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<title>Combi.Combi.subseq: Subsequence of a sequence as a fintype</title>
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<h1 class="libtitle">Library Combi.Combi.subseq: Subsequence of a sequence as a fintype</h1>
<div class="code">
</div>
<div class="doc">
</div>
<div class="code">
</div>
<div class="doc">
<a id="lab90"></a><h1 class="section">Subsequence of a sequence as a <span class="inlinecode"><span class="id" title="var">fintype</span></span></h1>
We define a sigma-type <span class="inlinecode"><span class="id" title="var">subseqs</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> for subsequence of a given sequence <span class="inlinecode"><span class="id" title="var">w</span></span>
We show that <span class="inlinecode"><span class="id" title="var">subseqs</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> is canonically a <span class="inlinecode"><span class="id" title="var">finType</span></span>. We define the three
following constructor
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">Subseqs</span></span> <span class="inlinecode"><span class="id" title="var">Pf</span></span> == construct a <span class="inlinecode"><span class="id" title="var">subseqs</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> from a proof <span class="inlinecode"><span class="id" title="var">subseq</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span>.
</li>
<li> <span class="inlinecode"><span class="id" title="var">sub_nil</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> == the empty sequence <span class="inlinecode">[::]</span> as a <span class="inlinecode"><span class="id" title="var">subseqs</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span>.
</li>
<li> <span class="inlinecode"><span class="id" title="var">sub_full</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> == <span class="inlinecode"><span class="id" title="var">w</span></span> as as a <span class="inlinecode"><span class="id" title="var">subseqs</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</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">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>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
</div>
<div class="doc">
TODO: these probably should be contributed to path.v
<div class="paragraph"> </div>
<a id="lab91"></a><h2 class="section">A few lemmas about <span class="inlinecode"><span class="id" title="var">subseq</span></span> and <span class="inlinecode"><span class="id" title="var">rcons</span></span></h2>
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a id="RCons" class="idref" href="#RCons"><span class="id" title="section">RCons</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> (<a id="RCons.T" class="idref" href="#RCons.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">eqType</span>).<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">w</span> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#RCons.T"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">l</span> : <a class="idref" href="Combi.Combi.subseq.html#RCons.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="subseq_rcons_eq" class="idref" href="#subseq_rcons_eq"><span class="id" title="lemma">subseq_rcons_eq</span></a> <a id="s:3" class="idref" href="#s:3"><span class="id" title="binder">s</span></a> <a id="w:4" class="idref" href="#w:4"><span class="id" title="binder">w</span></a> <a id="l:5" class="idref" href="#l:5"><span class="id" title="binder">l</span></a> : <span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#s:3"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#w:4"><span class="id" title="variable">w</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">subseq</span> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#s:3"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#l:5"><span class="id" title="variable">l</span></a>) (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#w:4"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.subseq.html#l:5"><span class="id" title="variable">l</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="subseq_rcons_neq" class="idref" href="#subseq_rcons_neq"><span class="id" title="lemma">subseq_rcons_neq</span></a> <a id="s:6" class="idref" href="#s:6"><span class="id" title="binder">s</span></a> <a id="si:7" class="idref" href="#si:7"><span class="id" title="binder">si</span></a> <a id="w:8" class="idref" href="#w:8"><span class="id" title="binder">w</span></a> <a id="wn:9" class="idref" href="#wn:9"><span class="id" title="binder">wn</span></a> :<br/>
<a class="idref" href="Combi.Combi.subseq.html#si:7"><span class="id" title="variable">si</span></a> <span class="id" title="notation">!=</span> <a class="idref" href="Combi.Combi.subseq.html#wn:9"><span class="id" title="variable">wn</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">subseq</span> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#s:6"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#si:7"><span class="id" title="variable">si</span></a>) (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#w:8"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.subseq.html#wn:9"><span class="id" title="variable">wn</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">subseq</span> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#s:6"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#si:7"><span class="id" title="variable">si</span></a>) <a class="idref" href="Combi.Combi.subseq.html#w:8"><span class="id" title="variable">w</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.subseq.html#RCons"><span class="id" title="section">RCons</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="SubseqSorted" class="idref" href="#SubseqSorted"><span class="id" title="section">SubseqSorted</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> (<a id="SubseqSorted.T" class="idref" href="#SubseqSorted.