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diff --git a/docs/htmldoc/mathcomp.character.classfun.html b/docs/htmldoc/mathcomp.character.classfun.html deleted file mode 100644 index f6b1f56..0000000 --- a/docs/htmldoc/mathcomp.character.classfun.html +++ /dev/null @@ -1,1956 +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.classfun</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.character.classfun</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 theory of class functions: - 'CF(G) == the type of class functions on G : {group gT}, i.e., - which map gT to the type algC of complex algebraics, - have support in G, and are constant on each conjugacy - class of G. 'CF(G) implements the FalgType interface of - finite-dimensional F-algebras. - The identity 1 : 'CF(G) is the indicator function of G, - and (later) the principal character. -> The %CF scope (cfun_scope) is bound to the 'CF(_) types. - 'CF(G)%VS == the (total) vector space of 'CF(G). - 'CF(G, A) == the subspace of functions in 'CF(G) with support in A. - phi x == the image of x : gT under phi : 'CF(G). - # [phi]%CF == the multiplicative order of phi : 'CF(G). - cfker phi == the kernel of phi : 'CF(G); note that cfker phi <| G. - cfaithful phi <=> phi : 'CF(G) is faithful (has a trivial kernel). - '1_A == the indicator function of A as a function of 'CF(G). - (Provided A <| G; G is determined by the context.) - phi^*%CF == the function conjugate to phi : 'CF(G). - cfAut u phi == the function conjugate to phi by an algC-automorphism u - phi^u The notation "_ ^u" is only reserved; it is up to - clients to set Notation "phi ^u" := (cfAut u phi). - ' [phi, psi] == the convolution of phi, psi : 'CF(G) over G, normalised - ' [phi, psi]_G by #|G| so that ' [1, 1]_G = 1 (G is usually inferred). - cfdotr psi phi == ' [phi, psi] (self-expanding). - ' [phi], ' [phi]_G == the squared norm ' [phi, phi] of phi : 'CF(G). - orthogonal R S <=> each phi in R : seq 'CF(G) is orthogonal to each psi in - S, i.e., ' [phi, psi] = 0. As 'CF(G) coerces to seq, one - can write orthogonal phi S and orthogonal phi psi. - pairwise_orthogonal S <=> the class functions in S are pairwise orthogonal - AND non-zero. - orthonormal S <=> S is pairwise orthogonal and all class functions in S - have norm 1. - isometry tau <-> tau : 'CF(D) -> 'CF(R) is an isometry, mapping - ' [, _ ]_D to ' [, _ ]_R. - {in CD, isometry tau, to CR} <-> in the domain CD, tau is an isometry - whose range is contained in CR. - cfReal phi <=> phi is real, i.e., phi^* == phi. - cfAut_closed u S <-> S : seq 'CF(G) is closed under conjugation by u. - cfConjC_closed S <-> S : seq 'CF(G) is closed under complex conjugation. - conjC_subset S1 S2 <-> S1 : seq 'CF(G) represents a subset of S2 closed - under complex conjugation. - := [/\ uniq S1, {subset S1 <= S2} & cfConjC_closed S1]. - 'Res[H] phi == the restriction of phi : 'CF(G) to a function of 'CF(H) - 'Res[H, G] phi 'Res[H] phi x = phi x if x \in H (when H \subset G), - 'Res phi 'Res[H] phi x = 0 if x \notin H. The syntax variants - allow H and G to be inferred; the default is to specify - H explicitly, and infer G from the type of phi. - 'Ind[G] phi == the class function of 'CF(G) induced by phi : 'CF(H), - 'Ind[G, H] phi when H \subset G. As with 'Res phi, both G and H can - 'Ind phi be inferred, though usually G isn't. - cfMorph phi == the class function in 'CF(G) that maps x to phi (f x), - where phi : 'CF(f @* G), provided G \subset 'dom f. - cfIsom isoGR phi == the class function in 'CF(R) that maps f x to phi x, - given isoGR : isom G R f, f : {morphism G >-> rT} and - phi : 'CF(G). - (phi %% H)%CF == special case of cfMorph phi, when phi : 'CF(G / H). - (phi / H)%CF == the class function in 'CF(G / H) that coincides with - phi : 'CF(G) on cosets of H \subset cfker phi. - For a group G that is a semidirect product (defG : K ><| H = G), we have - cfSdprod KxH phi == for phi : 'CF(H), the class function of 'CF(G) that - maps k * h to psi h when k \in K and h \in H. - For a group G that is a direct product (with KxH : K \x H = G), we have - cfDprodl KxH phi == for phi : 'CF(K), the class function of 'CF(G) that - maps k * h to phi k when k \in K and h \in H. - cfDprodr KxH psi == for psi : 'CF(H), the class function of 'CF(G) that - maps k * h to psi h when k \in K and h \in H. - cfDprod KxH phi psi == for phi : 'CF(K), psi : 'CF(H), the class function - of 'CF(G) that maps k * h to phi k * psi h (this is - the product of the two functions above). - Finally, given defG : \big[dprod/1](i | P i) A i = G, with G and A i - groups and i ranges over a finType, we have - cfBigdprodi defG phi == for phi : 'CF(A i) s.t. P i, the class function - of 'CF(G) that maps x to phi x_i, where x_i is the - (A i)-component of x : G. - cfBigdprod defG phi == for phi : forall i, 'CF(A i), the class function - of 'CF(G) that maps x to \prod(i | P i) phi i x_i, - where x_i is the (A i)-component of x : 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/> -<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">cfun_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Reserved Notation</span> "''CF' ( G , A )" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''CF' ( G , A )").<br/> -<span class="id" title="keyword">Reserved Notation</span> "''CF' ( G )" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''CF' ( G )").<br/> -<span class="id" title="keyword">Reserved Notation</span> "''1_' G" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">G</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''1_' G").<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Res[' H , G ]" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Res[' H ]" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''Res[' H ]").<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Res'" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Ind[' G , H ]" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Ind[' G ]" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''Ind[' G ]").<br/> -<span class="id" title="keyword">Reserved Notation</span> "''Ind'" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "'[ phi , psi ]_ G" (<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>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "'[ phi , psi ]"<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "'[hv' ''[' phi , '/ ' psi ] ']'").<br/> -<span class="id" title="keyword">Reserved Notation</span> "'[ phi ]_ G" (<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>).