">p</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invstd_is_std" class="idref" href="#invstd_is_std"><span class="id" title="lemma">invstd_is_std</span></a> <a id="p:289" class="idref" href="#p:289"><span class="id" title="binder">p</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#p:289"><span class="id" title="variable">p</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> (<a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#p:289"><span class="id" title="variable">p</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invseqE" class="idref" href="#invseqE"><span class="id" title="lemma">invseqE</span></a> <a id="s:290" class="idref" href="#s:290"><span class="id" title="binder">s</span></a> <a id="t1:291" class="idref" href="#t1:291"><span class="id" title="binder">t1</span></a> <a id="t2:292" class="idref" href="#t2:292"><span class="id" title="binder">t2</span></a> : <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:290"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t1:291"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#invseq"><span class="id" title="definition">invseq</span></a> <a class="idref" href="Combi.Combi.std.html#s:290"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.std.html#t2:292"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#t1:291"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#t2:292"><span class="id" title="variable">t2</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invstdK" class="idref" href="#invstdK"><span class="id" title="lemma">invstdK</span></a> <a id="s:293" class="idref" href="#s:293"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:293"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> (<a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#s:293"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#s:293"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="invstd_inj" class="idref" href="#invstd_inj"><span class="id" title="lemma">invstd_inj</span></a> <a id="s:294" class="idref" href="#s:294"><span class="id" title="binder">s</span></a> <a id="t:295" class="idref" href="#t:295"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#s:294"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#is_std"><span class="id" title="definition">is_std</span></a> <a class="idref" href="Combi.Combi.std.html#t:295"><span class="id" title="variable">t</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#s:294"><span class="id" title="variable">s</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="Combi.Combi.std.html#invstd"><span class="id" title="definition">invstd</span></a> <a class="idref" href="Combi.Combi.std.html#t:295"><span class="id" title="variable">t</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> <a class="idref" href="Combi.Combi.std.html#s:294"><span class="id" title="variable">s</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Combi.std.html#t:295"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#InvSeq"><span class="id" title="section">InvSeq</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="Examples" class="idref" href="#Examples"><span class="id" title="section">Examples</span></a>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a id="Examples.u" class="idref" href="#Examples.u"><span class="id" title="variable">u</span></a> := <span class="id" title="notation">[::</span> 4<span class="id" title="notation">;</span> 1<span class="id" title="notation">;</span> 2<span class="id" title="notation">;</span> 2<span class="id" title="notation">;</span> 5<span class="id" title="notation">;</span> 3<span class="id" title="notation">]</span>.<br/>
<span class="id" title="keyword">Let</span> <a id="Examples.v" class="idref" href="#Examples.v"><span class="id" title="variable">v</span></a> := <span class="id" title="notation">[::</span> 0<span class="id" title="notation">;</span> 4<span class="id" title="notation">;</span> 3<span class="id" title="notation">;</span> 3<span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">std</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">u</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">[::</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">4;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">0;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">1;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">2;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">5;</span></a> <a id="Unnamed_thm" class="idref" href="#Unnamed_thm"><span class="id" title="definition">3]</span></a>.<br/>
<span class="id" title="keyword">Goal</span> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">invstd</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">std</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">u</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">)</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">=</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">filter</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">gtn</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">size</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">u</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">))</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">invstd</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">std</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">(</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">u</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">++</span></a> <a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">v</span></a><a id="Unnamed_thm0" class="idref" href="#Unnamed_thm0"><span class="id" title="definition">)))</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.std.html#Examples"><span class="id" title="section">Examples</span></a>.<br/>
<br/>
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