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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>Combi.Combi.permuted: Listing the Permutations of a tuple or seq</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library Combi.Combi.permuted: Listing the Permutations of a tuple or seq</h1>
<div class="code">
</div>
<div class="doc">
</div>
<div class="code">
</div>
<div class="doc">
<a id="lab52"></a><h1 class="section">The list of the permuted tuple of a given tuple</h1>
<div class="paragraph"> </div>
The main goal is to show that, given a sequence <span class="inlinecode"><span class="id" title="var">s</span></span> over an <span class="inlinecode"><span class="id" title="var">eqType</span></span> there
are only finitely many sequences <span class="inlinecode"><span class="id" title="var">s'</span></span> which are a permutation of <span class="inlinecode"><span class="id" title="var">s</span></span> (that is
<span class="inlinecode"><span class="id" title="var">perm_eq</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">s'</span></span>) and to show that the number is a multinomial coefficient.
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">permuted_tuple</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> == a sequence of tuples containing (with duplicates) all
tuple <span class="inlinecode"><span class="id" title="var">t'</span></span> such that <span class="inlinecode"><span class="id" title="var">perm_eq</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode"><span class="id" title="var">t'</span></span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">permuted_seq</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> == a sequence of seqs containing (with duplicates) all seqs
<span class="inlinecode"><span class="id" title="var">s'</span></span> such that <span class="inlinecode"><span class="id" title="var">perm_eq</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">s'</span></span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">permuted</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> == sigma typle for tuple <span class="inlinecode"><span class="id" title="var">t'</span></span> such that <span class="inlinecode"><span class="id" title="var">perm_eq</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode"><span class="id" title="var">t'</span></span>. this
is canonically a <span class="inlinecode"><span class="id" title="var">fintype</span></span>, provided the type of the elements of
<span class="inlinecode"><span class="id" title="var">t</span></span> is a <span class="inlinecode"><span class="id" title="var">countType</span></span>.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode"><span class="id" title="var">permutedact</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> == the <span class="inlinecode"><span class="id" title="var">n</span></span> tuple <span class="inlinecode"><span class="id" title="var">t</span></span> permuted by the permutation <span class="inlinecode"><span class="id" title="var">s</span></span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">permuted_action</span></span> == the corresponding action of the symmetric group <span class="inlinecode">'<span class="id" title="var">S_n</span></span>
</li>
</ul>
<div class="paragraph"> </div>
The main result is the cardinality of the set of permuted of a tuple expressed
as a multinomial <span class="inlinecode"><span class="id" title="var">card_permuted_multinomial</span></span>.
</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>.<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> <span class="id" title="library">action</span> <span class="id" title="library">gproduct</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.SSRcomplements.sorted.html#"><span class="id" title="library">sorted</span></a> <a class="idref" href="Combi.Combi.partition.html#"><span class="id" title="library">partition</span></a> <a class="idref" href="Combi.Combi.composition.html#"><span class="id" title="library">composition</span></a> <a class="idref" href="Combi.Combi.multinomial.html#"><span class="id" title="library">multinomial</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Combi.SSRcomplements.permcomp.html#"><span class="id" title="library">permcomp</span></a> <a class="idref" href="Combi.SymGroup.cycles.html#"><span class="id" title="library">cycles</span></a>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab53"></a><h2 class="section">Enumeration of the permutation of a tuple</h2>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="Permuted" class="idref" href="#Permuted"><span class="id" title="section">Permuted</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a id="Permuted.T" class="idref" href="#Permuted.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">eqType</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="Permuted.SizeN" class="idref" href="#Permuted.SizeN"><span class="id" title="section">SizeN</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a id="Permuted.SizeN.n" class="idref" href="#Permuted.SizeN.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="permuted_tuple" class="idref" href="#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> (<a id="t:5" class="idref" href="#t:5"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>) :=<br/>