<br/> -<span class="id" title="keyword">Reserved Notation</span> "'[ phi ]" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''[' phi ]").<br/> -<span class="id" title="keyword">Reserved Notation</span> "phi ^u" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 3, <span class="id" title="var">format</span> "phi ^u").<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AlgC"><span class="id" title="section">AlgC</span></a>.<br/> -</div> - -<div class="doc"> - Arithmetic properties of group orders in the characteristic 0 field algC. -</div> -<div class="code"> - -<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/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</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.classfun.html#AlgC.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">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.classfun.html#AlgC.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="neq0CG"><span class="id" title="lemma">neq0CG</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><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.classfun.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="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.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.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="neq0CiG"><span class="id" title="lemma">neq0CiG</span></a> <span class="id" title="var">G</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><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.classfun.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.classfun.html#B"><span class="id" title="variable">B</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.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.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="gt0CG"><span class="id" title="lemma">gt0CG</span></a> <span class="id" title="var">G</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#292c223180a62926ca0f2279c23ce13c"><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.classfun.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="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.ssrnum.html#292c223180a62926ca0f2279c23ce13c"><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>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="gt0CiG"><span class="id" title="lemma">gt0CiG</span></a> <span class="id" title="var">G</span> <span class="id" title="var">B</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#292c223180a62926ca0f2279c23ce13c"><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.classfun.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.classfun.html#B"><span class="id" title="variable">B</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.ssrnum.html#292c223180a62926ca0f2279c23ce13c"><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>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="algC'G"><span class="id" title="lemma">algC'G</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#0928aaf0450c3a4c5521d7d3da15b6d8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#0928aaf0450c3a4c5521d7d3da15b6d8"><span class="id" title="notation">char</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#0928aaf0450c3a4c5521d7d3da15b6d8"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</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.classfun.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#AlgC"><span class="id" title="section">AlgC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Defs.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="is_class_fun"><span class="id" title="definition">is_class_fun</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.classfun.html#Defs.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">f</span> : <a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.character.classfun.html#Defs.gT"><span class="id" title="variable">gT</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.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>) :=<br/> - <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">x</span><a class="idref" href="mathcomp.ssreflect.fintype.html#7406769ad390d6c18d532b497e931ef0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#7406769ad390d6c18d532b497e931ef0"><span class="id" title="notation">∀</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#7406769ad390d6c18d532b497e931ef0"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</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.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.algebra.ssralg.html#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="intro_class_fun"><span class="id" title="lemma">intro_class_fun</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.classfun.html#Defs.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> :<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.classfun.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> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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#b4f176550f5b849a7fbba2ee164934d3"><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">x</span>, <a class="idref" href="mathcomp.character.classfun.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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.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#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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> 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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</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.character.classfun.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Defs.B"><span class="id" title="variable">B</span></a> : <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.classfun.html#Defs.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/> - -<br/> -<span class="id" title="keyword">Record</span> <a name="classfun"><span class="id" title="record">classfun</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predArgType"><span class="id" title="definition">predArgType</span></a> :=<br/> - <a name="Classfun"><span class="id" title="constructor">Classfun</span></a> {<a name="cfun_val"><span class="id" title="projection">cfun_val</span></a>; <span class="id" title="var">_</span> : <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.character.classfun.html#G"><span class="id" title="abbreviation">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_val"><span class="id" title="method">cfun_val</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">psi</span> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a>.