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| author | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
|---|---|---|
| committer | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
| commit | 6b59540a2460633df4e3d8347cb4dfe2fb3a3afb (patch) | |
| tree | 1239c1d5553d51a7d73f2f8b465f6a23178ff8a0 /docs/htmldoc/mathcomp.character.character.html | |
| parent | dd82aaeae7e9478efc178ce8430986649555b032 (diff) | |
removing everything but index which redirects to the new page
Diffstat (limited to 'docs/htmldoc/mathcomp.character.character.html')
| -rw-r--r-- | docs/htmldoc/mathcomp.character.character.html | 2209 |
1 files changed, 0 insertions, 2209 deletions
diff --git a/docs/htmldoc/mathcomp.character.character.html b/docs/htmldoc/mathcomp.character.character.html deleted file mode 100644 index 54edc9e..0000000 --- a/docs/htmldoc/mathcomp.character.character.html +++ /dev/null @@ -1,2209 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.character.character</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.character.character</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - This file contains the basic notions of character theory, based on Isaacs. - irr G == tuple of the elements of 'CF(G) that are irreducible - characters of G. - Nirr G == number of irreducible characters of G. - Iirr G == index type for the irreducible characters of G. - := 'I(Nirr G). - 'chi_i == the i-th element of irr G, for i : Iirr G. - 'chi[G]_i Note that 'chi_0 = 1, the principal character of G. - 'Chi_i == an irreducible representation that affords 'chi_i. - socle_of_Iirr i == the Wedderburn component of the regular representation - of G, corresponding to 'Chi_i. - Iirr_of_socle == the inverse of socle_of_Iirr (which is one-to-one). - phi. [A]%CF == the image of A \in group_ring G under phi : 'CF(G). - cfRepr rG == the character afforded by the representation rG of G. - cfReg G == the regular character, afforded by the regular - representation of G. - detRepr rG == the linear character afforded by the determinant of rG. - cfDet phi == the linear character afforded by the determinant of a - representation affording phi. - 'o(phi) == the "determinential order" of phi (the multiplicative - order of cfDet phi. - phi \is a character <=> phi : 'CF(G) is a character of G or 0. - i \in irr_constt phi <=> 'chi_i is an irreducible constituent of phi: phi - has a non-zero coordinate on 'chi_i over the basis irr G. - xi \is a linear_char xi <=> xi : 'CF(G) is a linear character of G. - 'Z(chi)%CF == the center of chi when chi is a character of G, i.e., - rcenter rG where rG is a representation that affords phi. - If phi is not a character then 'Z(chi)%CF = cfker phi. - aut_Iirr u i == the index of cfAut u 'chi_i in irr G. - conjC_Iirr i == the index of 'chi_i^*%CF in irr G. - morph_Iirr i == the index of cfMorph 'chi[f @* G]_i in irr G. - isom_Iirr isoG i == the index of cfIsom isoG 'chi[G]_i in irr R. - mod_Iirr i == the index of ('chi[G / H]_i %% H)%CF in irr G. - quo_Iirr i == the index of ('chi[G]_i / H)%CF in irr (G / H). - Ind_Iirr G i == the index of 'Ind[G, H] 'chi_i, provided it is an - irreducible character (such as when if H is the inertia - group of 'chi_i). - Res_Iirr H i == the index of 'Res[H, G] 'chi_i, provided it is an - irreducible character (such as when 'chi_i is linear). - sdprod_Iirr defG i == the index of cfSdprod defG 'chi_i in irr G, given - defG : K ><| H = G. - And, for KxK : K \x H = G. - dprodl_Iirr KxH i == the index of cfDprodl KxH 'chi[K]_i in irr G. - dprodr_Iirr KxH j == the index of cfDprodr KxH 'chi[H]_j in irr G. - dprod_Iirr KxH (i, j) == the index of cfDprod KxH 'chi[K]_i 'chi[H]_j. - inv_dprod_Iirr KxH == the inverse of dprod_Iirr KxH. - The following are used to define and exploit the character table: - character_table G == the character table of G, whose i-th row lists the - values taken by 'chi_i on the conjugacy classes - of G; this is a square Nirr G x NirrG matrix. - irr_class i == the conjugacy class of G with index i : Iirr G. - class_Iirr xG == the index of xG \in classes G, in Iirr G. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span> <span class="id" title="var">GRing.Theory</span> <span class="id" title="var">Num.Theory</span>.<br/> -<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AlgC"><span class="id" title="section">AlgC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="AlgC.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="groupC"><span class="id" title="lemma">groupC</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#group_closure_field"><span class="id" title="definition">group_closure_field</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#AlgC.gT"><span class="id" title="variable">gT</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#AlgC"><span class="id" title="section">AlgC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Tensor"><span class="id" title="section">Tensor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Tensor.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="trow"><span class="id" title="definition">trow</span></a> (<span class="id" title="var">n1</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/> - <span class="id" title="keyword">∀</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">rV</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#928a892a0c1438777aeb17535aec0378"><span class="id" title="notation">_n1</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>), <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'1</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <br/> - <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">fun</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a>1<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,(</span></a>1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">n'1</span><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">))</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) ⇒<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#lsubmx"><span class="id" title="definition">lsubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> 0 0 <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#rsubmx"><span class="id" title="definition">rsubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>))<br/> - <span class="id" title="keyword">else</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ 0).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trow0"><span class="id" title="lemma">trow0</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">B</span> : @<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> 0 <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="trowb"><span class="id" title="definition">trowb</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">B</span> <span class="id" title="var">A</span> := @<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trowbE"><span class="id" title="lemma">trowbE</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.character.character.html#trowb"><span class="id" title="definition">trowb</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> @<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trowb_is_linear"><span class="id" title="lemma">trowb_is_linear</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> (@<a class="idref" href="mathcomp.character.character.html#trowb"><span class="id" title="definition">trowb</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="keyword">Structure</span> <span class="id" title="var">trowb_linear</span> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">B</span> := <br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> (@<a class="idref" href="mathcomp.character.character.html#trowb_is_linear"><span class="id" title="lemma">trowb_is_linear</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trow_is_linear"><span class="id" title="lemma">trow_is_linear</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_n1</span></a>) : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.linear"><span class="id" title="abbreviation">linear</span></a> (@<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="keyword">Structure</span> <span class="id" title="var">trow_linear</span> <span class="id" title="var">n1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">A</span> := <br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.Linear"><span class="id" title="abbreviation">Linear</span></a> (@<a class="idref" href="mathcomp.character.character.html#trow_is_linear"><span class="id" title="lemma">trow_is_linear</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="tprod"><span class="id" title="definition">tprod</span></a> (<span class="id" title="var">m1</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">n1</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>),<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">m'1</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <br/> - <span class="id" title="keyword">return</span> <span class="id" title="keyword">∀</span> <span class="id" title="var">n1</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>),<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a><br/> - <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">fun</span> <span class="id" title="var">n1</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a>1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <span class="id" title="var">m'1</span><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">B</span> ⇒<br/> - (<a class="idref" href="mathcomp.algebra.matrix.html#col_mx"><span class="id" title="definition">col_mx</span></a> (<a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#usubmx"><span class="id" title="definition">usubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#dsubmx"><span class="id" title="definition">dsubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>))<br/> - <span class="id" title="keyword">else</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ 0).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dsumx_mul"><span class="id" title="lemma">dsumx_mul</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#dsubmx"><span class="id" title="definition">dsubmx</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#dsubmx"><span class="id" title="definition">dsubmx</span></a> (<a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="usumx_mul"><span class="id" title="lemma">usumx_mul</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#usubmx"><span class="id" title="definition">usubmx</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#usubmx"><span class="id" title="definition">usubmx</span></a> (<a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">M_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.algebra.matrix.html#5402b0dfe2a7ea661b91256aeeaf93da"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Tensor.trow_mul"><span class="id" title="variable">trow_mul</span></a> (<span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">p2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <br/> - (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2f65cfd766dcf020894d753750ad1a23"><span class="id" title="notation">rV_m1</span></a>) (<span class="id" title="var">B1</span>: <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B2</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#p2"><span class="id" title="variable">p2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) : <br/> - <a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.character.character.html#B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B2"><span class="id" title="variable">B2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#B2"><span class="id" title="variable">B2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tprodE"><span class="id" title="lemma">tprodE</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">n1</span> <span class="id" title="var">p1</span> (<span class="id" title="var">A1</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">A2</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#p1"><span class="id" title="variable">p1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>)<br/> - <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> <span class="id" title="var">p2</span> (<span class="id" title="var">B1</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B2</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#p2"><span class="id" title="variable">p2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> (<a class="idref" href="mathcomp.character.character.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#A2"><span class="id" title="variable">A2</span></a>) (<a class="idref" href="mathcomp.character.character.html#B1"><span class="id" title="variable">B1</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B2"><span class="id" title="variable">B2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> <a class="idref" href="mathcomp.character.character.html#A1"><span class="id" title="variable">A1</span></a> <a class="idref" href="mathcomp.character.character.html#B1"><span class="id" title="variable">B1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> <a class="idref" href="mathcomp.character.character.html#A2"><span class="id" title="variable">A2</span></a> <a class="idref" href="mathcomp.character.character.html#B2"><span class="id" title="variable">B2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Tensor.tprod_tr"><span class="id" title="variable">tprod_tr</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">n1</span> (<span class="id" title="var">A</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m1"><span class="id" title="variable">m1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <span class="id" title="var">m2</span> <span class="id" title="var">n2</span> (<span class="id" title="var">B</span> :<a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m2"><span class="id" title="variable">m2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#row_mx"><span class="id" title="definition">row_mx</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#trow"><span class="id" title="definition">trow</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#lsubmx"><span class="id" title="definition">lsubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">T</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">T</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.algebra.matrix.html#21ca0474978189f352b592dccdd75860"><span class="id" title="notation">T</span></a> (<a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> (<a class="idref" href="mathcomp.algebra.matrix.html#rsubmx"><span class="id" title="definition">rsubmx</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tprod1"><span class="id" title="lemma">tprod1</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) (1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">,</span></a><a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9c0a062cce31174bb4a1f05fb9cee844"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mxtrace_prod"><span class="id" title="lemma">mxtrace_prod</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> (<span class="id" title="var">A</span> :<a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">B</span> :<a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Tensor.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.matrix.html#1a8719a2ddc3d5ba380d052dba75ec31"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">tr</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">tr</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">tr</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Tensor"><span class="id" title="section">Tensor</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Representation sigma type and standard representations. -</div> -<div class="code"> -<span class="id" title="keyword">Section</span> <a name="StandardRepresentation"><span class="id" title="section">StandardRepresentation</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="StandardRepresentation.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="StandardRepresentation.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="StandardRepresentation.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Record</span> <a name="representation"><span class="id" title="record">representation</span></a> :=<br/> - <a name="Representation"><span class="id" title="constructor">Representation</span></a> {<a name="rdegree"><span class="id" title="projection">rdegree</span></a>; <a name="mx_repr_of_repr"><span class="id" title="projection">mx_repr_of_repr</span></a> :> <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#rdegree"><span class="id" title="method">rdegree</span></a>}.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mx_repr0"><span class="id" title="lemma">mx_repr0</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_repr"><span class="id" title="definition">mx_repr</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.G"><span class="id" title="variable">G</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.gT"><span class="id" title="variable">gT</span></a> ⇒ 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">M</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#StandardRepresentation.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.matrix.html#60bd2bc9fb9187afe5d7f780c1576e3c"><span class="id" title="notation">_0</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="grepr0"><span class="id" title="definition">grepr0</span></a> := <a class="idref" href="mathcomp.character.character.html#Representation"><span class="id" title="constructor">Representation</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#MxRepresentation"><span class="id" title="constructor">MxRepresentation</span></a> <a class="idref" href="mathcomp.character.character.html#mx_repr0"><span class="id" title="lemma">mx_repr0</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="add_mx_repr"><span class="id" title="lemma">add_mx_repr</span></a> (<span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> : <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a>) : <br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_repr"><span class="id" title="definition">mx_repr</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.G"><span class="id" title="variable">G</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">g</span> ⇒ <a class="idref" href="mathcomp.algebra.matrix.html#block_mx"><span class="id" title="definition">block_mx</span></a> (<a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="variable">g</span></a>) 0 0 (<a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="variable">g</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a> <span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> :=<br/> - <a class="idref" href="mathcomp.character.character.html#Representation"><span class="id" title="constructor">Representation</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#MxRepresentation"><span class="id" title="constructor">MxRepresentation</span></a> (<a class="idref" href="mathcomp.character.character.html#add_mx_repr"><span class="id" title="lemma">add_mx_repr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="StandardRepresentation.DsumRepr"><span class="id" title="section">DsumRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="StandardRepresentation.DsumRepr.