<span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="notation">[</span><span class="id" title="notation">tuple</span> <span class="id" title="definition">tnth</span> <a class="idref" href="Combi.Combi.permuted.html#t:5"><span class="id" title="variable">t</span></a> (<span class="id" title="definition">aperm</span> <a class="idref" href="Combi.Combi.permuted.html#i:8"><span class="id" title="variable">i</span></a> <a class="idref" href="Combi.Combi.permuted.html#p:6"><span class="id" title="variable">p</span></a>) <span class="id" title="notation">|</span> <a id="i:8" class="idref" href="#i:8"><span class="id" title="binder">i</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">|</span> <a id="p:6" class="idref" href="#p:6"><span class="id" title="binder"><span id="p:7" class="id"><span id="p:9" class="id"><span id="p:10" class="id">p</span></span></span></span></a> <a id="p:7" class="idref" href="#p:7"><span class="id" title="binder"><span id="p:10" class="id">:</span></span></a> <a id="p:7" class="idref" href="#p:7"><span class="id" title="binder"><span id="p:10" class="id">'</span></span></a><a id="p:7" class="idref" href="#p:7"><span class="id" title="binder"><span id="p:10" class="id">S_n</span></span></a> <span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_permuted_tuple" class="idref" href="#size_permuted_tuple"><span class="id" title="lemma">size_permuted_tuple</span></a> (<a id="t:11" class="idref" href="#t:11"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>) : <span class="id" title="definition">size</span> (<a class="idref" href="Combi.Combi.permuted.html#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> <a class="idref" href="Combi.Combi.permuted.html#t:11"><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="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="perm_eq_permuted_tuple" class="idref" href="#perm_eq_permuted_tuple"><span class="id" title="lemma">perm_eq_permuted_tuple</span></a> (<a id="s:12" class="idref" href="#s:12"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.permuted.html#Permuted.T"><span class="id" title="variable">T</span></a>) (<a id="H:13" class="idref" href="#H:13"><span class="id" title="binder">H</span></a> : <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.permuted.html#s:12"><span class="id" title="variable">s</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a>) :<br/>
<span class="id" title="keyword">forall</span> <a id="s1:14" class="idref" href="#s1:14"><span class="id" title="binder">s1</span></a>, <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#s:12"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.permuted.html#s1:14"><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><br/>
<a class="idref" href="Combi.Combi.permuted.html#s1:14"><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#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> <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="projection">tval</span> <a class="idref" href="Combi.Combi.permuted.html#t:15"><span class="id" title="variable">t</span></a> <span class="id" title="notation">|</span> <a id="t:15" class="idref" href="#t:15"><span class="id" title="binder"><span id="t:16" class="id">t</span></span></a> <span class="id" title="notation"><-</span> <a class="idref" href="Combi.Combi.permuted.html#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> (<span class="id" title="constructor">Tuple</span> <a class="idref" href="Combi.Combi.permuted.html#H:13"><span class="id" title="variable">H</span></a>)<span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="mem_enum_permuted" class="idref" href="#mem_enum_permuted"><span class="id" title="lemma">mem_enum_permuted</span></a> (<a id="s:18" class="idref" href="#s:18"><span class="id" title="binder">s</span></a> <a id="t:19" class="idref" href="#t:19"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>) :<br/>
<span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#s:18"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.permuted.html#t:19"><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.permuted.html#t:19"><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#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.permuted.html#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> <a class="idref" href="Combi.Combi.permuted.html#s:18"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="all_permuted" class="idref" href="#all_permuted"><span class="id" title="lemma">all_permuted</span></a> (<a id="s:20" class="idref" href="#s:20"><span class="id" title="binder">s</span></a> : <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>) :<br/>
<span class="id" title="definition">all</span> (<span class="id" title="keyword">fun</span> <a id="x:21" class="idref" href="#x:21"><span class="id" title="binder">x</span></a> : <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span> => <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#s:20"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.permuted.html#x:21"><span class="id" title="variable">x</span></a>) (<a class="idref" href="Combi.Combi.permuted.html#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> <a class="idref" href="Combi.Combi.permuted.html#s:20"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.permuted.html#Permuted.SizeN"><span class="id" title="section">SizeN</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="permuted_seq" class="idref" href="#permuted_seq"><span class="id" title="definition">permuted_seq</span></a> <a id="s:22" class="idref" href="#s:22"><span class="id" title="binder">s</span></a> :=<br/>
<span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="projection">tval</span> <a class="idref" href="Combi.Combi.permuted.html#t:23"><span class="id" title="variable">t</span></a> <span class="id" title="notation">|</span> <a id="t:23" class="idref" href="#t:23"><span class="id" title="binder"><span id="t:24" class="id">t</span></span></a> <span class="id" title="notation"><-</span> <a class="idref" href="Combi.Combi.permuted.html#permuted_tuple"><span class="id" title="definition">permuted_tuple</span></a> (<span class="id" title="constructor">Tuple</span> (<span class="id" title="lemma">eq_refl</span> (<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.permuted.html#s:22"><span class="id" title="variable">s</span></a>)))<span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_permuted_seq" class="idref" href="#size_permuted_seq"><span class="id" title="lemma">size_permuted_seq</span></a> <a id="s:25" class="idref" href="#s:25"><span class="id" title="binder">s</span></a> : <span class="id" title="definition">size</span> (<a class="idref" href="Combi.Combi.permuted.html#permuted_seq"><span class="id" title="definition">permuted_seq</span></a> <a class="idref" href="Combi.Combi.permuted.