<br/> -</div> - -<div class="doc"> - The default expansion lemma cfunE requires key = 0. -</div> -<div class="code"> -<span class="id" title="keyword">Fact</span> <a name="classfun_key"><span class="id" title="lemma">classfun_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="Cfun"><span class="id" title="definition">Cfun</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.classfun.html#classfun_key"><span class="id" title="lemma">classfun_key</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">flag</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> ⇒ <a class="idref" href="mathcomp.character.classfun.html#Classfun"><span class="id" title="constructor">Classfun</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_subType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c2d02b544d823cdc1e1e08de552cdba4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#c2d02b544d823cdc1e1e08de552cdba4"><span class="id" title="notation">subType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c2d02b544d823cdc1e1e08de552cdba4"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_val"><span class="id" title="projection">cfun_val</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#c2d02b544d823cdc1e1e08de552cdba4"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_eqMixin"><span class="id" title="definition">cfun_eqMixin</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b361a0fe0b43cea5c506ee5eccc55542"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#b361a0fe0b43cea5c506ee5eccc55542"><span class="id" title="notation">eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b361a0fe0b43cea5c506ee5eccc55542"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b361a0fe0b43cea5c506ee5eccc55542"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b361a0fe0b43cea5c506ee5eccc55542"><span class="id" title="notation"><:]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_eqMixin"><span class="id" title="definition">cfun_eqMixin</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_choiceMixin"><span class="id" title="definition">cfun_choiceMixin</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#035054ba987e1c05f2985518b41ec31f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#035054ba987e1c05f2985518b41ec31f"><span class="id" title="notation">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#035054ba987e1c05f2985518b41ec31f"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#035054ba987e1c05f2985518b41ec31f"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#035054ba987e1c05f2985518b41ec31f"><span class="id" title="notation"><:]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_choiceMixin"><span class="id" title="definition">cfun_choiceMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="fun_of_cfun"><span class="id" title="definition">fun_of_cfun</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#cfun_val"><span class="id" title="projection">cfun_val</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#Defs.gT"><span class="id" title="variable">gT</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>.<br/> -<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.character.classfun.html#fun_of_cfun"><span class="id" title="definition">fun_of_cfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#fun_of_cfun"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#fun_of_cfun"><span class="id" title="definition">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#fun_of_cfun"><span class="id" title="definition">>-></span></a> <a class="idref" href="mathcomp.character.classfun.html#fun_of_cfun"><span class="id" title="definition">Funclass</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunElock"><span class="id" title="lemma">cfunElock</span></a> <span class="id" title="var">k</span> <span class="id" title="var">f</span> <span class="id" title="var">fP</span> : @<a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#k"><span class="id" title="variable">k</span></a> (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="mathcomp.character.classfun.html#fP"><span class="id" title="variable">fP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunE"><span class="id" title="lemma">cfunE</span></a> <span class="id" title="var">f</span> <span class="id" title="var">fP</span> : @<a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 0 (<a class="idref" href="mathcomp.ssreflect.finfun.html#finfun"><span class="id" title="abbreviation">finfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="mathcomp.character.classfun.html#fP"><span class="id" title="variable">fP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunP"><span class="id" title="lemma">cfunP</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a> <a class="idref" href="mathcomp.character.classfun.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#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun0gen"><span class="id" title="lemma">cfun0gen</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_in_genP"><span class="id" title="lemma">cfun_in_genP</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.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.classfun.html#G"><span class="id" title="abbreviation">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="mathcomp.character.classfun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#8c08d4203604dbed63e7afa9b689d858"><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#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#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunJgen"><span class="id" title="lemma">cfunJgen</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#G"><span class="id" title="abbreviation">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#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_zero_subproof"><span class="id" title="lemma">cfun_zero_subproof</span></a> : <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.character.classfun.html#G"><span class="id" title="abbreviation">G</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.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">ffun</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.ssreflect.finfun.html#31493a873acc18a8368490ef56022c0c"><span class="id" title="notation">}</span></a>).