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<title>Combi.Basic.unitriginv: Uni-triangular Matrices</title>
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<h1 class="libtitle">Library Combi.Basic.unitriginv: Uni-triangular Matrices</h1>
<div class="code">
</div>
<div class="doc">
</div>
<div class="code">
</div>
<div class="doc">
<a id="lab251"></a><h1 class="section">Triangular matrix with 1 on the diagonal</h1>
<div class="paragraph"> </div>
We deal with "matrices" which are triangular for a <i>possibly partial</i> order
with 1 on the diagonal. The goal is to show that such a matrix is invertible
on any ring and to give formulas for the inverse. The matrices are given as a
function <span class="inlinecode"><span class="id" title="var">M</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">T</span></span> <span class="inlinecode">-></span> <span class="inlinecode"><span class="id" title="var">T</span></span> <span class="inlinecode">-></span> <span class="inlinecode"><span class="id" title="var">R</span></span> for a finite partially ordered type <span class="inlinecode"><span class="id" title="var">T</span></span> and a
commutative unit ring <span class="inlinecode"><span class="id" title="var">R</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">unitrig</span></span> <span class="inlinecode"><span class="id" title="var">M</span></span> == <span class="inlinecode"><span class="id" title="var">M</span></span> is unitriangular where <span class="inlinecode"><span class="id" title="var">M</span></span>.
</li>
<li> <span class="inlinecode"><span class="id" title="var">Mat</span></span> <span class="inlinecode"><span class="id" title="var">M</span></span> == transform <span class="inlinecode"><span class="id" title="var">M</span></span> to a usual mathcomp square matrix of order <span class="inlinecode">#|<span class="id" title="var">T</span>|</span>
</li>
<li> <span class="inlinecode"><span class="id" title="var">Minv</span></span> <span class="inlinecode"><span class="id" title="var">M</span></span> == the inverse of the matrix <span class="inlinecode"><span class="id" title="var">M</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
We show that such a matrix has determinant 1 (Lemma <span class="inlinecode"><span class="id" title="var">det_unitrig</span></span>) and is
therefore invertible. Moreover Lemma <span class="inlinecode"><span class="id" title="var">Minv_unitrig</span></span> says that the inverse
is unitriangular too.
</div>
<div class="code">
<span class="id" title="keyword">From</span> <span class="id" title="var">Corelib</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Setoids.Setoid.html#"><span class="id" title="library">Setoid</span></a>.<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">ssreflect</span> <span class="id" title="library">ssrfun</span> <span class="id" title="library">ssrbool</span> <span class="id" title="library">eqtype</span> <span class="id" title="library">ssrnat</span> <span class="id" title="library">order</span>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">mathcomp</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">fintype</span> <span class="id" title="library">bigop</span> <span class="id" title="library">ssralg</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">finset</span> <span class="id" title="library">fingroup</span> <span class="id" title="library">perm</span> <span class="id" title="library">matrix</span>.<br/>
<br/>
<span class="id" title="keyword">Require</span> <a class="idref" href="Combi.Basic.ordtype.html#"><span class="id" title="library">ordtype</span></a>.<br/>
<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>
<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="module">Order.TTheory</span>.<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="module">GRing.Theory</span>.<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="UniTriangular" class="idref" href="#UniTriangular"><span class="id" title="section">UniTriangular</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a id="UniTriangular.R" class="idref" href="#UniTriangular.R"><span class="id" title="variable">R</span></a> : <span class="id" title="abbreviation">comUnitRingType</span>.<br/>
<span class="id" title="keyword">Context</span> <a id="UniTriangular.disp" class="idref" href="#UniTriangular.disp"><span class="id" title="variable">disp</span></a> (<a id="UniTriangular.T" class="idref" href="#UniTriangular.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">finPOrderType</span> <a class="idref" href="Combi.Basic.unitriginv.html#disp:3"><span class="id" title="variable">disp</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">M</span> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="unitrig" class="idref" href="#unitrig"><span class="id" title="definition">unitrig</span></a> <a id="M:5" class="idref" href="#M:5"><span class="id" title="binder">M</span></a> :=<br/>
<span class="id" title="notation">[</span><span class="id" title="notation">forall</span> <a id="t:6" class="idref" href="#t:6"><span class="id" title="binder"><span id="t:7" class="id"><span id="t:8" class="id">t</span></span></span></a><span class="id" title="notation">,</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:5"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:6"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:6"><span class="id" title="variable">t</span></a> <span class="id" title="notation">==</span> 1<span class="id" title="notation">]</span> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#::bool_scope:x_'&&'_x"><span class="id" title="notation">&&</span></a> <span class="id" title="notation">[</span><span class="id" title="notation">forall</span> <a id="t:9" class="idref" href="#t:9"><span class="id" title="binder"><span id="t:13" class="id"><span id="t:14" class="id">t</span></span></span></a><span class="id" title="notation">,</span> <span class="id" title="notation">[</span><span class="id" title="notation">forall</span> <span class="id" title="notation">(</span><a id="u:10" class="idref" href="#u:10"><span class="id" title="binder"><span id="u:11" class="id"><span id="u:12" class="id">u</span></span></span></a> <span class="id" title="notation">|</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:5"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:10"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:9"><span class="id" title="variable">t</span></a> <span class="id" title="notation">!