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">eqType</span>) (<a id="SubseqSorted.leT" class="idref" href="#SubseqSorted.leT"><span class="id" title="variable">leT</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="Combi.Combi.subseq.html#T:11"><span class="id" title="variable">T</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="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="sorted_subseqP" class="idref" href="#sorted_subseqP"><span class="id" title="lemma">sorted_subseqP</span></a> <a id="s1:14" class="idref" href="#s1:14"><span class="id" title="binder">s1</span></a> <a id="s2:15" class="idref" href="#s2:15"><span class="id" title="binder">s2</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted.leT"><span class="id" title="variable">leT</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#irreflexive"><span class="id" title="definition">irreflexive</span></a> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted.leT"><span class="id" title="variable">leT</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">sorted</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="Combi.Combi.subseq.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> <span class="id" title="definition">sorted</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:15"><span class="id" title="variable">s2</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.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.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.subseq.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#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation"><=</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:15"><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> (<span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#s1:14"><span class="id" title="variable">s1</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:15"><span class="id" title="variable">s2</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSorted"><span class="id" title="section">SubseqSorted</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="SubseqSortedIn" class="idref" href="#SubseqSortedIn"><span class="id" title="section">SubseqSortedIn</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> (<a id="SubseqSortedIn.T" class="idref" href="#SubseqSortedIn.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">eqType</span>) (<a id="SubseqSortedIn.leT" class="idref" href="#SubseqSortedIn.leT"><span class="id" title="variable">leT</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="Combi.Combi.subseq.html#T:21"><span class="id" title="variable">T</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="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="sorted_subseq_inP" class="idref" href="#sorted_subseq_inP"><span class="id" title="lemma">sorted_subseq_inP</span></a> <a id="s1:24" class="idref" href="#s1:24"><span class="id" title="binder">s1</span></a> <a id="s2:25" class="idref" href="#s2:25"><span class="id" title="binder">s2</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#41940d7b27ea6f4cfc62b9919ed9d131"><span class="id" title="notation">{</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#41940d7b27ea6f4cfc62b9919ed9d131"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:25"><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#41940d7b27ea6f4cfc62b9919ed9d131"><span class="id" title="notation">&</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#41940d7b27ea6f4cfc62b9919ed9d131"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn.leT"><span class="id" title="variable">leT</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#41940d7b27ea6f4cfc62b9919ed9d131"><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.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:25"><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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#irreflexive"><span class="id" title="definition">irreflexive</span></a> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn.leT"><span class="id" title="variable">leT</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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/>
<span class="id" title="definition">sorted</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="Combi.Combi.subseq.html#s1:24"><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">sorted</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn.leT"><span class="id" title="variable">leT</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:25"><span class="id" title="variable">s2</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.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.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.subseq.html#s1:24"><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.subseq.html#s2:25"><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> (<span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#s1:24"><span class="id" title="variable">s1</span></a> <a class="idref" href="Combi.Combi.subseq.html#s2:25"><span class="id" title="variable">s2</span></a>).<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.subseq.html#SubseqSortedIn"><span class="id" title="section">SubseqSortedIn</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a id="lab92"></a><h1 class="section">Subsequence of a sequence as a <span class="inlinecode"><span class="id" title="var">fintype</span></span></h1>
<div class="paragraph"> </div>
We define a dependent type <span class="inlinecode"><span class="id" title="var">SubSeq</span></span> <span class="inlinecode"><span class="id" title="var">w</span></span> and provide it with a Canonical
<span class="inlinecode"><span class="id" title="var">finType</span></span> structure
</div>
<div class="code">
<br/>
<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">Variables</span> (<a id="FinType.T" class="idref" href="#FinType.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">choiceType</span>) (<a id="FinType.w" class="idref" href="#FinType.w"><span class="id" title="variable">w</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#T:27"><span class="id" title="variable">T</span></a>).<br/>