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mx_rsim_dadd"><span class="id" title="lemma">mx_rsim_dadd</span></a> (<span class="id" title="var">U</span> <span class="id" title="var">V</span> <span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>) (<span class="id" title="var">rU</span> <span class="id" title="var">rV</span> : <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a>)<br/> - (<span class="id" title="var">modU</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#U"><span class="id" title="variable">U</span></a>) (<span class="id" title="var">modV</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#V"><span class="id" title="variable">V</span></a>) (<span class="id" title="var">modW</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>) :<br/> - (<a class="idref" href="mathcomp.character.character.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> (<a class="idref" href="mathcomp.character.character.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b116c353d9d5a3e6e54e78df2da7c80e"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#V"><span class="id" title="variable">V</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#modU"><span class="id" title="variable">modU</span></a>) <a class="idref" href="mathcomp.character.character.html#rU"><span class="id" title="variable">rU</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#modV"><span class="id" title="variable">modV</span></a>) <a class="idref" href="mathcomp.character.character.html#rV"><span class="id" title="variable">rV</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#modW"><span class="id" title="variable">modW</span></a>) (<a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a> <a class="idref" href="mathcomp.character.character.html#rU"><span class="id" title="variable">rU</span></a> <a class="idref" href="mathcomp.character.character.html#rV"><span class="id" title="variable">rV</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mx_rsim_dsum"><span class="id" title="lemma">mx_rsim_dsum</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.character.character.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">U</span> <span class="id" title="var">rU</span> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_n</span></a>)<br/> - (<span class="id" title="var">modU</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> (<a class="idref" href="mathcomp.character.character.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)) (<span class="id" title="var">modW</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>) : <br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">S</span> := (<a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#ba43ca3989a0bfce795ffb9f5d1783ba"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">MS</span> <span class="id" title="tactic">in</span> (<a class="idref" href="mathcomp.character.character.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#f769dda5dbc6895d666659cb6e305422"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#mxdirect"><span class="id" title="abbreviation">mxdirect</span></a> <a class="idref" href="mathcomp.character.character.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> (<a class="idref" href="mathcomp.character.character.html#modU"><span class="id" title="variable">modU</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="mathcomp.character.character.html#rU"><span class="id" title="variable">rU</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#modW"><span class="id" title="variable">modW</span></a>) (<a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#grepr0"><span class="id" title="definition">grepr0</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#rU"><span class="id" title="variable">rU</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="muln_grepr"><span class="id" title="definition">muln_grepr</span></a> <span class="id" title="var">rW</span> <span class="id" title="var">k</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#grepr0"><span class="id" title="definition">grepr0</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#afef6bddeda988bbc365e556241d5732"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#rW"><span class="id" title="variable">rW</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mx_rsim_socle"><span class="id" title="lemma">mx_rsim_socle</span></a> (<span class="id" title="var">sG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#socleType"><span class="id" title="record">socleType</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a>) (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.character.character.html#sG"><span class="id" title="variable">sG</span></a>) (<span class="id" title="var">rW</span> : <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a>) :<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">modW</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mxmodule"><span class="id" title="definition">mxmodule</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#component_mx_module"><span class="id" title="lemma">component_mx_module</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr.rG"><span class="id" title="variable">rG</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#socle_base"><span class="id" title="definition">socle_base</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#socle_repr"><span class="id" title="definition">socle_repr</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>) <a class="idref" href="mathcomp.character.character.html#rW"><span class="id" title="variable">rW</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#submod_repr"><span class="id" title="definition">submod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#modW"><span class="id" title="variable">modW</span></a>) (<a class="idref" href="mathcomp.character.character.html#muln_grepr"><span class="id" title="definition">muln_grepr</span></a> <a class="idref" href="mathcomp.character.character.html#rW"><span class="id" title="variable">rW</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#socle_mult"><span class="id" title="definition">socle_mult</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.DsumRepr"><span class="id" title="section">DsumRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="StandardRepresentation.ProdRepr"><span class="id" title="section">ProdRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="StandardRepresentation.ProdRepr.n1"><span class="id" title="variable">n1</span></a> <a name="StandardRepresentation.ProdRepr.n2"><span class="id" title="variable">n2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="StandardRepresentation.ProdRepr.rG1"><span class="id" title="variable">rG1</span></a> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a>) (<a name="StandardRepresentation.ProdRepr.rG2"><span class="id" title="variable">rG2</span></a> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="prod_mx_repr"><span class="id" title="lemma">prod_mx_repr</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_repr"><span class="id" title="definition">mx_repr</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.G"><span class="id" title="variable">G</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">g</span> ⇒ <a class="idref" href="mathcomp.character.character.html#tprod"><span class="id" title="definition">tprod</span></a> (<a class="idref" href="mathcomp.character.character.html#StandardRepresentation.ProdRepr.rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="variable">g</span></a>) (<a class="idref" href="mathcomp.character.character.html#StandardRepresentation.ProdRepr.rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="variable">g</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="prod_repr"><span class="id" title="definition">prod_repr</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#MxRepresentation"><span class="id" title="constructor">MxRepresentation</span></a> <a class="idref" href="mathcomp.character.character.html#prod_mx_repr"><span class="id" title="lemma">prod_mx_repr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.ProdRepr"><span class="id" title="section">ProdRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="prod_repr_lin"><span class="id" title="lemma">prod_repr_lin</span></a> <span class="id" title="var">n2</span> (<span class="id" title="var">rG1</span> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> 1) (<span class="id" title="var">rG2</span> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <span class="id" title="keyword">let</span> <span class="id" title="var">cast_n2</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#esym"><span class="id" title="definition">esym</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#mul1n"><span class="id" title="lemma">mul1n</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.character.character.html#prod_repr"><span class="id" title="definition">prod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#castmx"><span class="id" title="definition">castmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#cast_n2"><span class="id" title="variable">cast_n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#cast_n2"><span class="id" title="variable">cast_n2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> 0 0 <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#StandardRepresentation"><span class="id" title="section">StandardRepresentation</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Char"><span class="id" title="section">Char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Char.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Char.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfRepr_subproof"><span class="id" title="lemma">cfRepr_subproof</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.character.classfun.html#is_class_fun"><span class="id" title="definition">is_class_fun</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">ffun</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">⇒</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">tr</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfRepr"><span class="id" title="definition">cfRepr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">rG</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 0 (@<a class="idref" href="mathcomp.character.character.html#cfRepr_subproof"><span class="id" title="lemma">cfRepr_subproof</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr1"><span class="id" title="lemma">cfRepr1</span></a> <span class="id" title="var">n</span> <span class="id" title="var">rG</span> : @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_sim"><span class="id" title="lemma">cfRepr_sim</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> <span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> :<br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr0"><span class="id" title="lemma">cfRepr0</span></a> : <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#grepr0"><span class="id" title="definition">grepr0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_dadd"><span class="id" title="lemma">cfRepr_dadd</span></a> <span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_dsum"><span class="id" title="lemma">cfRepr_dsum</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.character.character.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">rG</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#grepr0"><span class="id" title="definition">grepr0</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#60e57ff387b8a0840e944d0d03f215e2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0e493beb85c9c1b3ab2241ceeaa98b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_muln"><span class="id" title="lemma">cfRepr_muln</span></a> <span class="id" title="var">rG</span> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.character.html#muln_grepr"><span class="id" title="definition">muln_grepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Char.StandardRepr"><span class="id" title="section">StandardRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Char.StandardRepr.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Char.StandardRepr.rG"><span class="id" title="variable">rG</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>).<br/> -<span class="id" title="keyword">Let</span> <a name="Char.StandardRepr.sG"><span class="id" title="variable">sG</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#DecSocleType"><span class="id" title="lemma">DecSocleType</span></a> <a class="idref" href="mathcomp.character.character.html#Char.StandardRepr.rG"><span class="id" title="variable">rG</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Char.StandardRepr.iG"><span class="id" title="variable">iG</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#irrType"><span class="id" title="definition">irrType</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#DecSocleType"><span class="id" title="lemma">DecSocleType</span></a> <span class="id" title="var">_</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="standard_irr"><span class="id" title="definition">standard_irr</span></a> (<span class="id" title="var">W</span> : <a class="idref" href="mathcomp.character.character.html#Char.StandardRepr.sG"><span class="id" title="variable">sG</span></a>) := <a class="idref" href="mathcomp.character.mxrepresentation.html#irr_comp"><span class="id" title="definition">irr_comp</span></a> <a class="idref" href="mathcomp.character.character.html#Char.StandardRepr.iG"><span class="id" title="variable">iG</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#socle_repr"><span class="id" title="definition">socle_repr</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="standard_socle"><span class="id" title="definition">standard_socle</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#pick"><span class="id" title="definition">pick</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">W</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#standard_irr"><span class="id" title="definition">standard_irr</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="standard_irr_coef"><span class="id" title="definition">standard_irr_coef</span></a> <span class="id" title="var">i</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#oapp"><span class="id" title="abbreviation">oapp</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">W</span> ⇒ <a class="idref" href="mathcomp.character.mxrepresentation.html#socle_mult"><span class="id" title="definition">socle_mult</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="variable">W</span></a>) 0%<span class="id" title="var">N</span> (<a class="idref" href="mathcomp.character.character.html#soc"><span class="id" title="abbreviation">soc</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="standard_grepr"><span class="id" title="definition">standard_grepr</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#dadd_grepr"><span class="id" title="definition">dadd_grepr</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#grepr0"><span class="id" title="definition">grepr0</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">_i</span></a><br/> - <a class="idref" href="mathcomp.character.character.html#muln_grepr"><span class="id" title="definition">muln_grepr</span></a> (<a class="idref" href="mathcomp.character.character.html#Representation"><span class="id" title="constructor">Representation</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#socle_repr"><span class="id" title="definition">socle_repr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="mathcomp.character.character.html#standard_irr_coef"><span class="id" title="definition">standard_irr_coef</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mx_rsim_standard"><span class="id" title="lemma">mx_rsim_standard</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> <a class="idref" href="mathcomp.character.character.html#Char.StandardRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#standard_grepr"><span class="id" title="definition">standard_grepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Char.StandardRepr"><span class="id" title="section">StandardRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfReg"><span class="id" title="definition">cfReg</span></a> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#Char.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.classfun.html#483aeb200c980494b369f3b9c18f042e"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#483aeb200c980494b369f3b9c18f042e"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b54f5e35cb228bba5934c852e0951c39"><span class="id" title="notation">[1]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRegE"><span class="id" title="lemma">cfRegE</span></a> <span class="id" title="var">x</span> : @<a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Lemma (2.10). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfReprReg"><span class="id" title="lemma">cfReprReg</span></a> : <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#regular_repr"><span class="id" title="definition">regular_repr</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> <a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="xcfun"><span class="id" title="definition">xcfun</span></a> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">A</span> :=<br/> - (<a class="idref" href="mathcomp.character.mxrepresentation.html#gring_row"><span class="id" title="definition">gring_row</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#3bdb4e1060a71660067c77fac6c5f4a5"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#3bdb4e1060a71660067c77fac6c5f4a5"><span class="id" title="notation">col_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#3bdb4e1060a71660067c77fac6c5f4a5"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.matrix.html#3bdb4e1060a71660067c77fac6c5f4a5"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Char.