=</span> 0<span class="id" title="notation">)</span><span class="id" title="notation">,</span> (<a class="idref" href="Combi.Basic.unitriginv.html#t:9"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:10"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">]</span><span class="id" title="notation">]</span>.<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrigP" class="idref" href="#unitrigP"><span class="id" title="lemma">unitrigP</span></a> <a id="M:15" class="idref" href="#M:15"><span class="id" title="binder">M</span></a> :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Datatypes.html#reflect"><span class="id" title="inductive">reflect</span></a> (<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:16" class="idref" href="#t:16"><span class="id" title="binder">t</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#M:15"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:16"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:16"><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> 1<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:17" class="idref" href="#t:17"><span class="id" title="binder">t</span></a> <a id="u:18" class="idref" href="#u:18"><span class="id" title="binder">u</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#M:15"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:18"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:17"><span class="id" title="variable">t</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> (<a class="idref" href="Combi.Basic.unitriginv.html#t:17"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:18"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>)<br/>
(<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:15"><span class="id" title="variable">M</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig1" class="idref" href="#unitrig1"><span class="id" title="lemma">unitrig1</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> (<span class="id" title="keyword">fun</span> <a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> <a id="y:20" class="idref" href="#y:20"><span class="id" title="binder">y</span></a> => <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#x:19"><span class="id" title="variable">x</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Basic.unitriginv.html#y:20"><span class="id" title="variable">y</span></a><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>).<br/>
<br/>
</div>
<div class="doc">
TODO : construct the group of unitriangular matrix
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_suml" class="idref" href="#unitrig_suml"><span class="id" title="lemma">unitrig_suml</span></a> <a id="M:21" class="idref" href="#M:21"><span class="id" title="binder">M</span></a> (<a id="Mod:22" class="idref" href="#Mod:22"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:23" class="idref" href="#F:23"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:22"><span class="id" title="variable">Mod</span></a>) <a id="u:24" class="idref" href="#u:24"><span class="id" title="binder">u</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:21"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="t:25" class="idref" href="#t:25"><span class="id" title="binder"><span id="t:26" class="id"><span id="t:27" class="id">t</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:21"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:24"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:25"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:23"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:25"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="t:28" class="idref" href="#t:28"><span class="id" title="binder"><span id="t:29" class="id"><span id="t:30" class="id">t</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#t:28"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:24"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:21"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:24"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:28"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:23"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:28"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sum1l" class="idref" href="#unitrig_sum1l"><span class="id" title="lemma">unitrig_sum1l</span></a> <a id="M:34" class="idref" href="#M:34"><span class="id" title="binder">M</span></a> (<a id="Mod:35" class="idref" href="#Mod:35"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:36" class="idref" href="#F:36"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:35"><span class="id" title="variable">Mod</span></a>) <a id="u:37" class="idref" href="#u:37"><span class="id" title="binder">u</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:34"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="t:38" class="idref" href="#t:38"><span class="id" title="binder"><span id="t:39" class="id"><span id="t:40" class="id">t</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:34"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:37"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:38"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:36"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:38"><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.Basic.unitriginv.html#F:36"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:37"><span class="id" title="variable">u</span></a> <span class="id" title="notation">+</span> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="t:41" class="idref" href="#t:41"><span class="id" title="binder"><span id="t:42" class="id"><span id="t:43" class="id">t</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#t:41"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:37"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:34"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:37"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:41"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:36"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:41"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sumr" class="idref" href="#unitrig_sumr"><span