<br/>
<span class="id" title="keyword">Structure</span> <a id="subseqs" class="idref" href="#subseqs"><span class="id" title="record">subseqs</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">Subseqs</span> {<a id="subseqsval" class="idref" href="#subseqsval"><span class="id" title="projection">subseqsval</span></a> :> <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.T"><span class="id" title="variable">T</span></a>; <span class="id" title="var">_</span> : <span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#subseqsval:31"><span class="id" title="method">subseqsval</span></a> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>}.<br/>
<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> (<span class="id" title="var">s</span> : <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="to_mask_spec" class="idref" href="#to_mask_spec"><span class="id" title="lemma">to_mask_spec</span></a> <a id="s:32" class="idref" href="#s:32"><span class="id" title="binder">s</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="m:33" class="idref" href="#m:33"><span class="id" title="binder">m</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="definition">size</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.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="notation">bool</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> <span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:33"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.subseq.html#s:32"><span class="id" title="variable">s</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/>
<span class="id" title="keyword">Definition</span> <a id="to_mask" class="idref" href="#to_mask"><span class="id" title="definition">to_mask</span></a> <a id="s:34" class="idref" href="#s:34"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">m</span> <span class="id" title="var">_</span> := <a class="idref" href="Combi.Combi.subseq.html#to_mask_spec"><span class="id" title="lemma">to_mask_spec</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:34"><span class="id" title="variable">s</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">m</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="to_maskK" class="idref" href="#to_maskK"><span class="id" title="lemma">to_maskK</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="Combi.Combi.subseq.html#to_mask"><span class="id" title="definition">to_mask</span></a> (<span class="id" title="keyword">fun</span> <a id="m:35" class="idref" href="#m:35"><span class="id" title="binder">m</span></a> => <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">mask_subseq</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:35"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>)).<br/>
<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="subseqsP" class="idref" href="#subseqsP"><span class="id" title="lemma">subseqsP</span></a> <a id="s:36" class="idref" href="#s:36"><span class="id" title="binder">s</span></a> : <span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#s:36"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="sub_nil" class="idref" href="#sub_nil"><span class="id" title="definition">sub_nil</span></a> : <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> := <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">sub0seq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a id="sub_full" class="idref" href="#sub_full"><span class="id" title="definition">sub_full</span></a> : <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> := <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">subseq_refl</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="mask1E" class="idref" href="#mask1E"><span class="id" title="lemma">mask1E</span></a> <a id="x0:37" class="idref" href="#x0:37"><span class="id" title="binder">x0</span></a> <a id="i:38" class="idref" href="#i:38"><span class="id" title="binder">i</span></a> :<br/>
<a class="idref" href="Combi.Combi.subseq.html#i:38"><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.subseq.html#FinType.w"><span class="id" title="variable">w</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">mask</span> <span class="id" title="notation">[</span><span class="id" title="notation">tuple</span> <span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#x:39"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Combi.subseq.html#i:38"><span class="id" title="variable">i</span></a> <span class="id" title="notation">|</span> <a id="x:39" class="idref" href="#x:39"><span class="id" title="binder">x</span></a> <span class="id" title="notation"><</span> <span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a><span class="id" title="notation">]</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</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">nth</span> <a class="idref" href="Combi.Combi.subseq.html#x0:37"><span class="id" title="variable">x0</span></a> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:38"><span class="id" title="variable">i</span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="mask_injP" class="idref" href="#mask_injP"><span class="id" title="lemma">mask_injP</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.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<span class="id" title="keyword">fun</span> <a id="m:46" class="idref" href="#m:46"><span class="id" title="binder">m</span></a> : <span class="id" title="notation">(</span><span class="id" title="definition">size</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.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="notation">bool</span> => <span class="id" title="definition">mask</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:46"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>))<br/>