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> <span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.permuted.html#s:25"><span class="id" title="variable">s</span></a><span class="id" title="notation">)`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="eq_seqE" class="idref" href="#eq_seqE"><span class="id" title="lemma">eq_seqE</span></a> <a id="s:26" class="idref" href="#s:26"><span class="id" title="binder">s</span></a> <a id="s1:27" class="idref" href="#s1:27"><span class="id" title="binder">s1</span></a> : <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#s:26"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.permuted.html#s1:27"><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="Combi.Combi.permuted.html#s1:27"><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#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.permuted.html#permuted_seq"><span class="id" title="definition">permuted_seq</span></a> <a class="idref" href="Combi.Combi.permuted.html#s:26"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.permuted.html#Permuted"><span class="id" title="section">Permuted</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab54"></a><h2 class="section">Permutation of a tuple as a fintype</h2>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="FinType" class="idref" href="#FinType"><span class="id" title="section">FinType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a id="FinType.T" class="idref" href="#FinType.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">countType</span>.<br/>
<span class="id" title="keyword">Variable</span> <a id="FinType.n" class="idref" href="#FinType.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/>
<span class="id" title="keyword">Variable</span> <a id="FinType.w" class="idref" href="#FinType.w"><span class="id" title="variable">w</span></a> : <a class="idref" href="Combi.Combi.permuted.html#FinType.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>.<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a id="permuted" class="idref" href="#permuted"><span class="id" title="record">permuted</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#predArgType"><span class="id" title="definition">predArgType</span></a> :=<br/>
<span class="id" title="var">Permuted</span> { <a id="tpval" class="idref" href="#tpval"><span class="id" title="projection">tpval</span></a> :> <a class="idref" href="Combi.Combi.permuted.html#FinType.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>; <span class="id" title="var">_</span> : <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#FinType.w"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.permuted.html#tpval:35"><span class="id" title="method">tpval</span></a> }.<br/>
<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="permutedP" class="idref" href="#permutedP"><span class="id" title="lemma">permutedP</span></a> (<a id="p:36" class="idref" href="#p:36"><span class="id" title="binder">p</span></a> : <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a>) : <span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#FinType.w"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.permuted.html#p:36"><span class="id" title="variable">p</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.permuted.html#FinType"><span class="id" title="section">FinType</span></a>.<br/>
<br/>
#[<span class="id" title="var">export</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">permutedP</span> : <span class="id" title="var">core</span>.<br/>
<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="module">GroupScope</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab55"></a><h1 class="section">Action of <span class="inlinecode">'<span class="id" title="var">S_n</span></span> on permuted for and <span class="inlinecode"><span class="id" title="var">n</span>.-<span class="id" title="var">tuple</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span>.</h1>
<div class="paragraph"> </div>
There is no use defining an action on general tuple because most of the lemmas
on actions assume that the type acted upon is a <span class="inlinecode"><span class="id" title="var">finType</span></span>. We could require
moreover that the underlying type is a <span class="inlinecode"><span class="id" title="var">fintype</span></span> so that the set of tuple is a
<span class="inlinecode"><span class="id" title="var">fintype</span></span> too, but the use we have in mind is <span class="inlinecode"><span class="id" title="var">T</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">nat</span></span> allowing to deal with
the action on monomials. Instead of that, we decide to act only on the sigma
type <span class="inlinecode"><span class="id" title="var">permuted</span></span>.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Section</span> <a id="ActOnTuple" class="idref" href="#ActOnTuple"><span class="id" title="section">ActOnTuple</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a id="ActOnTuple.T" class="idref" href="#ActOnTuple.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">countType</span>) (<a id="ActOnTuple.n" class="idref" href="#ActOnTuple.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>) (<a id="ActOnTuple.w" class="idref" href="#ActOnTuple.w"><span class="id" title="variable">w</span></a> : <a class="idref" href="Combi.Combi.permuted.html#n:40"><span class="id" title="variable">n</span></a><span class="id" title="notation">.-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">T</span>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> (<span class="id" title="var">t</span> : <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a>).<br/>