class="id" title="lemma">unitrig_sumr</span></a> <a id="M:44" class="idref" href="#M:44"><span class="id" title="binder">M</span></a> (<a id="Mod:45" class="idref" href="#Mod:45"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:46" class="idref" href="#F:46"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:45"><span class="id" title="variable">Mod</span></a>) <a id="t:47" class="idref" href="#t:47"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:44"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="u:48" class="idref" href="#u:48"><span class="id" title="binder"><span id="u:49" class="id"><span id="u:50" class="id">u</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:44"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:48"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:47"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:46"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:48"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="u:51" class="idref" href="#u:51"><span class="id" title="binder"><span id="u:52" class="id"><span id="u:53" class="id">u</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#t:47"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:51"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:44"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:51"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:47"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:46"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:51"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sum1r" class="idref" href="#unitrig_sum1r"><span class="id" title="lemma">unitrig_sum1r</span></a> <a id="M:57" class="idref" href="#M:57"><span class="id" title="binder">M</span></a> (<a id="Mod:58" class="idref" href="#Mod:58"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:59" class="idref" href="#F:59"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:58"><span class="id" title="variable">Mod</span></a>) <a id="t:60" class="idref" href="#t:60"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:57"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="u:61" class="idref" href="#u:61"><span class="id" title="binder"><span id="u:62" class="id"><span id="u:63" class="id">u</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:57"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:61"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:60"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:59"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:61"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#F:59"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:60"><span class="id" title="variable">t</span></a> <span class="id" title="notation">+</span> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="u:64" class="idref" href="#u:64"><span class="id" title="binder"><span id="u:65" class="id"><span id="u:66" class="id">u</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#t:60"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:64"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:57"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:64"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:60"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:59"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:64"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sumlV" class="idref" href="#unitrig_sumlV"><span class="id" title="lemma">unitrig_sumlV</span></a> <a id="M:67" class="idref" href="#M:67"><span class="id" title="binder">M</span></a> (<a id="Mod:68" class="idref" href="#Mod:68"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:69" class="idref" href="#F:69"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:68"><span class="id" title="variable">Mod</span></a>) <a id="u:70" class="idref" href="#u:70"><span class="id" title="binder">u</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:67"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="t:71" class="idref" href="#t:71"><span class="id" title="binder"><span id="t:72" class="id"><span id="t:73" class="id">t</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:67"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:71"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:70"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:69"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:71"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="t:74" class="idref" href="#t:74"><span class="id" title="binder"><span id="t:75" class="id"><span id="t:76" class="id">t</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#u:70"><span class="id" title="variable">u</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:74"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:67"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:74"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:70"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:69"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:74"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sum1lV" class="idref" href="#unitrig_sum1lV"><span class="id" title="lemma">unitrig_sum1lV</span></a> <a id="M:80" class="idref" href="#M:80"><span class="id" title="binder">M</span></a> (<a id="Mod:81" class="idref" href="#Mod:81"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:82" class="idref" href="#F:82"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:81"><span class="id" title="variable">Mod</span></a>) <a id="u:83" class="idref" href="#u:83"><span class="id" title="binder">u</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:80"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="t:84" class="idref" href="#t:84"><span class="id" title="binder"><span id="t:85" class="id"><span id="t:86" class="id">t</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:80"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:84"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:83"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:82"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:84"><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.Basic.unitriginv.html#F:82"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:83"><span