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.matrix.html#3bdb4e1060a71660067c77fac6c5f4a5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_val"><span class="id" title="definition">enum_val</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">)</span></a>) 0 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_is_additive"><span class="id" title="lemma">xcfun_is_additive</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.additive"><span class="id" title="abbreviation">additive</span></a> (<a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - <span class="id" title="keyword">Canonical</span> <span class="id" title="var">xcfun_additive</span> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> (<a class="idref" href="mathcomp.character.character.html#xcfun_is_additive"><span class="id" title="lemma">xcfun_is_additive</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfunZr"><span class="id" title="lemma">xcfunZr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - In order to add a second canonical structure on xcfun -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="xcfun_r_head"><span class="id" title="definition">xcfun_r_head</span></a> <span class="id" title="var">k</span> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> := <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> := <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_rE"><span class="id" title="lemma">xcfun_rE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#xcfun_r"><span class="id" title="abbreviation">xcfun_r</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="xcfun_r_is_additive"><span class="id" title="lemma">xcfun_r_is_additive</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.additive"><span class="id" title="abbreviation">additive</span></a> (<a class="idref" href="mathcomp.character.character.html#xcfun_r"><span class="id" title="abbreviation">xcfun_r</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">xcfun_r_additive</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Additive.Exports.Additive"><span class="id" title="abbreviation">Additive</span></a> (<a class="idref" href="mathcomp.character.character.html#xcfun_r_is_additive"><span class="id" title="lemma">xcfun_r_is_additive</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfunZl"><span class="id" title="lemma">xcfunZl</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> (<a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_repr"><span class="id" title="lemma">xcfun_repr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">rG</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> (@<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>) <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">tr</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.mxrepresentation.html#gring_op"><span class="id" title="definition">gring_op</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.matrix.html#055f111b06ebab166375c628a8e0315f"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Char"><span class="id" title="section">Char</span></a>.<br/> -<span class="id" title="keyword">Notation</span> <a name="xcfun_r"><span class="id" title="abbreviation">xcfun_r</span></a> <span class="id" title="var">A</span> := (<a class="idref" href="mathcomp.character.character.html#xcfun_r_head"><span class="id" title="definition">xcfun_r_head</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <span class="id" title="var">A</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="c7fef873721a4be094199a25626252ca"><span class="id" title="notation">"</span></a>phi .[ A ]" := (<a class="idref" href="mathcomp.character.character.html#xcfun"><span class="id" title="definition">xcfun</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">A</span>) : <span class="id" title="var">cfun_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="pred_Nirr"><span class="id" title="definition">pred_Nirr</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a>@<a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">.-1</span></a>.<br/> -<span class="id" title="keyword">Notation</span> <a name="Nirr"><span class="id" title="abbreviation">Nirr</span></a> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#pred_Nirr"><span class="id" title="definition">pred_Nirr</span></a> <span class="id" title="var">G</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a>.<br/> -<span class="id" title="keyword">Notation</span> <a name="Iirr"><span class="id" title="abbreviation">Iirr</span></a> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">I_</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Nirr"><span class="id" title="abbreviation">Nirr</span></a> <span class="id" title="var">G</span><a class="idref" href="mathcomp.ssreflect.fintype.html#545d9d6249a673300f950a2a8b8a930b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IrrClassDef"><span class="id" title="section">IrrClassDef</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="IrrClassDef.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="IrrClassDef.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClassDef.sG"><span class="id" title="variable">sG</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#DecSocleType"><span class="id" title="lemma">DecSocleType</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#regular_repr"><span class="id" title="definition">regular_repr</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="NirrE"><span class="id" title="lemma">NirrE</span></a> : <a class="idref" href="mathcomp.character.character.html#Nirr"><span class="id" title="abbreviation">Nirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="Iirr_cast"><span class="id" title="lemma">Iirr_cast</span></a> : <a class="idref" href="mathcomp.character.character.html#Nirr"><span class="id" title="abbreviation">Nirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClassDef.offset"><span class="id" title="variable">offset</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#cast_ord"><span class="id" title="definition">cast_ord</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#esym"><span class="id" title="definition">esym</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr_cast"><span class="id" title="lemma">Iirr_cast</span></a>) (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_rank"><span class="id" title="definition">enum_rank</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#c49896f064298adaf11dc4abacd4c29b"><span class="id" title="notation">[1</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a><a class="idref" href="mathcomp.character.mxrepresentation.html#c49896f064298adaf11dc4abacd4c29b"><span class="id" title="notation">]</span></a>%<span class="id" title="var">irr</span>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="socle_of_Iirr"><span class="id" title="definition">socle_of_Iirr</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a>) : <a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#enum_val"><span class="id" title="definition">enum_val</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#cast_ord"><span class="id" title="definition">cast_ord</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr_cast"><span class="id" title="lemma">Iirr_cast</span></a> (<a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.offset"><span class="id" title="variable">offset</span></a>)).<br/> -<span class="id" title="keyword">Definition</span> <a name="irr_of_socle"><span class="id" title="definition">irr_of_socle</span></a> (<span class="id" title="var">Wi</span> : <a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a>) : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#cast_ord"><span class="id" title="definition">cast_ord</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#esym"><span class="id" title="definition">esym</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr_cast"><span class="id" title="lemma">Iirr_cast</span></a>) (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_rank"><span class="id" title="definition">enum_rank</span></a> <a class="idref" href="mathcomp.character.character.html#Wi"><span class="id" title="variable">Wi</span></a>) <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.offset"><span class="id" title="variable">offset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="socle_Iirr0"><span class="id" title="lemma">socle_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="abbreviation">W</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#c49896f064298adaf11dc4abacd4c29b"><span class="id" title="notation">[1</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a><a class="idref" href="mathcomp.character.mxrepresentation.html#c49896f064298adaf11dc4abacd4c29b"><span class="id" title="notation">]</span></a>%<span class="id" title="var">irr</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="socle_of_IirrK"><span class="id" title="lemma">socle_of_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="abbreviation">W</span></a> <a class="idref" href="mathcomp.character.character.html#irr_of_socle"><span class="id" title="definition">irr_of_socle</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_of_socleK"><span class="id" title="lemma">irr_of_socleK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#irr_of_socle"><span class="id" title="definition">irr_of_socle</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="abbreviation">W</span></a>.<br/> - <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">socle_of_IirrK</span> <span class="id" title="var">irr_of_socleK</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_of_socle_bij"><span class="id" title="lemma">irr_of_socle_bij</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">)}</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">bijective</span></a> <a class="idref" href="mathcomp.character.character.html#irr_of_socle"><span class="id" title="definition">irr_of_socle</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="socle_of_Iirr_bij"><span class="id" title="lemma">socle_of_Iirr_bij</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClassDef.sG"><span class="id" title="variable">sG</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">bijective</span></a> <a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="abbreviation">W</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#IrrClassDef"><span class="id" title="section">IrrClassDef</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">"</span></a>''Chi_' i" := (<a class="idref" href="mathcomp.character.mxrepresentation.html#irr_repr"><span class="id" title="definition">irr_repr</span></a> (<a class="idref" href="mathcomp.character.character.html#socle_of_Iirr"><span class="id" title="definition">socle_of_Iirr</span></a> <span class="id" title="var">i</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">i</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Chi_' i").<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="irr_key"><span class="id" title="lemma">irr_key</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a>. <br/> -<span class="id" title="keyword">Definition</span> <a name="irr_def"><span class="id" title="definition">irr_def</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.tuple.html#c3913abe839346eb60d82da74b0b1f67"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Nirr"><span class="id" title="abbreviation">Nirr</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#c3913abe839346eb60d82da74b0b1f67"><span class="id" title="notation">).-</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#c3913abe839346eb60d82da74b0b1f67"><span class="id" title="notation">tuple</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> :=<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">irr_of</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation">>></span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> (@<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">Chi_</span></a><a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#inord"><span class="id" title="definition">inord</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">)</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.ssreflect.tuple.html#1c05412e4f131fc504427f72854c7514"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#1c05412e4f131fc504427f72854c7514"><span class="id" title="notation">tuple</span></a> <a class="idref" href="mathcomp.ssreflect.tuple.html#1c05412e4f131fc504427f72854c7514"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.character.character.html#irr_of"><span class="id" title="variable">irr_of</span></a> (<a class="idref" href="mathcomp.character.character.html#Nirr"><span class="id" title="abbreviation">Nirr</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>)<a class="idref" href="mathcomp.ssreflect.tuple.html#1c05412e4f131fc504427f72854c7514"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="irr"><span class="id" title="definition">irr</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.character.character.html#irr_key"><span class="id" title="lemma">irr_key</span></a> <a class="idref" href="mathcomp.character.character.html#irr_def"><span class="id" title="definition">irr_def</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">"</span></a>''chi_' i" := (<a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">i</span>%<span class="id" title="var">R</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">i</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''chi_' i") : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">"</span></a>''chi[' G ]_ i" := (<a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <span class="id" title="var">G</span>) <span class="id" title="var">i</span>%<span class="id" title="var">R</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">i</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IrrClass"><span class="id" title="section">IrrClass</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="IrrClass.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="IrrClass.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>) (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">group_ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="congr_irr"><span class="id" title="lemma">congr_irr</span></a> <span class="id" title="var">i1</span> <span class="id" title="var">i2</span> : <a class="idref" href="mathcomp.character.character.html#i1"><span class="id" title="variable">i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#i2"><span class="id" title="variable">i2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i2</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Iirr1_neq0"><span class="id" title="lemma">Iirr1_neq0</span></a> : <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#inord"><span class="id" title="definition">inord</span></a> 1 <a class="idref" href="mathcomp.ssreflect.eqtype.html#228e85e3c31a939cba019f255574c875"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#228e85e3c31a939cba019f255574c875"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_nonprincipal_irr"><span class="id" title="lemma">has_nonprincipal_irr</span></a> : <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrRepr"><span class="id" title="lemma">irrRepr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">Chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr0"><span class="id" title="lemma">irr0</span></a> : <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun1_irr"><span class="id" title="lemma">cfun1_irr</span></a> : 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_irr"><span class="id" title="lemma">mem_irr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrP"><span class="id" title="lemma">irrP</span></a> <span class="id" title="var">xi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>) (<a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClass.sG"><span class="id" title="variable">sG</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#DecSocleType"><span class="id" title="lemma">DecSocleType</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#regular_repr"><span class="id" title="definition">regular_repr</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClass.C'G"><span class="id" title="variable">C'G</span></a> := <a class="idref" href="mathcomp.character.classfun.html#algC'G"><span class="id" title="lemma">algC'G</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClass.closG"><span class="id" title="variable">closG</span></a> := @<a class="idref" href="mathcomp.character.character.html#groupC"><span class="id" title="lemma">groupC</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr1_degree"><span class="id" title="lemma">irr1_degree</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#8b022bdd3e8afd1d802a570e34363ef0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#8b022bdd3e8afd1d802a570e34363ef0"><span class="id" title="notation">n_i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Cnat_irr1"><span class="id" title="lemma">Cnat_irr1</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.Cnat"><span class="id" title="definition">Cnat</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr1_gt0"><span class="id" title="lemma">irr1_gt0</span></a> <span class="id" title="var">i</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr1_neq0"><span class="id" title="lemma">irr1_neq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_neq0"><span class="id" title="lemma">irr_neq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> - <br/> -<span class="id" title="keyword">Definition</span> <a name="cfIirr"><span class="id" title="definition">cfIirr</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>, <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#locked_with"><span class="id" title="definition">locked_with</span></a> <a class="idref" href="mathcomp.character.character.html#cfIirr_key"><span class="id" title="lemma">cfIirr_key</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">B</span> <span class="id" title="var">chi</span> ⇒ <a class="idref" href="mathcomp.ssreflect.fintype.html#inord"><span class="id" title="definition">inord</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>))).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIirrE"><span class="id" title="lemma">cfIirrE</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIirrPE"><span class="id" title="lemma">cfIirrPE</span></a> <span class="id" title="var">J</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.character.character.html#J"><span class="id" title="variable">J</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.character.character.html#J"><span class="id" title="variable">J</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a>)<a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Corollary (2.7). -</div> -<div class="code"> -<span class="id" title="keyword">Corollary</span> <a name="irr_sum_square"><span class="id" title="lemma">irr_sum_square</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Lemma (2.11). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfReg_sum"><span class="id" title="lemma">cfReg_sum</span></a> : <a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClass.aG"><span class="id" title="variable">aG</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#regular_repr"><span class="id" title="definition">regular_repr</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="IrrClass.R_G"><span class="id" title="variable">R_G</span></a> := <a class="idref" href="mathcomp.character.mxrepresentation.html#group_ring"><span class="id" title="definition">group_ring</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_annihilate"><span class="id" title="lemma">xcfun_annihilate</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#49783294b9f6804d8e1991317fec1b3a"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#49783294b9f6804d8e1991317fec1b3a"><span class="id" title="notation">R_j</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfunG"><span class="id" title="lemma">xcfunG</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">.[</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.aG"><span class="id" title="variable">aG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_mul_id"><span class="id" title="lemma">xcfun_mul_id</span></a> <span class="id" title="var">i</span> <span class="id" title="var">A</span> :<br/> - (<a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.mxalgebra.html#b07e6617bc8db0b83b350e09f8851b51"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.R_G"><span class="id" title="variable">R_G</span></a>)%<span class="id" title="var">MS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">e_i</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="xcfun_id"><span class="id" title="lemma">xcfun_id</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">).