<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="wp" class="idref" href="#wp"><span class="id" title="abbreviation">wp</span></a> := (<a class="idref" href="Combi.Combi.permuted.html#Permuted"><span class="id" title="constructor">Permuted</span></a> (<span class="id" title="lemma">perm_refl</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="permutedact_subproof" class="idref" href="#permutedact_subproof"><span class="id" title="lemma">permutedact_subproof</span></a> <a id="t:43" class="idref" href="#t:43"><span class="id" title="binder">t</span></a> (<a id="s:44" class="idref" href="#s:44"><span class="id" title="binder">s</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span>) :<br/>
<span class="id" title="definition">perm_eq</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">[</span><span class="id" title="notation">tuple</span> <span class="id" title="definition">tnth</span> <a class="idref" href="Combi.Combi.permuted.html#t:43"><span class="id" title="variable">t</span></a> (<a class="idref" href="Combi.Combi.permuted.html#s:44"><span class="id" title="variable">s</span></a><span class="id" title="notation">^-1</span> <span class="id" title="notation">i</span>) <span class="id" title="notation">|</span> <a id="i:45" class="idref" href="#i:45"><span class="id" title="binder">i</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">]</span>.<br/>
<span class="id" title="keyword">Definition</span> <a id="permutedact" class="idref" href="#permutedact"><span class="id" title="definition">permutedact</span></a> <a id="t:46" class="idref" href="#t:46"><span class="id" title="binder">t</span></a> <a id="s:47" class="idref" href="#s:47"><span class="id" title="binder">s</span></a> := <a class="idref" href="Combi.Combi.permuted.html#Permuted"><span class="id" title="constructor">Permuted</span></a> (<a class="idref" href="Combi.Combi.permuted.html#permutedact_subproof"><span class="id" title="lemma">permutedact_subproof</span></a> <a class="idref" href="Combi.Combi.permuted.html#t:46"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Combi.permuted.html#s:47"><span class="id" title="variable">s</span></a>).<br/>
<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> "t # s" := (<a class="idref" href="Combi.Combi.permuted.html#permutedact"><span class="id" title="definition">permutedact</span></a> <span class="id" title="var">t</span> <span class="id" title="var">s</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">format</span> "t # s").<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="permutedact_is_action" class="idref" href="#permutedact_is_action"><span class="id" title="lemma">permutedact_is_action</span></a> : <span class="id" title="definition">is_action</span> <span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <span class="id" title="notation">'</span><span class="id" title="notation">S_n</span><span class="id" title="notation">]</span> <a class="idref" href="Combi.Combi.permuted.html#permutedact"><span class="id" title="definition">permutedact</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <a id="permuted_action" class="idref" href="#permuted_action"><span class="id" title="definition">permuted_action</span></a> := <span class="id" title="constructor">Action</span> <a class="idref" href="Combi.Combi.permuted.html#permutedact_is_action"><span class="id" title="lemma">permutedact_is_action</span></a>.<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="pact" class="idref" href="#pact"><span class="id" title="abbreviation">pact</span></a> := <a class="idref" href="Combi.Combi.permuted.html#permuted_action"><span class="id" title="definition">permuted_action</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="permuted_action_trans" class="idref" href="#permuted_action_trans"><span class="id" title="lemma">permuted_action_trans</span></a> :<br/>
<span class="id" title="notation">[</span><span class="id" title="notation">transitive</span> <span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <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">,</span> <span class="id" title="notation">on</span> <span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.permuted.html#pact"><span class="id" title="abbreviation">pact</span></a><span class="id" title="notation">]</span>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab56"></a><h2 class="section">The stabilizer of a tuple under permutation</h2>
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="stab_tuple_prod" class="idref" href="#stab_tuple_prod"><span class="id" title="lemma">stab_tuple_prod</span></a> :<br/>
<span class="id" title="notation">'</span><span class="id" title="notation">C</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.permuted.html#wp"><span class="id" title="abbreviation">wp</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.permuted.html#pact"><span class="id" title="abbreviation">pact</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><br/>
(<span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:49" class="idref" href="#x:49"><span class="id" title="binder"><span id="x:50" class="id"><span id="x:51" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="definition">Sym_group</span> <span class="id" title="notation">[</span><span class="id" title="notation">set</span> <a id="i:52" class="idref" href="#i:52"><span class="id" title="binder">i</span></a> <span class="id" title="notation">|</span> <span class="id" title="definition">tnth</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.permuted.html#i:52"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> <span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:49"><span class="id" title="variable">x</span></a><span class="id" title="notation">]</span>)%<span class="id" title="var">G</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="stab_tuple_dprod" class="idref" href="#stab_tuple_dprod"><span class="id" title="lemma">stab_tuple_dprod</span></a> :<br/>