(<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Subseqs_maskK" class="idref" href="#Subseqs_maskK"><span class="id" title="lemma">Subseqs_maskK</span></a> :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</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.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<span class="id" title="keyword">fun</span> <a id="m:50" class="idref" href="#m:50"><span class="id" title="binder">m</span></a> => <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">mask_subseq</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:50"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>)) <a class="idref" href="Combi.Combi.subseq.html#to_mask"><span class="id" title="definition">to_mask</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="enum_subseqsE" class="idref" href="#enum_subseqsE"><span class="id" title="lemma">enum_subseqsE</span></a> :<br/>
<span class="id" title="definition">perm_eq</span><br/>
(<span class="id" title="abbreviation">enum</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.subseq.html#subseqs"><span class="id" title="record">subseqs</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>)<br/>
(<span class="id" title="definition">undup</span> <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">mask_subseq</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:51"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>) <span class="id" title="notation">|</span> <a id="m:51" class="idref" href="#m:51"><span class="id" title="binder"><span id="m:52" class="id"><span id="m:53" class="id"><span id="m:54" class="id">m</span></span></span></span></a> <a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">:</span></span></a> <a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">(</span></span></a><a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">size</span></span></a> <a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">w</span></span></a><a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">).-</span></span></a><a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">tuple</span></span></a> <a id="m:52" class="idref" href="#m:52"><span class="id" title="binder"><span id="m:54" class="id">bool</span></span></a><span class="id" title="notation">]</span>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="val_enum_subseqs" class="idref" href="#val_enum_subseqs"><span class="id" title="lemma">val_enum_subseqs</span></a> :<br/>
<span class="id" title="definition">perm_eq</span><br/>
(<span class="id" title="definition">map</span> <span class="id" title="abbreviation">val</span> (<span class="id" title="abbreviation">enum</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.subseq.html#subseqs"><span class="id" title="record">subseqs</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>))<br/>
(<span class="id" title="definition">undup</span> <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="definition">mask</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:55"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">|</span> <a id="m:55" class="idref" href="#m:55"><span class="id" title="binder"><span id="m:56" class="id"><span id="m:57" class="id"><span id="m:58" class="id">m</span></span></span></span></a> <a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">:</span></span></a> <a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">(</span></span></a><a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">size</span></span></a> <a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">w</span></span></a><a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">).-</span></span></a><a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">tuple</span></span></a> <a id="m:56" class="idref" href="#m:56"><span class="id" title="binder"><span id="m:58" class="id">bool</span></span></a><span class="id" title="notation">]</span>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="seq_masks_uniq" class="idref" href="#seq_masks_uniq"><span class="id" title="lemma">seq_masks_uniq</span></a> :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</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> <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <span class="id" title="definition">mask</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:59"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a> <span class="id" title="notation">|</span> <a id="m:59" class="idref" href="#m:59"><span class="id" title="binder"><span id="m:60" class="id"><span id="m:61" class="id"><span id="m:62" class="id">m</span></span></span></span></a> <a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">:</span></span></a> <a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">(</span></span></a><a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">size</span></span></a> <a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">w</span></span></a><a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">).-</span></span></a><a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">tuple</span></span></a> <a id="m:60" class="idref" href="#m:60"><span class="id" title="binder"><span id="m:62" class="id">bool</span></span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="subseqs_masks_uniq" class="idref" href="#subseqs_masks_uniq"><span class="id" title="lemma">subseqs_masks_uniq</span></a> :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</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="definition">uniq</span> <span class="id" title="notation">[</span><span class="id" title="notation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Subseqs"><span class="id" title="constructor">Subseqs</span></a> (<span class="id" title="lemma">mask_subseq</span> (<span class="id" title="abbreviation">val</span> <a class="idref" href="Combi.Combi.subseq.html#m:63"><span class="id" title="variable">m</span></a>) <a class="idref" href="Combi.Combi.subseq.html#FinType.w"><span class="id" title="variable">w</span></a>) <span class="id" title="notation">|</span> <a id="m:63" class="idref" href="#m:63"><span class="id" title="binder"><span id="m:64" class="id"><span id="m:65" class="id"><span id="m:66" class="id">m</span></span></span></span></a> <a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">:</span></span></a> <a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">(</span></span></a><a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">size</span></span></a> <a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">w</span></span></a><a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">).