[</span></a><a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">e_j</span></a><a class="idref" href="mathcomp.character.character.html#c7fef873721a4be094199a25626252ca"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_free"><span class="id" title="lemma">irr_free</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_inj"><span class="id" title="lemma">irr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrK"><span class="id" title="lemma">irrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.tuple.html#tnth"><span class="id" title="definition">tnth</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>)) (@<a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_eq1"><span class="id" title="lemma">irr_eq1</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_irr_eq1"><span class="id" title="lemma">cforder_irr_eq1</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_basis"><span class="id" title="lemma">irr_basis</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#basis_of"><span class="id" title="definition">basis_of</span></a> <a class="idref" href="mathcomp.character.classfun.html#ade764dfeb1c233c7a245d960646d8b8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#ade764dfeb1c233c7a245d960646d8b8"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#ade764dfeb1c233c7a245d960646d8b8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#ade764dfeb1c233c7a245d960646d8b8"><span class="id" title="notation">)</span></a>%<span class="id" title="var">VS</span> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_sum_nth_irr"><span class="id" title="lemma">eq_sum_nth_irr</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#82d810f9f90b79e8fe98d90a63070c32"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#82d810f9f90b79e8fe98d90a63070c32"><span class="id" title="notation">)`</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#82d810f9f90b79e8fe98d90a63070c32"><span class="id" title="notation">_i</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Theorem (2.8). -</div> -<div class="code"> -<span class="id" title="keyword">Theorem</span> <a name="cfun_irr_sum"><span class="id" title="lemma">cfun_irr_sum</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">a</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_standard"><span class="id" title="lemma">cfRepr_standard</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.character.html#standard_grepr"><span class="id" title="definition">standard_grepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>)<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#standard_irr_coef"><span class="id" title="definition">standard_irr_coef</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> (<a class="idref" href="mathcomp.character.character.html#W"><span class="id" title="abbreviation">W</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_inj"><span class="id" title="lemma">cfRepr_inj</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> <span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> :<br/> - @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_rsimP"><span class="id" title="lemma">cfRepr_rsimP</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> <span class="id" title="var">rG1</span> <span class="id" title="var">rG2</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#mx_rsim"><span class="id" title="inductive">mx_rsim</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>) (@<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> @<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_reprP"><span class="id" title="lemma">irr_reprP</span></a> <span class="id" title="var">xi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">rG</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_irreducible"><span class="id" title="definition">mx_irreducible</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>)<br/> - (<a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Theorem (2.12). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Wedderburn_id_expansion"><span class="id" title="lemma">Wedderburn_id_expansion</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#a2283ad47979c7821fe6d70d159dc0f5"><span class="id" title="notation">e_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#IrrClass.aG"><span class="id" title="variable">aG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#IrrClass"><span class="id" title="section">IrrClass</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IsChar"><span class="id" title="section">IsChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IsChar.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="character"><span class="id" title="definition">character</span></a> {<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>} :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">qualify</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">a</span></a> <span class="id" title="var">phi</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">∀</span></a> <span class="id" title="var">i</span><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.vector.html#coord"><span class="id" title="definition">coord</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.Cnat"><span class="id" title="definition">Cnat</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Fact</span> <a name="character_key"><span class="id" title="lemma">character_key</span></a> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>). <br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">character_keyed</span> <span class="id" title="var">G</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> (<a class="idref" href="mathcomp.character.character.html#character_key"><span class="id" title="lemma">character_key</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="IsChar.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">phi</span> <span class="id" title="var">chi</span> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_char"><span class="id" title="lemma">irr_char</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun1_char"><span class="id" title="lemma">cfun1_char</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun0_char"><span class="id" title="lemma">cfun0_char</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="add_char"><span class="id" title="lemma">add_char</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.addr_closed"><span class="id" title="abbreviation">addr_closed</span></a> (@<a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">character_addrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.AddrPred"><span class="id" title="definition">AddrPred</span></a> <a class="idref" href="mathcomp.character.character.html#add_char"><span class="id" title="lemma">add_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char_sum_irrP"><span class="id" title="lemma">char_sum_irrP</span></a> {<span class="id" title="var">phi</span>} :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>) (<a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char_sum_irr"><span class="id" title="lemma">char_sum_irr</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#706f4f0208bba5d79e26d335c76ea034"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#706f4f0208bba5d79e26d335c76ea034"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#706f4f0208bba5d79e26d335c76ea034"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#706f4f0208bba5d79e26d335c76ea034"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#706f4f0208bba5d79e26d335c76ea034"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Cnat_char1"><span class="id" title="lemma">Cnat_char1</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.Cnat"><span class="id" title="definition">Cnat</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char1_ge0"><span class="id" title="lemma">char1_ge0</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char1_eq0"><span class="id" title="lemma">char1_eq0</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char1_gt0"><span class="id" title="lemma">char1_gt0</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#eb5186e6835d7e27cbb4c691b2f398bb"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char_reprP"><span class="id" title="lemma">char_reprP</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.character.html#representation"><span class="id" title="record">representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>)<br/> - (<a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_char"><span class="id" title="lemma">cfRepr_char</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) : <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfReg_char"><span class="id" title="lemma">cfReg_char</span></a> : <a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_prod"><span class="id" title="lemma">cfRepr_prod</span></a> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> (<span class="id" title="var">rG1</span> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n1"><span class="id" title="variable">n1</span></a>) (<span class="id" title="var">rG2</span> : <a class="idref" href="mathcomp.character.character.html#reprG"><span class="id" title="abbreviation">reprG</span></a> <a class="idref" href="mathcomp.character.character.html#n2"><span class="id" title="variable">n2</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.character.html#prod_repr"><span class="id" title="definition">prod_repr</span></a> <a class="idref" href="mathcomp.character.character.html#rG1"><span class="id" title="variable">rG1</span></a> <a class="idref" href="mathcomp.character.character.html#rG2"><span class="id" title="variable">rG2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mul_char"><span class="id" title="lemma">mul_char</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.mulr_closed"><span class="id" title="abbreviation">mulr_closed</span></a> (@<a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="mathcomp.character.character.html#IsChar.G"><span class="id" title="variable">G</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">char_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.character.character.html#mul_char"><span class="id" title="lemma">mul_char</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">char_semiringPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.SemiringPred"><span class="id" title="definition">SemiringPred</span></a> <a class="idref" href="mathcomp.character.character.html#mul_char"><span class="id" title="lemma">mul_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#IsChar"><span class="id" title="section">IsChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AutChar"><span class="id" title="section">AutChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="AutChar.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="AutChar.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_map"><span class="id" title="lemma">cfRepr_map</span></a> <span class="id" title="var">u</span> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#map_repr"><span class="id" title="definition">map_repr</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_char"><span class="id" title="lemma">cfAut_char</span></a> <span class="id" title="var">u</span> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjC_char"><span class="id" title="lemma">cfConjC_char</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_char1"><span class="id" title="lemma">cfAut_char1</span></a> <span class="id" title="var">u</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_irr1"><span class="id" title="lemma">cfAut_irr1</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> : (<a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a>) 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjC_char1"><span class="id" title="lemma">cfConjC_char1</span></a> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">CF</span> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjC_irr1"><span class="id" title="lemma">cfConjC_irr1</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#AutChar.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">)^*</span></a>%<span class="id" title="var">CF</span> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#AutChar"><span class="id" title="section">AutChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Linear"><span class="id" title="section">Linear</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Linear.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Linear.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="linear_char"><span class="id" title="definition">linear_char</span></a> {<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>} :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">qualify</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">a</span></a> <span class="id" title="var">phi</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Linear.OneChar"><span class="id" title="section">OneChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Linear.OneChar.CFxi"><span class="id" title="variable">CFxi</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char1"><span class="id" title="lemma">lin_char1</span></a>: <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_charW"><span class="id" title="lemma">lin_charW</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun1_lin_char"><span class="id" title="lemma">cfun1_lin_char</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">(</span></a>1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_charM"><span class="id" title="lemma">lin_charM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_prod"><span class="id" title="lemma">lin_char_prod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.character.character.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.gT"><span class="id" title="variable">gT</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ad6525fbcd46e6e7df3fb31cfc13e89c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#edca584f226f01d7a05a12e4ceba1caf"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Linear.OneChar.xiMV"><span class="id" title="variable">xiMV</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_neq0"><span class="id" title="lemma">lin_char_neq0</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_charV"><span class="id" title="lemma">lin_charV</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">)^-1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_charX"><span class="id" title="lemma">lin_charX</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_unity_root"><span class="id" title="lemma">lin_char_unity_root</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="normC_lin_char"><span class="id" title="lemma">normC_lin_char</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_charV_conj"><span class="id" title="lemma">lin_charV_conj</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_irr"><span class="id" title="lemma">lin_char_irr</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mul_conjC_lin_char"><span class="id" title="lemma">mul_conjC_lin_char</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_unitr"><span class="id" title="lemma">lin_char_unitr</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.unit"><span class="id" title="definition">GRing.unit</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="invr_lin_char"><span class="id" title="lemma">invr_lin_char</span></a> : <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="fful_lin_char_inj"><span class="id" title="lemma">fful_lin_char_inj</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Linear.OneChar"><span class="id" title="section">OneChar</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_lin_char"><span class="id" title="lemma">cfAut_lin_char</span></a> <span class="id" title="var">u</span> (<span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjC_lin_char"><span class="id" title="lemma">cfConjC_lin_char</span></a> (<span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_Iirr_abelian"><span class="id" title="lemma">card_Iirr_abelian</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_Iirr_cyclic"><span class="id" title="lemma">card_Iirr_cyclic</span></a> : <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char_abelianP"><span class="id" title="lemma">char_abelianP</span></a> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a>, <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>) (<a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_repr_lin_char"><span class="id" title="lemma">irr_repr_lin_char</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.mxrepresentation.html#irr_repr"><span class="id" title="definition">irr_repr</span></a> (<a class="idref" href="mathcomp.character.character.html#socle_of_Iirr"><span class="id" title="definition">socle_of_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="linear_char_key"><span class="id" title="lemma">linear_char_key</span></a> <span class="id" title="var">B</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred_key"><span class="id" title="inductive">pred_key</span></a> (@<a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>). <br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">linear_char_keted</span> <span class="id" title="var">B</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#KeyedQualifier"><span class="id" title="definition">KeyedQualifier</span></a> (<a class="idref" href="mathcomp.character.character.html#linear_char_key"><span class="id" title="lemma">linear_char_key</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a>).<br/> -<span class="id" title="keyword">Fact</span> <a name="linear_char_divr"><span class="id" title="lemma">linear_char_divr</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.divr_closed"><span class="id" title="abbreviation">divr_closed</span></a> (@<a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lin_char_mulrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.MulrPred"><span class="id" title="definition">MulrPred</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char_divr"><span class="id" title="lemma">linear_char_divr</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">lin_char_divrPred</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Pred.Exports.DivrPred"><span class="id" title="definition">DivrPred</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char_divr"><span class="id" title="lemma">linear_char_divr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_cyclic_lin"><span class="id" title="lemma">irr_cyclic_lin</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_prime_lin"><span class="id" title="lemma">irr_prime_lin</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Linear.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Linear"><span class="id" title="section">Linear</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="OrthogonalityRelations"><span class="id" title="section">OrthogonalityRelations</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="OrthogonalityRelations.aT"><span class="id" title="variable">aT</span></a> <a name="OrthogonalityRelations.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Lemma (2.15) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="repr_rsim_diag"><span class="id" title="lemma">repr_rsim_diag</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">f</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">let</span> <span class="id" title="var">chi</span> := <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <span class="comment">(*a*)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">B</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#invmx"><span class="id" title="definition">invmx</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#diag_mx"><span class="id" title="definition">diag_mx</span></a> <a class="idref" href="mathcomp.character.character.