<br/> - <span class="id" title="keyword">Definition</span> <a name="cfun_zero"><span class="id" title="definition">cfun_zero</span></a> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 0 <a class="idref" href="mathcomp.character.classfun.html#cfun_zero_subproof"><span class="id" title="lemma">cfun_zero_subproof</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_comp_subproof"><span class="id" title="lemma">cfun_comp_subproof</span></a> <span class="id" title="var">f</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</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 <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#is_class_fun"><span class="id" title="definition">is_class_fun</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">G</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.character.classfun.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</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="cfun_comp"><span class="id" title="definition">cfun_comp</span></a> <span class="id" title="var">f</span> <span class="id" title="var">f0</span> <span class="id" title="var">phi</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.classfun.html#cfun_comp_subproof"><span class="id" title="lemma">cfun_comp_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f0"><span class="id" title="variable">f0</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_opp"><span class="id" title="definition">cfun_opp</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfun_comp"><span class="id" title="definition">cfun_comp</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.oppr0"><span class="id" title="definition">oppr0</span></a> <span class="id" title="var">_</span>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_add_subproof"><span class="id" title="lemma">cfun_add_subproof</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <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.character.classfun.html#G"><span class="id" title="abbreviation">G</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.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</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="cfun_add"><span class="id" title="definition">cfun_add</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.classfun.html#cfun_add_subproof"><span class="id" title="lemma">cfun_add_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_indicator_subproof"><span class="id" title="lemma">cfun_indicator_subproof</span></a> (<span class="id" title="var">A</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.classfun.html#Defs.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="mathcomp.character.classfun.html#is_class_fun"><span class="id" title="definition">is_class_fun</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">G</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.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.classfun.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.classfun.html#G"><span class="id" title="abbreviation">G</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#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.classfun.html#G"><span class="id" title="abbreviation">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#A"><span class="id" title="variable">A</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><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="cfun_indicator"><span class="id" title="definition">cfun_indicator</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 1 (<a class="idref" href="mathcomp.character.classfun.html#cfun_indicator_subproof"><span class="id" title="lemma">cfun_indicator_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun1Egen"><span class="id" title="lemma">cfun1Egen</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">_G</span></a> <a class="idref" href="mathcomp.character.classfun.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#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#G"><span class="id" title="abbreviation">G</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">Fact</span> <a name="cfun_mul_subproof"><span class="id" title="lemma">cfun_mul_subproof</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <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.character.classfun.html#G"><span class="id" title="abbreviation">G</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.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</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="cfun_mul"><span class="id" title="definition">cfun_mul</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.classfun.html#cfun_mul_subproof"><span class="id" title="lemma">cfun_mul_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_unit"><span class="id" title="definition">cfun_unit</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">pred</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#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#9b40a7420e06ba2a775d87b43bd1c69f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#9b40a7420e06ba2a775d87b43bd1c69f"><span class="id" title="notation">∀</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#9b40a7420e06ba2a775d87b43bd1c69f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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<a class="idref" href="mathcomp.ssreflect.fintype.html#9b40a7420e06ba2a775d87b43bd1c69f"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#fbeb549dcb6350fb8ceb1bda39acce60"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_inv"><span class="id" title="definition">cfun_inv</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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.character.classfun.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#cfun_unit"><span class="id" title="definition">cfun_unit</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.character.classfun.html#cfun_comp"><span class="id" title="definition">cfun_comp</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.invr0"><span class="id" title="definition">invr0</span></a> <span class="id" title="var">_</span>) <a class="idref" href="mathcomp.