class="id" title="variable">u</span></a> <span class="id" title="notation">+</span> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="t:87" class="idref" href="#t:87"><span class="id" title="binder"><span id="t:88" class="id"><span id="t:89" class="id">t</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#u:83"><span class="id" title="variable">u</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:87"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:80"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:87"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:83"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:82"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:87"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sumrV" class="idref" href="#unitrig_sumrV"><span class="id" title="lemma">unitrig_sumrV</span></a> <a id="M:90" class="idref" href="#M:90"><span class="id" title="binder">M</span></a> (<a id="Mod:91" class="idref" href="#Mod:91"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:92" class="idref" href="#F:92"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:91"><span class="id" title="variable">Mod</span></a>) <a id="t:93" class="idref" href="#t:93"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:90"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="u:94" class="idref" href="#u:94"><span class="id" title="binder"><span id="u:95" class="id"><span id="u:96" class="id">u</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:90"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:93"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:94"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:92"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:94"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="u:97" class="idref" href="#u:97"><span class="id" title="binder"><span id="u:98" class="id"><span id="u:99" class="id">u</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#u:97"><span class="id" title="variable">u</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:93"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:90"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:93"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:97"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:92"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:97"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="unitrig_sum1rV" class="idref" href="#unitrig_sum1rV"><span class="id" title="lemma">unitrig_sum1rV</span></a> <a id="M:103" class="idref" href="#M:103"><span class="id" title="binder">M</span></a> (<a id="Mod:104" class="idref" href="#Mod:104"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.R"><span class="id" title="variable">R</span></a>) (<a id="F:105" class="idref" href="#F:105"><span class="id" title="binder">F</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.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.Basic.unitriginv.html#Mod:104"><span class="id" title="variable">Mod</span></a>) <a id="t:106" class="idref" href="#t:106"><span class="id" title="binder">t</span></a> :<br/>
<a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#M:103"><span class="id" title="variable">M</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">sum_</span><span class="id" title="notation">(</span><a id="u:107" class="idref" href="#u:107"><span class="id" title="binder"><span id="u:108" class="id"><span id="u:109" class="id">u</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:103"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:106"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:107"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:105"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:107"><span class="id" title="variable">u</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#F:105"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:106"><span class="id" title="variable">t</span></a> <span class="id" title="notation">+</span> <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="u:110" class="idref" href="#u:110"><span class="id" title="binder"><span id="u:111" class="id"><span id="u:112" class="id">u</span></span></span></a> <span class="id" title="notation">|</span> (<a class="idref" href="Combi.Basic.unitriginv.html#u:110"><span class="id" title="variable">u</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:106"><span class="id" title="variable">t</span></a>)%<span class="id" title="var">O</span><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#M:103"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:106"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:110"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:105"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:110"><span class="id" title="variable">u</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Basic.unitriginv.html#UniTriangular"><span class="id" title="section">UniTriangular</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a id="TriangularInv" class="idref" href="#TriangularInv"><span class="id" title="section">TriangularInv</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> (<a id="TriangularInv.R" class="idref" href="#TriangularInv.R"><span class="id" title="variable">R</span></a> : <span class="id" title="abbreviation">comUnitRingType</span>) <a id="TriangularInv.disp" class="idref" href="#TriangularInv.disp"><span class="id" title="variable">disp</span></a> (<a id="TriangularInv.T" class="idref" href="#TriangularInv.T"><span class="id" title="variable">T</span></a> : <span class="id" title="abbreviation">finPOrderType</span> <a class="idref" href="Combi.Basic.unitriginv.html#disp:114"><span class="id" title="variable">disp</span></a>).<br/>
<span class="id" title="keyword">Variable</span> <a id="TriangularInv.M" class="idref" href="#TriangularInv.M"><span class="id" title="variable">M</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.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.Basic.unitriginv.html#TriangularInv.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.Basic.unitriginv.html#TriangularInv.R"><span class="id" title="variable">R</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span> : <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.T"><span class="id" title="variable">T</span></a>.<br/>