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <span class="comment">(*b*)</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#e"><span class="id" title="variable">e</span></a> 0 <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#e"><span class="id" title="variable">e</span></a> 0 <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <span class="comment">(*c*)</span> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#e"><span class="id" title="variable">e</span></a> 0 <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <span class="comment">(*d*)</span> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="OrthogonalityRelations.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Lemma (2.15) (d). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="char_inv"><span class="id" title="lemma">char_inv</span></a> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_inv"><span class="id" title="lemma">irr_inv</span></a> <span class="id" title="var">i</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Theorem (2.13). -</div> -<div class="code"> -<span class="id" title="keyword">Theorem</span> <a name="generalized_orthogonality_relation"><span class="id" title="lemma">generalized_orthogonality_relation</span></a> <span class="id" title="var">y</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Corollary (2.14). -</div> -<div class="code"> -<span class="id" title="keyword">Corollary</span> <a name="first_orthogonality_relation"><span class="id" title="lemma">first_orthogonality_relation</span></a> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - The character table. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="irr_class"><span class="id" title="definition">irr_class</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#enum_val"><span class="id" title="definition">enum_val</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#cast_ord"><span class="id" title="definition">cast_ord</span></a> (<a class="idref" href="mathcomp.character.character.html#NirrE"><span class="id" title="lemma">NirrE</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="class_Iirr"><span class="id" title="definition">class_Iirr</span></a> <span class="id" title="var">xG</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#cast_ord"><span class="id" title="definition">cast_ord</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#esym"><span class="id" title="definition">esym</span></a> (<a class="idref" href="mathcomp.character.character.html#NirrE"><span class="id" title="lemma">NirrE</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>)) (<a class="idref" href="mathcomp.ssreflect.fintype.html#enum_rank_in"><span class="id" title="definition">enum_rank_in</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#classes1"><span class="id" title="lemma">classes1</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#xG"><span class="id" title="variable">xG</span></a>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="character_table"><span class="id" title="definition">character_table</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">matrix_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">,</span></a> <span class="id" title="var">j</span><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> (<a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="abbreviation">g</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_classP"><span class="id" title="lemma">irr_classP</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="abbreviation">c</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="repr_irr_classK"><span class="id" title="lemma">repr_irr_classK</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="abbreviation">g</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#38a288b01c62a2a6a720c34fc1fffe2c"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="abbreviation">c</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_classK"><span class="id" title="lemma">irr_classK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="abbreviation">c</span></a> <a class="idref" href="mathcomp.character.character.html#iC"><span class="id" title="abbreviation">iC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="class_IirrK"><span class="id" title="lemma">class_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#iC"><span class="id" title="abbreviation">iC</span></a> <a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="abbreviation">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reindex_irr_class"><span class="id" title="lemma">reindex_irr_class</span></a> <span class="id" title="var">R</span> <span class="id" title="var">idx</span> (<span class="id" title="var">op</span> : @<a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.com_law"><span class="id" title="record">Monoid.com_law</span></a> <a class="idref" href="mathcomp.character.character.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.character.html#idx"><span class="id" title="variable">idx</span></a>) <span class="id" title="var">F</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">(</span></a><span class="id" title="var">xG</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.character.character.html#xG"><span class="id" title="variable">xG</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#op"><span class="id" title="variable">op</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">/</span></a><a class="idref" href="mathcomp.character.character.html#idx"><span class="id" title="variable">idx</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#379a79a86133b2d1cd9cb43efa183ecb"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="abbreviation">c</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - The explicit value of the inverse is needed for the proof of the second - orthogonality relation. -</div> -<div class="code"> -<span class="id" title="keyword">Let</span> <a name="OrthogonalityRelations.X'"><span class="id" title="variable">X'</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">matrix_</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">,</span></a> <span class="id" title="var">j</span><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="abbreviation">g</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_j</span></a> (<a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="abbreviation">g</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a><a class="idref" href="mathcomp.algebra.matrix.html#9b7ac910045fe3e3a8253dae2e2bc494"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="OrthogonalityRelations.XX'_1"><span class="id" title="variable">XX'_1</span></a>: <a class="idref" href="mathcomp.character.character.html#X"><span class="id" title="abbreviation">X</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.algebra.matrix.html#b2b431de65e6c1e23c1ae3a60262ea15"><span class="id" title="notation">m</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.X'"><span class="id" title="variable">X'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="character_table_unit"><span class="id" title="lemma">character_table_unit</span></a> : <a class="idref" href="mathcomp.character.character.html#X"><span class="id" title="abbreviation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#unitmx"><span class="id" title="definition">unitmx</span></a>.<br/> - <span class="id" title="keyword">Let</span> <a name="OrthogonalityRelations.uX"><span class="id" title="variable">uX</span></a> := <a class="idref" href="mathcomp.character.character.html#character_table_unit"><span class="id" title="lemma">character_table_unit</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Theorem (2.18). -</div> -<div class="code"> -<span class="id" title="keyword">Theorem</span> <a name="second_orthogonality_relation"><span class="id" title="lemma">second_orthogonality_relation</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <br/> - <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#38a288b01c62a2a6a720c34fc1fffe2c"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_irr_mem_classP"><span class="id" title="lemma">eq_irr_mem_classP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#38a288b01c62a2a6a720c34fc1fffe2c"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Theorem (6.32) (due to Brauer). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="card_afix_irr_classes"><span class="id" title="lemma">card_afix_irr_classes</span></a> (<span class="id" title="var">ito</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>)) (<span class="id" title="var">cto</span> : <a class="idref" href="mathcomp.fingroup.action.html#action"><span class="id" title="record">action</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.A"><span class="id" title="variable">A</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">a</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#cto"><span class="id" title="variable">cto</span></a><a class="idref" href="mathcomp.fingroup.action.html#915e8fb7bcea89ddadab4deff6ea659e"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#cto"><span class="id" title="variable">cto</span></a> (<a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#38a288b01c62a2a6a720c34fc1fffe2c"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#ito"><span class="id" title="variable">ito</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#2f26cd6b59b1ae1fb2356f20ddb57cf1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#2f26cd6b59b1ae1fb2356f20ddb57cf1"><span class="id" title="notation">Fix_ito</span></a><a class="idref" href="mathcomp.fingroup.action.html#2f26cd6b59b1ae1fb2356f20ddb57cf1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#2f26cd6b59b1ae1fb2356f20ddb57cf1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">Fix_</span></a><a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#cto"><span class="id" title="variable">cto</span></a><a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.action.html#2bb94d3d01f9b00932888c009233b945"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#OrthogonalityRelations"><span class="id" title="section">OrthogonalityRelations</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="InnerProduct"><span class="id" title="section">InnerProduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="InnerProduct.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="InnerProduct.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_irr"><span class="id" title="lemma">cfdot_irr</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_irr"><span class="id" title="lemma">cfnorm_irr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_orthonormal"><span class="id" title="lemma">irr_orthonormal</span></a> : <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="coord_cfdot"><span class="id" title="lemma">coord_cfdot</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.algebra.vector.html#coord"><span class="id" title="definition">coord</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_sum_cfdot"><span class="id" title="lemma">cfun_sum_cfdot</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_sum_irr"><span class="id" title="lemma">cfdot_sum_irr</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Cnat_cfdot_char_irr"><span class="id" title="lemma">Cnat_cfdot_char_irr</span></a> <span class="id" title="var">i</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.Cnat"><span class="id" title="definition">Cnat</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_char_r"><span class="id" title="lemma">cfdot_char_r</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">chi</span> : <br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Cnat_cfdot_char"><span class="id" title="lemma">Cnat_cfdot_char</span></a> <span class="id" title="var">chi</span> <span class="id" title="var">xi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.Cnat"><span class="id" title="definition">Cnat</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdotC_char"><span class="id" title="lemma">cfdotC_char</span></a> <span class="id" title="var">chi</span> <span class="id" title="var">xi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrEchar"><span class="id" title="lemma">irrEchar</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrWchar"><span class="id" title="lemma">irrWchar</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irrWnorm"><span class="id" title="lemma">irrWnorm</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mul_lin_irr"><span class="id" title="lemma">mul_lin_irr</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_scaled_irr"><span class="id" title="lemma">eq_scaled_irr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_signed_irr"><span class="id" title="lemma">eq_signed_irr</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a>-1<a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_scale_irr"><span class="id" title="lemma">eq_scale_irr</span></a> <span class="id" title="var">a</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_addZ_irr"><span class="id" title="lemma">eq_addZ_irr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="var">r</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">[||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b9d6728f52abb7d090bb2a83817306bd"><span class="id" title="notation">[&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b9d6728f52abb7d090bb2a83817306bd"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b9d6728f52abb7d090bb2a83817306bd"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#6e3f5b731a46299b833a2834f381d536"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_subZnat_irr"><span class="id" title="lemma">eq_subZnat_irr</span></a> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="var">r</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#InnerProduct.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_r</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">[||</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">[||</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1d455c3124a700217b6ab4d86cca79e1"><span class="id" title="notation">]</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#InnerProduct"><span class="id" title="section">InnerProduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IrrConstt"><span class="id" title="section">IrrConstt</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="IrrConstt.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="IrrConstt.G"><span class="id" title="variable">G</span></a> <a name="IrrConstt.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char1_ge_norm"><span class="id" title="lemma">char1_ge_norm</span></a> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="max_cfRepr_norm_scalar"><span class="id" title="lemma">max_cfRepr_norm_scalar</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="max_cfRepr_mx1"><span class="id" title="lemma">max_cfRepr_mx1</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1<a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="irr_constt"><span class="id" title="definition">irr_constt</span></a> (<span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">phi</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_consttE"><span class="id" title="lemma">irr_consttE</span></a> <span class="id" title="var">i</span> <span class="id" title="var">phi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">_G</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_charP"><span class="id" title="lemma">constt_charP</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a>) <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">chi'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#chi'"><span class="id" title="variable">chi'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#chi'"><span class="id" title="variable">chi'</span></a>)<br/> - (<a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_sum_constt"><span class="id" title="lemma">cfun_sum_constt</span></a> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="neq0_has_constt"><span class="id" title="lemma">neq0_has_constt</span></a> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_irr"><span class="id" title="lemma">constt_irr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char1_ge_constt"><span class="id" title="lemma">char1_ge_constt</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a>) <span class="id" title="var">chi</span> : <br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_ortho_char"><span class="id" title="lemma">constt_ortho_char</span></a> (<span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#IrrConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#IrrConstt"><span class="id" title="section">IrrConstt</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Kernel"><span class="id" title="section">Kernel</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Kernel.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Kernel.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">phi</span> <span class="id" title="var">chi</span> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_repr"><span class="id" title="lemma">cfker_repr</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) : <br/> - <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#rker"><span class="id" title="definition">rker</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfkerEchar"><span class="id" title="lemma">cfkerEchar</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_nzcharE"><span class="id" title="lemma">cfker_nzcharE</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfkerEirr"><span class="id" title="lemma">cfkerEirr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">set</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#9e3f1d0cf47c39e3927b1f03a0797327"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_irr0"><span class="id" title="lemma">cfker_irr0</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfaithful_reg"><span class="id" title="lemma">cfaithful_reg</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> (<a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfkerE"><span class="id" title="lemma">cfkerE</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#17be462b372c78f64262e282e98f2452"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#17be462b372c78f64262e282e98f2452"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#17be462b372c78f64262e282e98f2452"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#17be462b372c78f64262e282e98f2452"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#17be462b372c78f64262e282e98f2452"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="TI_cfker_irr"><span class="id" title="lemma">TI_cfker_irr</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#4ba1db7734680ecf1e41cd64093d9c3e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#4ba1db7734680ecf1e41cd64093d9c3e"><span class="id" title="notation">bigcap_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#b54f5e35cb228bba5934c852e0951c39"><span class="id" title="notation">[1]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_constt"><span class="id" title="lemma">cfker_constt</span></a> <span class="id" title="var">i</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Kernel.KerLin"><span class="id" title="section">KerLin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Kernel.KerLin.lin_xi"><span class="id" title="variable">lin_xi</span></a> : <a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Kernel.KerLin.Nxi"><span class="id" title="variable">Nxi</span></a>: <a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_der1"><span class="id" title="lemma">lin_char_der1</span></a> : <a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_lin_char"><span class="id" title="lemma">cforder_lin_char</span></a> : <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#exponent"><span class="id" title="definition">exponent</span></a> (<a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_lin_char_dvdG"><span class="id" title="lemma">cforder_lin_char_dvdG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_lin_char_gt0"><span class="id" title="lemma">cforder_lin_char_gt0</span></a> : (0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#Kernel.KerLin.xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Kernel.KerLin"><span class="id" title="section">KerLin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Kernel"><span class="id" title="section">Kernel</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Restrict"><span class="id" title="section">Restrict</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Restrict.