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_scale"><span class="id" title="definition">cfun_scale</span></a> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.character.classfun.html#cfun_comp"><span class="id" title="definition">cfun_comp</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.mulr0"><span class="id" title="definition">mulr0</span></a> <a class="idref" href="mathcomp.character.classfun.html#a"><span class="id" title="variable">a</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_addA"><span class="id" title="lemma">cfun_addA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add"><span class="id" title="definition">cfun_add</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_addC"><span class="id" title="lemma">cfun_addC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add"><span class="id" title="definition">cfun_add</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_add0"><span class="id" title="lemma">cfun_add0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_zero"><span class="id" title="definition">cfun_zero</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add"><span class="id" title="definition">cfun_add</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_addN"><span class="id" title="lemma">cfun_addN</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_inverse"><span class="id" title="definition">left_inverse</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_zero"><span class="id" title="definition">cfun_zero</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_opp"><span class="id" title="definition">cfun_opp</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add"><span class="id" title="definition">cfun_add</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_zmodMixin"><span class="id" title="definition">cfun_zmodMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Exports.ZmodMixin"><span class="id" title="abbreviation">ZmodMixin</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_addA"><span class="id" title="lemma">cfun_addA</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_addC"><span class="id" title="lemma">cfun_addC</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add0"><span class="id" title="lemma">cfun_add0</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_addN"><span class="id" title="lemma">cfun_addN</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_zmodType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Zmodule.Exports.ZmodType"><span class="id" title="abbreviation">ZmodType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_zmodMixin"><span class="id" title="definition">cfun_zmodMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="muln_cfunE"><span class="id" title="lemma">muln_cfunE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.character.classfun.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sum_cfunE"><span class="id" title="lemma">sum_cfunE</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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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#classfun"><span class="id" title="record">classfun</span></a>) <span class="id" title="var">x</span> :<br/> - (<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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.character.classfun.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#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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_mulA"><span class="id" title="lemma">cfun_mulA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mul"><span class="id" title="definition">cfun_mul</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_mulC"><span class="id" title="lemma">cfun_mulC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mul"><span class="id" title="definition">cfun_mul</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_mul1"><span class="id" title="lemma">cfun_mul1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">_G</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mul"><span class="id" title="definition">cfun_mul</span></a>.<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_mulD"><span class="id" title="lemma">cfun_mulD</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_distributive"><span class="id" title="definition">left_distributive</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mul"><span class="id" title="definition">cfun_mul</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_add"><span class="id" title="definition">cfun_add</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_nz1"><span class="id" title="lemma">cfun_nz1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><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.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_ringMixin"><span class="id" title="definition">cfun_ringMixin</span></a> :=<br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.ComRingMixin"><span class="id" title="abbreviation">ComRingMixin</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mulA"><span class="id" title="lemma">cfun_mulA</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mulC"><span class="id" title="lemma">cfun_mulC</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mul1"><span class="id" title="lemma">cfun_mul1</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mulD"><span class="id" title="lemma">cfun_mulD</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_nz1"><span class="id" title="lemma">cfun_nz1</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_ringType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.RingType"><span class="id" title="abbreviation">RingType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_ringMixin"><span class="id" title="definition">cfun_ringMixin</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_comRingType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.ComRingType"><span class="id" title="abbreviation">ComRingType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mulC"><span class="id" title="lemma">cfun_mulC</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="expS_cfunE"><span class="id" title="lemma">expS_cfunE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>) <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_mulV"><span class="id" title="lemma">cfun_mulV</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.character.classfun.html#cfun_unit"><span class="id" title="definition">cfun_unit</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#left_inverse"><span class="id" title="definition">left_inverse</span></a> 1 <a class="idref" href="mathcomp.character.classfun.html#cfun_inv"><span class="id" title="definition">cfun_inv</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3609d85e23333c9e68741ad96b416eec"><span class="id" title="notation">R</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/> -<span class="id" title="keyword">Fact</span> <a name="cfun_unitP"><span class="id" title="lemma">cfun_unitP</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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#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> 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.classfun.