<span class="id" title="notation">'</span><span class="id" title="notation">C</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.permuted.html#wp"><span class="id" title="abbreviation">wp</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.permuted.html#pact"><span class="id" title="abbreviation">pact</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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><span class="id" title="abbreviation">dprod</span><span class="id" title="notation">/</span>1<span class="id" title="notation">]</span><span class="id" title="notation">_</span><span class="id" title="notation">(</span><a id="x:57" class="idref" href="#x:57"><span class="id" title="binder"><span id="x:58" class="id"><span id="x:59" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="definition">Sym</span> <span class="id" title="notation">[</span><span class="id" title="notation">set</span> <a id="i:60" class="idref" href="#i:60"><span class="id" title="binder">i</span></a> <span class="id" title="notation">|</span> <span class="id" title="definition">tnth</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.permuted.html#i:60"><span class="id" title="variable">i</span></a> <span class="id" title="notation">==</span> <span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:57"><span class="id" title="variable">x</span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Close</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="card_stab_tuple" class="idref" href="#card_stab_tuple"><span class="id" title="lemma">card_stab_tuple</span></a> :<br/>
<span class="id" title="notation">#|</span>(<span class="id" title="notation">'</span><span class="id" title="notation">C</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.permuted.html#wp"><span class="id" title="abbreviation">wp</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.permuted.html#pact"><span class="id" title="abbreviation">pact</span></a><span class="id" title="notation">]</span>)%<span class="id" title="var">G</span><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> <span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:62" class="idref" href="#x:62"><span class="id" title="binder"><span id="x:63" class="id"><span id="x:64" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:62"><span class="id" title="variable">x</span></a>) <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="card_permuted_prod" class="idref" href="#card_permuted_prod"><span class="id" title="lemma">card_permuted_prod</span></a> :<br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><span class="id" title="notation">|</span> <span class="id" title="notation">*</span> <span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:65" class="idref" href="#x:65"><span class="id" title="binder"><span id="x:66" class="id"><span id="x:67" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:65"><span class="id" title="variable">x</span></a>) <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</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.permuted.html#ActOnTuple.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="dvdn_card_permuted" class="idref" href="#dvdn_card_permuted"><span class="id" title="lemma">dvdn_card_permuted</span></a> :<br/>
<span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:68" class="idref" href="#x:68"><span class="id" title="binder"><span id="x:69" class="id"><span id="x:70" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:68"><span class="id" title="variable">x</span></a>) <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)`!</span> <span class="id" title="notation">%|</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="card_permuted_seq_sub" class="idref" href="#card_permuted_seq_sub"><span class="id" title="lemma">card_permuted_seq_sub</span></a> :<br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><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.permuted.html#ActOnTuple.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span> <span class="id" title="notation">%/</span> <span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:71" class="idref" href="#x:71"><span class="id" title="binder"><span id="x:72" class="id"><span id="x:73" class="id">x</span></span></span></a> <span class="id" title="notation">:</span> <span class="id" title="record">seq_sub</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.permuted.html#x:71"><span class="id" title="variable">x</span></a>) <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="card_permuted" class="idref" href="#card_permuted"><span class="id" title="lemma">card_permuted</span></a> :<br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><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.permuted.html#ActOnTuple.n"><span class="id" title="variable">n</span></a><span class="id" title="notation">`!</span> <span class="id" title="notation">%/</span> <span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="x:74" class="idref" href="#x:74"><span class="id" title="binder"><span id="x:75" class="id"><span id="x:76" class="id">x</span></span></span></a> <span class="id" title="notation"><-</span> <span class="id" title="definition">undup</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> <a class="idref" href="Combi.Combi.permuted.html#x:74"><span class="id" title="variable">x</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">)`!</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="size_count_mem_undup" class="idref" href="#size_count_mem_undup"><span class="id" title="lemma">size_count_mem_undup</span></a> (<a id="s:77" class="idref" href="#s:77"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.permuted.html#s:77"><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="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="j:78" class="idref" href="#j:78"><span class="id" title="binder"><span id="j:79" class="id"><span id="j:80" class="id">j</span></span></span></a> <span class="id" title="notation"><-</span> <span class="id" title="definition">undup</span> <a class="idref" href="Combi.Combi.permuted.html#s:77"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <span class="id" title="abbreviation">count_mem</span> <a class="idref" href="Combi.Combi.permuted.html#j:78"><span class="id" title="variable">j</span></a> <a class="idref" href="Combi.Combi.permuted.html#s:77"><span class="id" title="variable">s</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a id="card_permuted_multinomial" class="idref" href="#card_permuted_multinomial"><span class="id" title="lemma">card_permuted_multinomial</span></a> :<br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><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.