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Restrict.G"><span class="id" title="variable">G</span></a> <a name="Restrict.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_sub"><span class="id" title="lemma">cfRepr_sub</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) (<span class="id" title="var">sHG</span> : <a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#subg_repr"><span class="id" title="definition">subg_repr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#sHG"><span class="id" title="variable">sHG</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_char"><span class="id" title="lemma">cfRes_char</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_eq0"><span class="id" title="lemma">cfRes_eq0</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_lin_char"><span class="id" title="lemma">cfRes_lin_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Res_irr_neq0"><span class="id" title="lemma">Res_irr_neq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_lin_lin"><span class="id" title="lemma">cfRes_lin_lin</span></a> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_irr_irr"><span class="id" title="lemma">cfRes_irr_irr</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Res_Iirr0"><span class="id" title="lemma">Res_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_Res_IirrE"><span class="id" title="lemma">lin_Res_IirrE</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Restrict.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Restrict.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Restrict"><span class="id" title="section">Restrict</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MoreConstt"><span class="id" title="section">MoreConstt</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MoreConstt.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MoreConstt.G"><span class="id" title="variable">G</span></a> <a name="MoreConstt.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_Ind_Res"><span class="id" title="lemma">constt_Ind_Res</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_Res_ge_constt"><span class="id" title="lemma">cfdot_Res_ge_constt</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <br/> - <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#MoreConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_Res_trans"><span class="id" title="lemma">constt_Res_trans</span></a> <span class="id" title="var">j</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#MoreConstt.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#MoreConstt.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#MoreConstt"><span class="id" title="section">MoreConstt</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Morphim"><span class="id" title="section">Morphim</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Morphim.aT"><span class="id" title="variable">aT</span></a> <a name="Morphim.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Morphim.G"><span class="id" title="variable">G</span></a> <a name="Morphim.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="Morphim.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.character.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRepr_morphim"><span class="id" title="lemma">cfRepr_morphim</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rfG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> (<a class="idref" href="mathcomp.character.character.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) <span class="id" title="var">sGD</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> (<a class="idref" href="mathcomp.character.mxrepresentation.html#morphim_repr"><span class="id" title="definition">morphim_repr</span></a> <a class="idref" href="mathcomp.character.character.html#rfG"><span class="id" title="variable">rfG</span></a> <a class="idref" href="mathcomp.character.character.html#sGD"><span class="id" title="variable">sGD</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> (<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rfG"><span class="id" title="variable">rfG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_char"><span class="id" title="lemma">cfMorph_char</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_lin_char"><span class="id" title="lemma">cfMorph_lin_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_charE"><span class="id" title="lemma">cfMorph_charE</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_lin_charE"><span class="id" title="lemma">cfMorph_lin_charE</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_irr"><span class="id" title="lemma">cfMorph_irr</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> (<a class="idref" href="mathcomp.character.character.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="morph_Iirr"><span class="id" title="definition">morph_Iirr</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_Iirr0"><span class="id" title="lemma">morph_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#morph_Iirr"><span class="id" title="definition">morph_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Morphim.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.character.character.html#Morphim.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Morphim.D"><span class="id" title="variable">D</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_IirrE"><span class="id" title="lemma">morph_IirrE</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#morph_Iirr"><span class="id" title="definition">morph_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_Iirr_inj"><span class="id" title="lemma">morph_Iirr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#morph_Iirr"><span class="id" title="definition">morph_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morph_Iirr_eq0"><span class="id" title="lemma">morph_Iirr_eq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#morph_Iirr"><span class="id" title="definition">morph_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Morphim"><span class="id" title="section">Morphim</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Isom"><span class="id" title="section">Isom</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Isom.aT"><span class="id" title="variable">aT</span></a> <a name="Isom.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Isom.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="Isom.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="Isom.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="Isom.isoGR"><span class="id" title="variable">isoGR</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.f"><span class="id" title="variable">f</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Isom.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_char"><span class="id" title="lemma">cfIsom_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_lin_char"><span class="id" title="lemma">cfIsom_lin_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_irr"><span class="id" title="lemma">cfIsom_irr</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_IirrE"><span class="id" title="lemma">isom_IirrE</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#Isom.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_Iirr_inj"><span class="id" title="lemma">isom_Iirr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_Iirr_eq0"><span class="id" title="lemma">isom_Iirr_eq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_Iirr0"><span class="id" title="lemma">isom_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Isom"><span class="id" title="section">Isom</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IsomInv"><span class="id" title="section">IsomInv</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="IsomInv.aT"><span class="id" title="variable">aT</span></a> <a name="IsomInv.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="IsomInv.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="IsomInv.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="IsomInv.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="IsomInv.isoGR"><span class="id" title="variable">isoGR</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.f"><span class="id" title="variable">f</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_IirrK"><span class="id" title="lemma">isom_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.isoGR"><span class="id" title="variable">isoGR</span></a>) (<a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#isom_sym"><span class="id" title="lemma">isom_sym</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.isoGR"><span class="id" title="variable">isoGR</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isom_IirrKV"><span class="id" title="lemma">isom_IirrKV</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#isom_sym"><span class="id" title="lemma">isom_sym</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.isoGR"><span class="id" title="variable">isoGR</span></a>)) (<a class="idref" href="mathcomp.character.character.html#isom_Iirr"><span class="id" title="definition">isom_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#IsomInv.isoGR"><span class="id" title="variable">isoGR</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#IsomInv"><span class="id" title="section">IsomInv</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Sdprod"><span class="id" title="section">Sdprod</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Sdprod.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Sdprod.K"><span class="id" title="variable">K</span></a> <a name="Sdprod.H"><span class="id" title="variable">H</span></a> <a name="Sdprod.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Sdprod.defG"><span class="id" title="variable">defG</span></a> : <a class="idref" href="mathcomp.character.character.html#Sdprod.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#4c7b411f14f1faa861c7c0ade82faf76"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.G"><span class="id" title="variable">G</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Sdprod.nKG"><span class="id" title="variable">nKG</span></a>: <a class="idref" href="mathcomp.character.character.html#Sdprod.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Sdprod.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_char"><span class="id" title="lemma">cfSdprod_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_lin_char"><span class="id" title="lemma">cfSdprod_lin_char</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_irr"><span class="id" title="lemma">cfSdprod_irr</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> <span class="id" title="var">j</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_IirrE"><span class="id" title="lemma">sdprod_IirrE</span></a> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_IirrK"><span class="id" title="lemma">sdprod_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Iirr_inj"><span class="id" title="lemma">sdprod_Iirr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Iirr_eq0"><span class="id" title="lemma">sdprod_Iirr_eq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Iirr0"><span class="id" title="lemma">sdprod_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Res_sdprod_irr"><span class="id" title="lemma">Res_sdprod_irr</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Sdprod.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Res_IirrE"><span class="id" title="lemma">sdprod_Res_IirrE</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Sdprod.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Sdprod.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_Res_IirrK"><span class="id" title="lemma">sdprod_Res_IirrK</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Sdprod.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#sdprod_Iirr"><span class="id" title="definition">sdprod_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Sdprod.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Sdprod"><span class="id" title="section">Sdprod</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DProd"><span class="id" title="section">DProd</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="DProd.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="DProd.G"><span class="id" title="variable">G</span></a> <a name="DProd.K"><span class="id" title="variable">K</span></a> <a name="DProd.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="DProd.KxH"><span class="id" title="variable">KxH</span></a> : <a class="idref" href="mathcomp.character.character.html#DProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodKl_abelian"><span class="id" title="lemma">cfDprodKl_abelian</span></a> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">)^~</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>) <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodKr_abelian"><span class="id" title="lemma">cfDprodKr_abelian</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>) <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodl_char"><span class="id" title="lemma">cfDprodl_char</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodr_char"><span class="id" title="lemma">cfDprodr_char</span></a> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_char"><span class="id" title="lemma">cfDprod_char</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_eq1"><span class="id" title="lemma">cfDprod_eq1</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodl_lin_char"><span class="id" title="lemma">cfDprodl_lin_char</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodr_lin_char"><span class="id" title="lemma">cfDprodr_lin_char</span></a> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_lin_char"><span class="id" title="lemma">cfDprod_lin_char</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodl_irr"><span class="id" title="lemma">cfDprodl_irr</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodr_irr"><span class="id" title="lemma">cfDprodr_irr</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprodl_IirrE"><span class="id" title="lemma">dprodl_IirrE</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodl_IirrK"><span class="id" title="lemma">dprodl_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.K"><span class="id" title="variable">K</span></a>).<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodl_Iirr_eq0"><span class="id" title="lemma">dprodl_Iirr_eq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodl_Iirr0"><span class="id" title="lemma">dprodl_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> <span class="id" title="var">j</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprodr_IirrE"><span class="id" title="lemma">dprodr_IirrE</span></a> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodr_IirrK"><span class="id" title="lemma">dprodr_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#Res_Iirr"><span class="id" title="definition">Res_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.H"><span class="id" title="variable">H</span></a>).<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodr_Iirr_eq0"><span class="id" title="lemma">dprodr_Iirr_eq0</span></a> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="dprodr_Iirr0"><span class="id" title="lemma">dprodr_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_irr"><span class="id" title="lemma">cfDprod_irr</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <span class="id" title="var">ij</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_ij</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_ij</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_IirrE"><span class="id" title="lemma">dprod_IirrE</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_IirrEl"><span class="id" title="lemma">dprod_IirrEl</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_IirrEr"><span class="id" title="lemma">dprod_IirrEr</span></a> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.character.html#DProd.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr_inj"><span class="id" title="lemma">dprod_Iirr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr0"><span class="id" title="lemma">dprod_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr0l"><span class="id" title="lemma">dprod_Iirr0l</span></a> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#dprodr_Iirr"><span class="id" title="definition">dprodr_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr0r"><span class="id" title="lemma">dprod_Iirr0r</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#dprodl_Iirr"><span class="id" title="definition">dprodl_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr_eq0"><span class="id" title="lemma">dprod_Iirr_eq0</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_dprod_irr"><span class="id" title="lemma">cfdot_dprod_irr</span></a> <span class="id" title="var">i1</span> <span class="id" title="var">i2</span> <span class="id" title="var">j1</span> <span class="id" title="var">j2</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i1"><span class="id" title="variable">i1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j1"><span class="id" title="variable">j1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i2"><span class="id" title="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j2"><span class="id" title="variable">j2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i1"><span class="id" title="variable">i1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#i2"><span class="id" title="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j1"><span class="id" title="variable">j1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#j2"><span class="id" title="variable">j2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_Iirr_onto"><span class="id" title="lemma">dprod_Iirr_onto</span></a> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#codom"><span class="id" title="definition">codom</span></a> <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="inv_dprod_Iirr"><span class="id" title="definition">inv_dprod_Iirr</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#iinv"><span class="id" title="definition">iinv</span></a> (<a class="idref" href="mathcomp.character.character.html#dprod_Iirr_onto"><span class="id" title="lemma">dprod_Iirr_onto</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_IirrK"><span class="id" title="lemma">dprod_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#inv_dprod_Iirr"><span class="id" title="definition">inv_dprod_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="inv_dprod_IirrK"><span class="id" title="lemma">inv_dprod_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.character.character.html#inv_dprod_Iirr"><span class="id" title="definition">inv_dprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="inv_dprod_Iirr0"><span class="id" title="lemma">inv_dprod_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#inv_dprod_Iirr"><span class="id" title="definition">inv_dprod_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#DProd"><span class="id" title="section">DProd</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dprod_IirrC"><span class="id" title="lemma">dprod_IirrC</span></a> (<span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>)<br/> - (<span class="id" title="var">KxH</span> : <a class="idref" href="mathcomp.character.character.