<br/>
<span class="id" title="keyword">Hypothesis</span> <a id="TriangularInv.Munitrig" class="idref" href="#TriangularInv.Munitrig"><span class="id" title="variable">Munitrig</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a>.<br/>
<br/>
#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Notation</span> <a id="n" class="idref" href="#n"><span class="id" title="abbreviation">n</span></a> := <span class="id" title="notation">#|</span><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.ssr.ssrbool.html#fa3b33ae9d0a52de608b305a09f3b881"><span class="id" title="notation">{:</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.T"><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#fa3b33ae9d0a52de608b305a09f3b881"><span class="id" title="notation">}</span></a><span class="id" title="notation">|</span>.<br/>
<span class="id" title="keyword">Definition</span> <a id="Mat" class="idref" href="#Mat"><span class="id" title="definition">Mat</span></a> : <span class="id" title="notation">'</span><span class="id" title="notation">M</span><span class="id" title="notation">[</span><a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.R"><span class="id" title="variable">R</span></a><span class="id" title="notation">]</span><span class="id" title="notation">_n</span> := <span class="id" title="notation">\</span><span class="id" title="notation">matrix_</span><span class="id" title="notation">(</span><a id="i:120" class="idref" href="#i:120"><span class="id" title="binder">i</span></a><span class="id" title="notation">,</span> <a id="j:121" class="idref" href="#j:121"><span class="id" title="binder">j</span></a> <span class="id" title="notation"><</span> <a class="idref" href="Combi.Basic.unitriginv.html#n"><span class="id" title="abbreviation">n</span></a><span class="id" title="notation">)</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a> (<span class="id" title="definition">enum_val</span> <a class="idref" href="Combi.Basic.unitriginv.html#i:120"><span class="id" title="variable">i</span></a>) (<span class="id" title="definition">enum_val</span> <a class="idref" href="Combi.Basic.unitriginv.html#j:121"><span class="id" title="variable">j</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="det_unitrig" class="idref" href="#det_unitrig"><span class="id" title="lemma">det_unitrig</span></a> : <span class="id" title="notation">\</span><span class="id" title="notation">det</span> <a class="idref" href="Combi.Basic.unitriginv.html#Mat"><span class="id" title="definition">Mat</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> 1.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a id="Minv" class="idref" href="#Minv"><span class="id" title="definition">Minv</span></a> <a id="t:125" class="idref" href="#t:125"><span class="id" title="binder">t</span></a> <a id="u:126" class="idref" href="#u:126"><span class="id" title="binder">u</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.R"><span class="id" title="variable">R</span></a> := <span class="id" title="definition">invmx</span> <a class="idref" href="Combi.Basic.unitriginv.html#Mat"><span class="id" title="definition">Mat</span></a> (<span class="id" title="definition">enum_rank</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:125"><span class="id" title="variable">t</span></a>) (<span class="id" title="definition">enum_rank</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:126"><span class="id" title="variable">u</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minvl" class="idref" href="#Minvl"><span class="id" title="lemma">Minvl</span></a> <a id="t:127" class="idref" href="#t:127"><span class="id" title="binder">t</span></a> <a id="u:128" class="idref" href="#u:128"><span class="id" title="binder">u</span></a> : <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="v:129" class="idref" href="#v:129"><span class="id" title="binder"><span id="v:130" class="id"><span id="v:131" class="id">v</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:127"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#v:129"><span class="id" title="variable">v</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">*</span> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#v:129"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:128"><span class="id" title="variable">u</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> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#u:128"><span class="id" title="variable">u</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:127"><span class="id" title="variable">t</span></a><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minvr" class="idref" href="#Minvr"><span class="id" title="lemma">Minvr</span></a> <a id="t:132" class="idref" href="#t:132"><span class="id" title="binder">t</span></a> <a id="u:133" class="idref" href="#u:133"><span class="id" title="binder">u</span></a> : <span class="id" title="notation">\</span><span class="id" title="notation">sum_</span><span class="id" title="notation">(</span><a id="v:134" class="idref" href="#v:134"><span class="id" title="binder"><span id="v:135" class="id"><span id="v:136" class="id">v</span></span></span></a> <span class="id" title="notation">:</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.T"><span class="id" title="variable">T</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:132"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#v:134"><span class="id" title="variable">v</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">*</span> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#v:134"><span class="id" title="variable">v</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:133"><span class="id" title="variable">u</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> <span class="id" title="notation">(</span><a class="idref" href="Combi.Basic.unitriginv.html#u:133"><span class="id" title="variable">u</span></a> <span class="id" title="notation">==</span> <a class="idref" href="Combi.Basic.unitriginv.html#t:132"><span class="id" title="variable">t</span></a><span class="id" title="notation">)%:</span><span class="id" title="notation">R</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minv_trig" class="idref" href="#Minv_trig"><span class="id" title="lemma">Minv_trig</span></a> <a id="t:137" class="idref" href="#t:137"><span class="id" title="binder">t</span></a> <a id="u:138" class="idref" href="#u:138"><span class="id" title="binder">u</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:138"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:137"><span class="id" title="variable">t</span></a> <span class="id" title="notation">!