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#cfun_unit"><span class="id" title="definition">cfun_unit</span></a>.<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_inv0id"><span class="id" title="lemma">cfun_inv0id</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#790b887fcb3f1d578b2c7a5460f7aeb5"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#790b887fcb3f1d578b2c7a5460f7aeb5"><span class="id" title="notation">predC</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_unit"><span class="id" title="definition">cfun_unit</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#790b887fcb3f1d578b2c7a5460f7aeb5"><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> <a class="idref" href="mathcomp.character.classfun.html#cfun_inv"><span class="id" title="definition">cfun_inv</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</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">Definition</span> <a name="cfun_unitMixin"><span class="id" title="definition">cfun_unitMixin</span></a> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Exports.ComUnitRingMixin"><span class="id" title="abbreviation">ComUnitRingMixin</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_mulV"><span class="id" title="lemma">cfun_mulV</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_unitP"><span class="id" title="lemma">cfun_unitP</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_inv0id"><span class="id" title="lemma">cfun_inv0id</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_unitRingType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.UnitRingType"><span class="id" title="abbreviation">UnitRingType</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_unitMixin"><span class="id" title="definition">cfun_unitMixin</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_comUnitRingType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#2dfeb3fb2088b370ad93742d4f23a0dc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2dfeb3fb2088b370ad93742d4f23a0dc"><span class="id" title="notation">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#2dfeb3fb2088b370ad93742d4f23a0dc"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2dfeb3fb2088b370ad93742d4f23a0dc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_scaleA"><span class="id" title="lemma">cfun_scaleA</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#cfun_scale"><span class="id" title="definition">cfun_scale</span></a> <a class="idref" href="mathcomp.character.classfun.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfun_scale"><span class="id" title="definition">cfun_scale</span></a> <a class="idref" href="mathcomp.character.classfun.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.character.classfun.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#cfun_scale"><span class="id" title="definition">cfun_scale</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_scale1"><span class="id" title="lemma">cfun_scale1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.character.classfun.html#cfun_scale"><span class="id" title="definition">cfun_scale</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_scaleDr"><span class="id" title="lemma">cfun_scaleDr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scale"><span class="id" title="definition">cfun_scale</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a87d5ea2e207e69e5e474db24f56d4cb"><span class="id" title="notation">R</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_scaleDl"><span class="id" title="lemma">cfun_scaleDl</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#e69c60b553f06d3463460a9f4cee3c01"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e69c60b553f06d3463460a9f4cee3c01"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scale"><span class="id" title="definition">cfun_scale</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e69c60b553f06d3463460a9f4cee3c01"><span class="id" title="notation">:</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.ssr.ssrfun.html#e69c60b553f06d3463460a9f4cee3c01"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#e69c60b553f06d3463460a9f4cee3c01"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_lmodMixin"><span class="id" title="definition">cfun_lmodMixin</span></a> :=<br/> - <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodMixin"><span class="id" title="abbreviation">LmodMixin</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scaleA"><span class="id" title="lemma">cfun_scaleA</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scale1"><span class="id" title="lemma">cfun_scale1</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scaleDr"><span class="id" title="lemma">cfun_scaleDr</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scaleDl"><span class="id" title="lemma">cfun_scaleDl</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_lmodType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodType"><span class="id" title="abbreviation">LmodType</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.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_lmodMixin"><span class="id" title="definition">cfun_lmodMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_scaleAl"><span class="id" title="lemma">cfun_scaleAl</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.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.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><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#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - <span class="id" title="keyword">Fact</span> <a name="cfun_scaleAr"><span class="id" title="lemma">cfun_scaleAr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.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.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><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#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.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_lalgType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lalgebra.Exports.LalgType"><span class="id" title="abbreviation">LalgType</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.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scaleAl"><span class="id" title="lemma">cfun_scaleAl</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_algType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Algebra.Exports.AlgType"><span class="id" title="abbreviation">AlgType</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.