-</span></span></a><a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">tuple</span></span></a> <a id="m:64" class="idref" href="#m:64"><span class="id" title="binder"><span id="m:66" class="id">bool</span></span></a><span class="id" title="notation">]</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.subseq.html#FinType"><span class="id" title="section">FinType</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="Bigop" class="idref" href="#Bigop"><span class="id" title="section">Bigop</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> {<a id="Bigop.R" class="idref" href="#Bigop.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>} {<a id="Bigop.idx" class="idref" href="#Bigop.idx"><span class="id" title="variable">idx</span></a> : <a class="idref" href="Combi.Combi.subseq.html#R:67"><span class="id" title="variable">R</span></a>} {<a id="Bigop.op" class="idref" href="#Bigop.op"><span class="id" title="variable">op</span></a> : <span class="id" title="abbreviation">Monoid.com_law</span> <a class="idref" href="Combi.Combi.subseq.html#idx:68"><span class="id" title="variable">idx</span></a>}.<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> "1" := <a class="idref" href="Combi.Combi.subseq.html#Bigop.idx"><span class="id" title="variable">idx</span></a>.<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> "'*%M'" := <a class="idref" href="Combi.Combi.subseq.html#Bigop.op"><span class="id" title="variable">op</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> "x * y" := (<a class="idref" href="Combi.Combi.subseq.html#Bigop.op"><span class="id" title="variable">op</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
<span class="id" title="keyword">Context</span> {<a id="Bigop.T" class="idref" href="#Bigop.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">choiceType</span>}.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs" class="idref" href="#big_subseqs"><span class="id" title="lemma">big_subseqs</span></a> (<a id="F:71" class="idref" href="#F:71"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="s:72" class="idref" href="#s:72"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#s:72"><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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:73" class="idref" href="#i:73"><span class="id" title="binder"><span id="i:74" class="id"><span id="i:75" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:72"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:71"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:73"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="m:76" class="idref" href="#m:76"><span class="id" title="binder"><span id="m:77" class="id"><span id="m:78" class="id">m</span></span></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.subseq.html#s:72"><span class="id" title="variable">s</span></a><span class="id" title="notation">).-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">bool</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:71"><span class="id" title="variable">F</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:76"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:72"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs_cond" class="idref" href="#big_subseqs_cond"><span class="id" title="lemma">big_subseqs_cond</span></a> (<a id="P:79" class="idref" href="#P:79"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> (<span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>)) (<a id="F:80" class="idref" href="#F:80"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="s:81" class="idref" href="#s:81"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">uniq</span> <a class="idref" href="Combi.Combi.subseq.html#s:81"><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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:82" class="idref" href="#i:82"><span class="id" title="binder"><span id="i:83" class="id"><span id="i:84" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:81"><span class="id" title="variable">s</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:79"><span class="id" title="variable">P</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:82"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:80"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:82"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="m:85" class="idref" href="#m:85"><span class="id" title="binder"><span id="m:86" class="id"><span id="m:87" class="id">m</span></span></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.subseq.html#s:81"><span class="id" title="variable">s</span></a><span class="id" title="notation">).-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">bool</span> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:79"><span class="id" title="variable">P</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:85"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:81"><span class="id" title="variable">s</span></a>)<span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:80"><span class="id" title="variable">F</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:85"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:81"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs0" class="idref" href="#big_subseqs0"><span class="id" title="lemma">big_subseqs0</span></a> (<a id="F:90" class="idref" href="#F:90"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) :<br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:91" class="idref" href="#i:91"><span class="id" title="binder"><span