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">'</span></a><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">C</span></a><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">[</span></a><span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="abbreviation">count_mem</span> <a class="idref" href="Combi.Combi.permuted.html#x:84"><span class="id" title="variable">x</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">|</span> <a id="x:84" class="idref" href="#x:84"><span class="id" title="binder"><span id="x:85" class="id">x</span></span></a> <span class="id" title="notation"><-</span> <span class="id" title="definition">undup</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">Corollary</span> <a id="card_permuted_multinomial_subset" class="idref" href="#card_permuted_multinomial_subset"><span class="id" title="lemma">card_permuted_multinomial_subset</span></a> (<a id="s:86" class="idref" href="#s:86"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.T"><span class="id" title="variable">T</span></a>) :<br/>
<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.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</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.permuted.html#s:86"><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="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.permuted.html#s:86"><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><br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span><span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.permuted.html#permuted"><span class="id" title="record">permuted</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span><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.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">'</span></a><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">C</span></a><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">[</span></a><span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="abbreviation">count_mem</span> <a class="idref" href="Combi.Combi.permuted.html#x:87"><span class="id" title="variable">x</span></a> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">|</span> <a id="x:87" class="idref" href="#x:87"><span class="id" title="binder"><span id="x:88" class="id">x</span></span></a> <span class="id" title="notation"><-</span> <a class="idref" href="Combi.Combi.permuted.html#s:86"><span class="id" title="variable">s</span></a><span class="id" title="notation">]</span><a class="idref" href="Combi.Combi.multinomial.html#1fe8f419670f016387192cea55d3d07e"><span class="id" title="notation">]</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.permuted.html#ActOnTuple"><span class="id" title="section">ActOnTuple</span></a>.<br/>
<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Combi.SSRcomplements.sorted.html#"><span class="id" title="library">sorted</span></a>.<br/>
<span class="id" title="keyword">Import</span> <a class="idref" href="Combi.SSRcomplements.sorted.html#LeqGeqOrder"><span class="id" title="module">LeqGeqOrder</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="card_preim_part_of_compn" class="idref" href="#card_preim_part_of_compn"><span class="id" title="lemma">card_preim_part_of_compn</span></a> <a id="n:89" class="idref" href="#n:89"><span class="id" title="binder">n</span></a> (<a id="sh:90" class="idref" href="#sh:90"><span class="id" title="binder">sh</span></a> : <a class="idref" href="Combi.Combi.partition.html#:::'''P_'_x"><span class="id" title="notation">'</span></a><a class="idref" href="Combi.Combi.partition.html#:::'''P_'_x"><span class="id" title="notation">P_n</span></a>) :<br/>
<span class="id" title="notation">#|</span><span class="id" title="notation">[</span><span class="id" title="notation">set</span> <a id="c:91" class="idref" href="#c:91"><span class="id" title="binder">c</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.composition.html#partn_of_compn"><span class="id" title="definition">partn_of_compn</span></a> <a class="idref" href="Combi.Combi.permuted.html#c:91"><span class="id" title="variable">c</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.permuted.html#sh:90"><span class="id" title="variable">sh</span></a><span class="id" title="notation">]</span><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><br/>
<span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.permuted.html#sh:90"><span class="id" title="variable">sh</span></a><span class="id" title="notation">)`!</span> <span class="id" title="notation">%/</span> <span class="id" title="notation">\</span><span class="id" title="notation">prod_</span><span class="id" title="notation">(</span><a id="i:92" class="idref" href="#i:92"><span class="id" title="binder"><span id="i:93" class="id"><span id="i:94" class="id">i</span></span></span></a> <span class="id" title="notation"><-</span> <span class="id" title="definition">iota</span> 1 <a class="idref" href="Combi.Combi.permuted.html#n:89"><span class="id" title="variable">n</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><span class="id" title="abbreviation">count_mem</span> <a class="idref" href="Combi.Combi.permuted.html#i:92"><span class="id" title="variable">i</span></a> <a class="idref" href="Combi.Combi.permuted.html#sh:90"><span class="id" title="variable">sh</span></a><span class="id" title="notation">)`!</span>.<br/>
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