=</span> 0 <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a> (<a class="idref" href="Combi.Basic.unitriginv.html#t:137"><span class="id" title="variable">t</span></a> <span class="id" title="notation"><=</span> <a class="idref" href="Combi.Basic.unitriginv.html#u:138"><span class="id" title="variable">u</span></a>)%<span class="id" title="var">O</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minv_uni" class="idref" href="#Minv_uni"><span class="id" title="lemma">Minv_uni</span></a> <a id="t:144" class="idref" href="#t:144"><span class="id" title="binder">t</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:144"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:144"><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> 1.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minv_unitrig" class="idref" href="#Minv_unitrig"><span class="id" title="lemma">Minv_unitrig</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#unitrig"><span class="id" title="definition">unitrig</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minv_lincombl" class="idref" href="#Minv_lincombl"><span class="id" title="lemma">Minv_lincombl</span></a> (<a id="Mod:153" class="idref" href="#Mod:153"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.R"><span class="id" title="variable">R</span></a>) (<a id="F:154" class="idref" href="#F:154"><span class="id" title="binder">F</span></a> <a id="G:155" class="idref" href="#G:155"><span class="id" title="binder">G</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.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.Basic.unitriginv.html#Mod:153"><span class="id" title="variable">Mod</span></a>) :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:156" class="idref" href="#t:156"><span class="id" title="binder">t</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#F:154"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:156"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_u</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:156"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:157"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#G:155"><span class="id" title="variable">G</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:157"><span class="id" title="variable">u</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:160" class="idref" href="#t:160"><span class="id" title="binder">t</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#G:155"><span class="id" title="variable">G</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:160"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_u</span> <a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:160"><span class="id" title="variable">t</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:161"><span class="id" title="variable">u</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:154"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:161"><span class="id" title="variable">u</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a id="Minv_lincombr" class="idref" href="#Minv_lincombr"><span class="id" title="lemma">Minv_lincombr</span></a> (<a id="Mod:164" class="idref" href="#Mod:164"><span class="id" title="binder">Mod</span></a> : <span class="id" title="abbreviation">lmodType</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.R"><span class="id" title="variable">R</span></a>) (<a id="F:165" class="idref" href="#F:165"><span class="id" title="binder">F</span></a> <a id="G:166" class="idref" href="#G:166"><span class="id" title="binder">G</span></a> : <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.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.Basic.unitriginv.html#Mod:164"><span class="id" title="variable">Mod</span></a>) :<br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:167" class="idref" href="#t:167"><span class="id" title="binder">t</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#F:165"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:167"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_u</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv.M"><span class="id" title="variable">M</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:168"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:167"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#G:166"><span class="id" title="variable">G</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:168"><span class="id" title="variable">u</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">-></span></a><br/>
<a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <a id="t:171" class="idref" href="#t:171"><span class="id" title="binder">t</span></a>, <a class="idref" href="Combi.Basic.unitriginv.html#G:166"><span class="id" title="variable">G</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:171"><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> <span class="id" title="notation">\</span><span class="id" title="notation">sum_u</span> <a class="idref" href="Combi.Basic.unitriginv.html#Minv"><span class="id" title="definition">Minv</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:172"><span class="id" title="variable">u</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#t:171"><span class="id" title="variable">t</span></a> <span class="id" title="notation">*:</span> <a class="idref" href="Combi.Basic.unitriginv.html#F:165"><span class="id" title="variable">F</span></a> <a class="idref" href="Combi.Basic.unitriginv.html#u:172"><span class="id" title="variable">u</span></a><a class="idref" href="http://rocq-prover.org/doc/V9.1.1/stdlib//Corelib.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Combi.Basic.unitriginv.html#TriangularInv"><span class="id" title="section">TriangularInv</span></a>.<br/>
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