id="i:92" class="id"><span id="i:93" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <span class="id" title="notation">[::]</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:90"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:91"><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.subseq.html#F:90"><span class="id" title="variable">F</span></a> <span class="id" title="notation">[::]</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs_cons" class="idref" href="#big_subseqs_cons"><span class="id" title="lemma">big_subseqs_cons</span></a> (<a id="F:100" class="idref" href="#F:100"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="a:101" class="idref" href="#a:101"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) (<a id="s:102" class="idref" href="#s:102"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">uniq</span> (<a class="idref" href="Combi.Combi.subseq.html#a:101"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#s:102"><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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:103" class="idref" href="#i:103"><span class="id" title="binder"><span id="i:104" class="id"><span id="i:105" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> (<a class="idref" href="Combi.Combi.subseq.html#a:101"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#s:102"><span class="id" title="variable">s</span></a>)<span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:100"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:103"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:106" class="idref" href="#i:106"><span class="id" title="binder"><span id="i:107" class="id"><span id="i:108" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:102"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:100"><span class="id" title="variable">F</span></a> (<a class="idref" href="Combi.Combi.subseq.html#a:101"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#i:106"><span class="id" title="variable">i</span></a>) <a class="idref" href="Combi.Combi.subseq.html#e0b08a12bdad0f54dff37beff51ec9d4"><span class="id" title="notation">*</span></a> <span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:109" class="idref" href="#i:109"><span class="id" title="binder"><span id="i:110" class="id"><span id="i:111" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:102"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:100"><span class="id" title="variable">F</span></a> (<a class="idref" href="Combi.Combi.subseq.html#i:109"><span class="id" title="variable">i</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs_cons_cond" class="idref" href="#big_subseqs_cons_cond"><span class="id" title="lemma">big_subseqs_cons_cond</span></a><br/>
(<a id="F:131" class="idref" href="#F:131"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="P:132" class="idref" href="#P:132"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> (<span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>)) (<a id="a:133" class="idref" href="#a:133"><span class="id" title="binder">a</span></a> : <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) (<a id="s:134" class="idref" href="#s:134"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">uniq</span> (<a class="idref" href="Combi.Combi.subseq.html#a:133"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#s:134"><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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:135" class="idref" href="#i:135"><span class="id" title="binder"><span id="i:136" class="id"><span id="i:137" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> (<a class="idref" href="Combi.Combi.subseq.html#a:133"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#s:134"><span class="id" title="variable">s</span></a>) <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:132"><span class="id" title="variable">P</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:135"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:131"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:135"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:138" class="idref" href="#i:138"><span class="id" title="binder"><span id="i:139" class="id"><span id="i:140" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:134"><span class="id" title="variable">s</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:132"><span class="id" title="variable">P</span></a> (<a class="idref" href="Combi.Combi.subseq.html#a:133"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#i:138"><span class="id" title="variable">i</span></a>)<span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:131"><span class="id" title="variable">F</span></a> (<a class="idref" href="Combi.Combi.subseq.html#a:133"><span class="id" title="variable">a</span></a> <span class="id" title="notation">::</span> <a class="idref" href="Combi.Combi.subseq.html#i:138"><span class="id" title="variable">i</span></a>) <a class="idref" href="Combi.Combi.subseq.html#e0b08a12bdad0f54dff37beff51ec9d4"><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><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:141" class="idref" href="#i:141"><span class="id" title="binder"><span id="i:142" class="id"><span id="i:143" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:134"><span class="id" title="variable">s</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:132"><span class="id" title="variable">P</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:141"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:131"><span class="id" title="variable">F</span></a> (<a class="idref" href="Combi.Combi.subseq.html#i:141"><span class="id" title="variable">i</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs_undup" class="idref" href="#big_subseqs_undup"><span class="id" title="lemma">big_subseqs_undup</span></a> (<a id="F:147" class="idref" href="#F:147"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="s:148" class="idref" href="#s:148"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">idempotent_op</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.op"><span class="id" title="variable">op</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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:149" class="idref" href="#i:149"><span class="id" title="binder"><span id="i:150" class="id"><span id="i:151" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:148"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:147"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:149"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="m:152" class="idref" href="#m:152"><span class="id" title="binder"><span id="m:153" class="id"><span id="m:154" class="id">m</span></span></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.subseq.html#s:148"><span class="id" title="variable">s</span></a><span class="id" title="notation">).