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>) (<span class="id" title="var">HxK</span> : <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.character.character.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>) <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#dprod_Iirr"><span class="id" title="definition">dprod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#HxK"><span class="id" title="variable">HxK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="BigDprod"><span class="id" title="section">BigDprod</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="BigDprod.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="BigDprod.I"><span class="id" title="variable">I</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="BigDprod.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.character.character.html#I"><span class="id" title="variable">I</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="BigDprod.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.character.character.html#BigDprod.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="BigDprod.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="BigDprod.defG"><span class="id" title="variable">defG</span></a> : <a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">/</span></a>1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#1871917561e26284874cb982a8cc32df"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="BigDprod.sAG"><span class="id" title="variable">sAG</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_char"><span class="id" title="lemma">cfBigdprodi_char</span></a> <span class="id" title="var">i</span> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_charE"><span class="id" title="lemma">cfBigdprodi_charE</span></a> <span class="id" title="var">i</span> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod_char"><span class="id" title="lemma">cfBigdprod_char</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_lin_char"><span class="id" title="lemma">cfBigdprodi_lin_char</span></a> <span class="id" title="var">i</span> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_lin_charE"><span class="id" title="lemma">cfBigdprodi_lin_charE</span></a> <span class="id" title="var">i</span> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod_lin_char"><span class="id" title="lemma">cfBigdprod_lin_char</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_irr"><span class="id" title="lemma">cfBigdprodi_irr</span></a> <span class="id" title="var">i</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> (<a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod_irr"><span class="id" title="lemma">cfBigdprod_irr</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> (<a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod_eq1"><span class="id" title="lemma">cfBigdprod_eq1</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">∀</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="mathcomp.ssreflect.fintype.html#90a753c4c9a43b6ba4178e7bc1e47801"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod_Res_lin"><span class="id" title="lemma">cfBigdprod_Res_lin</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.defG"><span class="id" title="variable">defG</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#BigDprod.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodKlin"><span class="id" title="lemma">cfBigdprodKlin</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">defG</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodKabelian"><span class="id" title="lemma">cfBigdprodKabelian</span></a> <span class="id" title="var">Iphi</span> (<span class="id" title="var">phi</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Iphi"><span class="id" title="variable">Iphi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.character.html#BigDprod.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.character.character.html#BigDprod.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">defG</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Iphi"><span class="id" title="variable">Iphi</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#BigDprod"><span class="id" title="section">BigDprod</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Aut"><span class="id" title="section">Aut</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Aut.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Aut.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d531732ed602c7af62b88c7cfce824e5"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_charAut"><span class="id" title="lemma">conjC_charAut</span></a> <span class="id" title="var">u</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_irrAut"><span class="id" title="lemma">conjC_irrAut</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_aut_char"><span class="id" title="lemma">cfdot_aut_char</span></a> <span class="id" title="var">u</span> (<span class="id" title="var">phi</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) : <br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_aut_irr"><span class="id" title="lemma">cfdot_aut_irr</span></a> <span class="id" title="var">u</span> <span class="id" title="var">phi</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_irr"><span class="id" title="lemma">cfAut_irr</span></a> <span class="id" title="var">u</span> <span class="id" title="var">chi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjC_irr"><span class="id" title="lemma">cfConjC_irr</span></a> <span class="id" title="var">i</span> : (<a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">)^*</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_aut_closed"><span class="id" title="lemma">irr_aut_closed</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.character.classfun.html#cfAut_closed"><span class="id" title="definition">cfAut_closed</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> (<a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="aut_IirrE"><span class="id" title="lemma">aut_IirrE</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="conjC_Iirr"><span class="id" title="definition">conjC_Iirr</span></a> := <a class="idref" href="mathcomp.character.character.html#aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_IirrE"><span class="id" title="lemma">conjC_IirrE</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#conjC_Iirr"><span class="id" title="definition">conjC_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">)^*</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_IirrK"><span class="id" title="lemma">conjC_IirrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#involutive"><span class="id" title="definition">involutive</span></a> <a class="idref" href="mathcomp.character.character.html#conjC_Iirr"><span class="id" title="definition">conjC_Iirr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="aut_Iirr0"><span class="id" title="lemma">aut_Iirr0</span></a> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.character.character.html#aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_Iirr0"><span class="id" title="lemma">conjC_Iirr0</span></a> : <a class="idref" href="mathcomp.character.character.html#conjC_Iirr"><span class="id" title="definition">conjC_Iirr</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="aut_Iirr_eq0"><span class="id" title="lemma">aut_Iirr_eq0</span></a> <span class="id" title="var">u</span> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_Iirr_eq0"><span class="id" title="lemma">conjC_Iirr_eq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#conjC_Iirr"><span class="id" title="definition">conjC_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#28a3089bb29d95d7bdc98c2c73b31552"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#28a3089bb29d95d7bdc98c2c73b31552"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Aut.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="aut_Iirr_inj"><span class="id" title="lemma">aut_Iirr_inj</span></a> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.character.character.html#aut_Iirr"><span class="id" title="definition">aut_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Aut"><span class="id" title="section">Aut</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Coset"><span class="id" title="section">Coset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Coset.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#Coset.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_char"><span class="id" title="lemma">cfQuo_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_lin_char"><span class="id" title="lemma">cfQuo_lin_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_char"><span class="id" title="lemma">cfMod_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_lin_char"><span class="id" title="lemma">cfMod_lin_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_charE"><span class="id" title="lemma">cfMod_charE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_lin_charE"><span class="id" title="lemma">cfMod_lin_charE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_charE"><span class="id" title="lemma">cfQuo_charE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_lin_charE"><span class="id" title="lemma">cfQuo_lin_charE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_irr"><span class="id" title="lemma">cfMod_irr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> (<a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mod_Iirr0"><span class="id" title="lemma">mod_Iirr0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mod_IirrE"><span class="id" title="lemma">mod_IirrE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mod_Iirr_eq0"><span class="id" title="lemma">mod_Iirr_eq0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#28a3089bb29d95d7bdc98c2c73b31552"><span class="id" title="notation">==</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#28a3089bb29d95d7bdc98c2c73b31552"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_irr"><span class="id" title="lemma">cfQuo_irr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>(<a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> (<a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr"><span class="id" title="definition">irr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quo_Iirr0"><span class="id" title="lemma">quo_Iirr0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> (0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quo_IirrE"><span class="id" title="lemma">quo_IirrE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quo_Iirr_eq0"><span class="id" title="lemma">quo_Iirr_eq0</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mod_IirrK"><span class="id" title="lemma">mod_IirrK</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (@<a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>) (@<a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quo_IirrK"><span class="id" title="lemma">quo_IirrK</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quo_IirrKeq"><span class="id" title="lemma">quo_IirrKeq</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> (<a class="idref" href="mathcomp.character.character.html#quo_Iirr"><span class="id" title="definition">quo_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mod_Iirr_bij"><span class="id" title="lemma">mod_Iirr_bij</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">on</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">bijective</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.character.character.html#mod_Iirr"><span class="id" title="definition">mod_Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7a12faf7e5f15d58335f3e4b34342ab7"><span class="id" title="notation">)}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sum_norm_irr_quo"><span class="id" title="lemma">sum_norm_irr_quo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#de3e30c288f66ee879ea2b40e81e186c"><span class="id" title="notation">sum_i</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> 2<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f43f2e9c8e0cc7a634fe022790373569"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> 2.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cap_cfker_normal"><span class="id" title="lemma">cap_cfker_normal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_reg_quo"><span class="id" title="lemma">cfker_reg_quo</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.character.html#cfReg"><span class="id" title="definition">cfReg</span></a> (<a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Coset"><span class="id" title="section">Coset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DerivedGroup"><span class="id" title="section">DerivedGroup</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="DerivedGroup.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#DerivedGroup.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lin_irr_der1"><span class="id" title="lemma">lin_irr_der1</span></a> <span class="id" title="var">G</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subGcfker"><span class="id" title="lemma">subGcfker</span></a> <span class="id" title="var">G</span> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_prime_injP"><span class="id" title="lemma">irr_prime_injP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0).<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.23)(a). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cap_cfker_lin_irr"><span class="id" title="lemma">cap_cfker_lin_irr</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ed4f70cb75d6b4771a0be60d14037c7b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.23)(b) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="card_lin_irr"><span class="id" title="lemma">card_lin_irr</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a>.<br/> -</div> - -<div class="doc"> - Alternative: use the equivalent result in modular representation theory -transitivity #|@socle_of_Iirr _ G @^-1: linear_irr _|; last first. - rewrite (on_card_preimset (socle_of_Iirr_bij _)). - by rewrite card_linear_irr ?algC'G; last apply: groupC. -by apply: eq_card => i; rewrite !inE /lin_char irr_char irr1_degree -eqC_nat. - -</div> -<div class="code"> - -<br/> -</div> - -<div class="doc"> - A non-trivial solvable group has a nonprincipal linear character. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="solvable_has_lin_char"><span class="id" title="lemma">solvable_has_lin_char</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 1.<br/> - -<br/> -</div> - -<div class="doc"> - A combinatorial group isommorphic to the linear characters. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="lin_char_group"><span class="id" title="lemma">lin_char_group</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><span class="id" title="var">linG</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">cF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#linG"><span class="id" title="variable">linG</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">|</span></a> <br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#linG"><span class="id" title="variable">linG</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">u</span>, <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span>, <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1%<span class="id" title="var">R</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">:</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#v"><span class="id" title="variable">v</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">>-></span></a> (<a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#v"><span class="id" title="variable">v</span></a>)%<span class="id" title="var">R</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">k</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">:</span></a> <span class="id" title="var">u</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">/</span></a> (<a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">:</span></a> <span class="id" title="var">u</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#766fd55608aa0e125ed6f55c83bcc09a"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#4e5a4c91ec0aa12de06dfe1cc07ea126"><span class="id" title="notation">^-1</span></a>%<span class="id" title="var">CF</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">}</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">mono</span></a> <a class="idref" href="mathcomp.character.character.html#cF"><span class="id" title="variable">cF</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">:</span></a> <span class="id" title="var">u</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#96bead841b87c72f1ae1be942b541e72"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfExp_prime_transitive"><span class="id" title="lemma">cfExp_prime_transitive</span></a> <span class="id" title="var">G</span> (<span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">k</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.24). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="card_subcent1_coset"><span class="id" title="lemma">card_subcent1_coset</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#53eecfec39d0fcb81098259cffd7ca5c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#53eecfec39d0fcb81098259cffd7ca5c"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#53eecfec39d0fcb81098259cffd7ca5c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#53eecfec39d0fcb81098259cffd7ca5c"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#53eecfec39d0fcb81098259cffd7ca5c"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#DerivedGroup"><span class="id" title="section">DerivedGroup</span></a>.<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - Determinant characters and determinential order. -</div> -<div class="code"> -<span class="id" title="keyword">Section</span> <a name="DetOrder"><span class="id" title="section">DetOrder</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="DetOrder.