-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">bool</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:147"><span class="id" title="variable">F</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:152"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:148"><span class="id" title="variable">s</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="big_subseqs_undup_cond" class="idref" href="#big_subseqs_undup_cond"><span class="id" title="lemma">big_subseqs_undup_cond</span></a> (<a id="F:155" class="idref" href="#F:155"><span class="id" title="binder">F</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><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.subseq.html#Bigop.R"><span class="id" title="variable">R</span></a>) (<a id="P:156" class="idref" href="#P:156"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> (<span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>)) (<a id="s:157" class="idref" href="#s:157"><span class="id" title="binder">s</span></a> : <span class="id" title="abbreviation">seq</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.T"><span class="id" title="variable">T</span></a>) :<br/>
<span class="id" title="definition">idempotent_op</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop.op"><span class="id" title="variable">op</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">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="i:158" class="idref" href="#i:158"><span class="id" title="binder"><span id="i:159" class="id"><span id="i:160" class="id">i</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Combi.subseq.html#subseqs"><span class="id" title="record">subseqs</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:157"><span class="id" title="variable">s</span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:156"><span class="id" title="variable">P</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:158"><span class="id" title="variable">i</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:155"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Combi.subseq.html#i:158"><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><br/>
<span class="id" title="notation">\</span><span class="id" title="notation">big</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">*%</span></a><a class="idref" href="Combi.Combi.subseq.html#478e4b1a8f8c5b55efa0a5ec074d0671"><span class="id" title="notation">M</span></a><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="m:161" class="idref" href="#m:161"><span class="id" title="binder"><span id="m:162" class="id"><span id="m:163" class="id">m</span></span></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.subseq.html#s:157"><span class="id" title="variable">s</span></a><span class="id" title="notation">).-</span><span class="id" title="notation">tuple</span> <span class="id" title="notation">bool</span> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Combi.subseq.html#P:156"><span class="id" title="variable">P</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:161"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:157"><span class="id" title="variable">s</span></a>)<span class="id" title="notation">)</span> <a class="idref" href="Combi.Combi.subseq.html#F:155"><span class="id" title="variable">F</span></a> (<span class="id" title="definition">mask</span> <a class="idref" href="Combi.Combi.subseq.html#m:161"><span class="id" title="variable">m</span></a> <a class="idref" href="Combi.Combi.subseq.html#s:157"><span class="id" title="variable">s</span></a>).<br/>
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<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Combi.subseq.html#Bigop"><span class="id" title="section">Bigop</span></a>.<br/>
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<a id="lab93"></a><h1 class="section">Relating sub sequences of <span class="inlinecode"><span class="id" title="keyword">iota</span></span> and being sorted</h1>
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<span class="id" title="keyword">Lemma</span> <a id="sorted_subseq_iota_rcons" class="idref" href="#sorted_subseq_iota_rcons"><span class="id" title="lemma">sorted_subseq_iota_rcons</span></a> <a id="s:167" class="idref" href="#s:167"><span class="id" title="binder">s</span></a> <a id="n:168" class="idref" href="#n:168"><span class="id" title="binder">n</span></a> :<br/>
<span class="id" title="definition">subseq</span> <a class="idref" href="Combi.Combi.subseq.html#s:167"><span class="id" title="variable">s</span></a> (<span class="id" title="definition">iota</span> 0 <a class="idref" href="Combi.Combi.subseq.html#n:168"><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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="definition">sorted</span> <span class="id" title="definition">ltn</span> (<span class="id" title="definition">rcons</span> <a class="idref" href="Combi.Combi.subseq.html#s:167"><span class="id" title="variable">s</span></a> <a class="idref" href="Combi.Combi.subseq.html#n:168"><span class="id" title="variable">n</span></a>).<br/>
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