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="DetOrder.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DetOrder.DetRepr"><span class="id" title="section">DetRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="DetOrder.DetRepr.n"><span class="id" title="variable">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="DetOrder.DetRepr.rG"><span class="id" title="variable">rG</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#be36f4c61e9a82f836d531a63f34e6c2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#be36f4c61e9a82f836d531a63f34e6c2"><span class="id" title="notation">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#be36f4c61e9a82f836d531a63f34e6c2"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#be36f4c61e9a82f836d531a63f34e6c2"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#DetOrder.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="det_repr_mx"><span class="id" title="definition">det_repr_mx</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.matrix.html#2a5412586d59ba16d2c60c55e120c7ee"><span class="id" title="notation">M_1</span></a> := <a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6b0b476e80941db501b42be981d1cdf0"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6b0b476e80941db501b42be981d1cdf0"><span class="id" title="notation">det</span></a> <a class="idref" href="mathcomp.algebra.matrix.html#6b0b476e80941db501b42be981d1cdf0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#DetOrder.DetRepr.rG"><span class="id" title="variable">rG</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.matrix.html#6b0b476e80941db501b42be981d1cdf0"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.matrix.html#850c060d75891e97ece38bfec139b8ea"><span class="id" title="notation">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="det_is_repr"><span class="id" title="lemma">det_is_repr</span></a> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_repr"><span class="id" title="definition">mx_repr</span></a> <a class="idref" href="mathcomp.character.character.html#DetOrder.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#det_repr_mx"><span class="id" title="definition">det_repr_mx</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">det_repr</span> := <a class="idref" href="mathcomp.character.mxrepresentation.html#MxRepresentation"><span class="id" title="constructor">MxRepresentation</span></a> <a class="idref" href="mathcomp.character.character.html#det_is_repr"><span class="id" title="lemma">det_is_repr</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="detRepr"><span class="id" title="definition">detRepr</span></a> := <a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#det_repr"><span class="id" title="definition">det_repr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="detRepr_lin_char"><span class="id" title="lemma">detRepr_lin_char</span></a> : <a class="idref" href="mathcomp.character.character.html#detRepr"><span class="id" title="definition">detRepr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#DetOrder.DetRepr"><span class="id" title="section">DetRepr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDet"><span class="id" title="definition">cfDet</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#bba809eef925bb2b4e421c8b99ce8372"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bba809eef925bb2b4e421c8b99ce8372"><span class="id" title="notation">prod_i</span></a> <a class="idref" href="mathcomp.character.character.html#detRepr"><span class="id" title="definition">detRepr</span></a> <a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#862ee9dddb7503f68e86ae1b7d9d8242"><span class="id" title="notation">Chi_i</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.truncC"><span class="id" title="definition">truncC</span></a> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#DetOrder.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDet_lin_char"><span class="id" title="lemma">cfDet_lin_char</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetD"><span class="id" title="lemma">cfDetD</span></a> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">:</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#40d800f6f36c47cb5f4f2f42555867a8"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDet0"><span class="id" title="lemma">cfDet0</span></a> : <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetMn"><span class="id" title="lemma">cfDetMn</span></a> <span class="id" title="var">k</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">:</span></a> <span class="id" title="var">phi</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.character.character.html#k"><span class="id" title="variable">k</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8bf6fdbe8b0c22b67e58fa5cd9937190"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetRepr"><span class="id" title="lemma">cfDetRepr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> (<a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> @<a class="idref" href="mathcomp.character.character.html#detRepr"><span class="id" title="definition">detRepr</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDet_id"><span class="id" title="lemma">cfDet_id</span></a> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDet_order"><span class="id" title="definition">cfDet_order</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDet_order_lin"><span class="id" title="definition">cfDet_order_lin</span></a> <span class="id" title="var">xi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet_order"><span class="id" title="definition">cfDet_order</span></a> <a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.character.html#xi"><span class="id" title="variable">xi</span></a><a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDet_order_dvdG"><span class="id" title="definition">cfDet_order_dvdG</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.character.html#cfDet_order"><span class="id" title="definition">cfDet_order</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#DetOrder.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#DetOrder"><span class="id" title="section">DetOrder</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="8adcf036f3cd1d43410cc3f8faabe076"><span class="id" title="notation">"</span></a>''o' ( phi )" := (<a class="idref" href="mathcomp.character.character.html#cfDet_order"><span class="id" title="definition">cfDet_order</span></a> <span class="id" title="var">phi</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''o' ( phi )") : <span class="id" title="var">cfun_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="CfDetOps"><span class="id" title="section">CfDetOps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetRes"><span class="id" title="lemma">cfDetRes</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> (<a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">Res</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">cfDet</span></a> <a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetMorph"><span class="id" title="lemma">cfDetMorph</span></a> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">D</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.character.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>)<br/> - (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> (<a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDetIsom"><span class="id" title="lemma">cfDetIsom</span></a> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>)<br/> - (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.character.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">isoGR</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a>) <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.character.html#isoGR"><span class="id" title="variable">isoGR</span></a> (<a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDet_mul_lin"><span class="id" title="lemma">cfDet_mul_lin</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">lambda</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :<br/> - <a class="idref" href="mathcomp.character.character.html#lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> (<a class="idref" href="mathcomp.character.character.html#lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#lambda"><span class="id" title="variable">lambda</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.truncC"><span class="id" title="definition">truncC</span></a> (<a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>) <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.character.html#cfDet"><span class="id" title="definition">cfDet</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#CfDetOps"><span class="id" title="section">CfDetOps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfcenter"><span class="id" title="definition">cfcenter</span></a> (<span class="id" title="var">gT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) := <br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">set</span></a> <span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.character.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">"</span></a>''Z' ( phi )" := (<a class="idref" href="mathcomp.character.character.html#cfcenter"><span class="id" title="definition">cfcenter</span></a> <span class="id" title="var">phi</span>) : <span class="id" title="var">cfun_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Center"><span class="id" title="section">Center</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Center.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Center.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">phi</span> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#Center.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.27)(a). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_repr"><span class="id" title="lemma">cfcenter_repr</span></a> <span class="id" title="var">n</span> (<span class="id" title="var">rG</span> : <a class="idref" href="mathcomp.character.mxrepresentation.html#mx_representation"><span class="id" title="record">mx_representation</span></a> <a class="idref" href="mathcomp.character.character.html#algCF"><span class="id" title="abbreviation">algCF</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.character.html#n"><span class="id" title="variable">n</span></a>) : <br/> - <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#cfRepr"><span class="id" title="definition">cfRepr</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.mxrepresentation.html#rcenter"><span class="id" title="definition">rcenter</span></a> <a class="idref" href="mathcomp.character.character.html#rG"><span class="id" title="variable">rG</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is part of Isaacs (2.27)(b). -</div> -<div class="code"> -<span class="id" title="keyword">Fact</span> <a name="cfcenter_group_set"><span class="id" title="lemma">cfcenter_group_set</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">CF</span>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfcenter_group</span> <span class="id" title="var">f</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (<a class="idref" href="mathcomp.character.character.html#cfcenter_group_set"><span class="id" title="lemma">cfcenter_group_set</span></a> <a class="idref" href="mathcomp.character.character.html#f"><span class="id" title="variable">f</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="char_cfcenterE"><span class="id" title="lemma">char_cfcenterE</span></a> <span class="id" title="var">chi</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">CF</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="irr_cfcenterE"><span class="id" title="lemma">irr_cfcenterE</span></a> <span class="id" title="var">i</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">`|</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.character.character.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is also Isaacs (2.27)(b). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_sub"><span class="id" title="lemma">cfcenter_sub</span></a> <span class="id" title="var">phi</span> : (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_center_normal"><span class="id" title="lemma">cfker_center_normal</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_normal"><span class="id" title="lemma">cfcenter_normal</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.27)(c). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_Res"><span class="id" title="lemma">cfcenter_Res</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">chi1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#chi1"><span class="id" title="variable">chi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#linear_char"><span class="id" title="definition">linear_char</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.character.html#chi1"><span class="id" title="variable">chi1</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.27)(d). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_cyclic"><span class="id" title="lemma">cfcenter_cyclic</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.27)(e). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_subset_center"><span class="id" title="lemma">cfcenter_subset_center</span></a> <span class="id" title="var">chi</span> : <br/> - (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.27)(f). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_eq_center"><span class="id" title="lemma">cfcenter_eq_center</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a>) : <br/> - (<a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>%<span class="id" title="var">g</span>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.28). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cap_cfcenter_irr"><span class="id" title="lemma">cap_cfcenter_irr</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#4ba1db7734680ecf1e41cd64093d9c3e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#4ba1db7734680ecf1e41cd64093d9c3e"><span class="id" title="notation">bigcap_i</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.29). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_Res_lerif"><span class="id" title="lemma">cfnorm_Res_lerif</span></a> <span class="id" title="var">H</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.30). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="irr1_bound"><span class="id" title="lemma">irr1_bound</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a><br/> - <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.31). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="irr1_abelian_bound"><span class="id" title="lemma">irr1_abelian_bound</span></a> (<span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Iirr"><span class="id" title="abbreviation">Iirr</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a>) :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> (<a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> 1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#663140372ac3b275aae871b74b140513"><span class="id" title="notation">^+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.32)(a). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="irr_faithful_center"><span class="id" title="lemma">irr_faithful_center</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfcenter_fful_irr"><span class="id" title="lemma">cfcenter_fful_irr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a><a class="idref" href="mathcomp.character.character.html#ae199d7c7d964620f7672d2c74e0edb0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs (2.32)(b). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="pgroup_cyclic_faithful"><span class="id" title="lemma">pgroup_cyclic_faithful</span></a> (<span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <br/> - <a class="idref" href="mathcomp.character.character.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> <a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">chi</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Center.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.character.html#7d5ba9be6923d4bf4a568a8a939b7ab0"><span class="id" title="notation">_i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Center"><span class="id" title="section">Center</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Induced"><span class="id" title="section">Induced</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Induced.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Induced.G"><span class="id" title="variable">G</span></a> <a name="Induced.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.character.character.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">CF</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_char"><span class="id" title="lemma">cfInd_char</span></a> <span class="id" title="var">chi</span> : <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_eq0"><span class="id" title="lemma">cfInd_eq0</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Ind_irr_neq0"><span class="id" title="lemma">Ind_irr_neq0</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Ind_Iirr"><span class="id" title="definition">Ind_Iirr</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.character.html#cfIirr"><span class="id" title="definition">cfIirr</span></a> (<a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_cfRes_irr"><span class="id" title="lemma">constt_cfRes_irr</span></a> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">j</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constt_cfInd_irr"><span class="id" title="lemma">constt_cfInd_irr</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">j</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.character.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.character.html#irr_constt"><span class="id" title="definition">irr_constt</span></a> (<a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_Res"><span class="id" title="lemma">cfker_Res</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">Res</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs Lemma (5.11). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfker_Ind"><span class="id" title="lemma">cfker_Ind</span></a> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.character.character.html#character"><span class="id" title="definition">character</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#gcore"><span class="id" title="definition">gcore</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#chi"><span class="id" title="variable">chi</span></a>) <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_Ind_irr"><span class="id" title="lemma">cfker_Ind_irr</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.character.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#gcore"><span class="id" title="definition">gcore</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.character.html#fc972dbc606652894cd5958d13eb0ca3"><span class="id" title="notation">chi_i</span></a>) <a class="idref" href="mathcomp.character.character.html#Induced.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.character.html#Induced"><span class="id" title="section">Induced</span></a>.<br/> - -<br/> -</div> -</div> - -<div id="footer"> -<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> -</div> - -</div> - -</body> -</html>
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