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_scaleAr"><span class="id" title="lemma">cfun_scaleAr</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_unitAlgType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#53130370ad22aac4f3ee8434dbc4850d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#53130370ad22aac4f3ee8434dbc4850d"><span class="id" title="notation">unitAlgType</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#53130370ad22aac4f3ee8434dbc4850d"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#53130370ad22aac4f3ee8434dbc4850d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Defs.Automorphism"><span class="id" title="section">Automorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Defs.Automorphism.u"><span class="id" title="variable">u</span></a> : <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">Definition</span> <a name="cfAut"><span class="id" title="definition">cfAut</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfun_comp"><span class="id" title="definition">cfun_comp</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.rmorph0"><span class="id" title="definition">rmorph0</span></a> <a class="idref" href="mathcomp.character.classfun.html#Defs.Automorphism.u"><span class="id" title="variable">u</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_cfun1i"><span class="id" title="lemma">cfAut_cfun1i</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">_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.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">_A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutZ"><span class="id" title="lemma">cfAutZ</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.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#Defs.Automorphism.u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_is_rmorphism"><span class="id" title="lemma">cfAut_is_rmorphism</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.rmorphism"><span class="id" title="abbreviation">rmorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfAut_additive</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.classfun.html#cfAut_is_rmorphism"><span class="id" title="lemma">cfAut_is_rmorphism</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfAut_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.RMorphism"><span class="id" title="abbreviation">RMorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut_is_rmorphism"><span class="id" title="lemma">cfAut_is_rmorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_cfun1"><span class="id" title="lemma">cfAut_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</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> 1. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_scalable"><span class="id" title="lemma">cfAut_scalable</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.scalable_for"><span class="id" title="abbreviation">scalable_for</span></a> (<a class="idref" href="mathcomp.character.classfun.html#Defs.Automorphism.u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#11ebad41b70994075d9152ef8d0a15b3"><span class="id" title="notation">\;</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#eb2b32cc2f63f97454a307a8ee8d68cc"><span class="id" title="notation">*:%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#eb2b32cc2f63f97454a307a8ee8d68cc"><span class="id" title="notation">R</span></a>) <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a>.<br/> - <span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfAut_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Linear.Exports.AddLinear"><span class="id" title="abbreviation">AddLinear</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut_scalable"><span class="id" title="lemma">cfAut_scalable</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfAut_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfAut_closed"><span class="id" title="definition">cfAut_closed</span></a> (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</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.classfun.html#S"><span class="id" title="variable">S</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">phi</span>, <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.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#S"><span class="id" title="variable">S</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.classfun.html#Defs.Automorphism"><span class="id" title="section">Automorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfReal"><span class="id" title="definition">cfReal</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfConjC_subset"><span class="id" title="definition">cfConjC_subset</span></a> (<span class="id" title="var">S1</span> <span class="id" title="var">S2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a>) :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><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">{</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.classfun.html#S1"><span class="id" title="variable">S1</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.classfun.html#S2"><span class="id" title="variable">S2</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfAut_closed"><span class="id" title="definition">cfAut_closed</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfun_vect_iso"><span class="id" title="lemma">cfun_vect_iso</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.axiom"><span class="id" title="abbreviation">Vector.axiom</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.classfun.html#G"><span class="id" title="abbreviation">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_vectMixin"><span class="id" title="definition">cfun_vectMixin</span></a> := <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.VectMixin"><span class="id" title="abbreviation">VectMixin</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_vect_iso"><span class="id" title="lemma">cfun_vect_iso</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_vectType</span> := <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.VectType"><span class="id" title="abbreviation">VectType</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.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfun_vectMixin"><span class="id" title="definition">cfun_vectMixin</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_FalgType</span> := <a class="idref" href="mathcomp.field.falgebra.html#8fcc6f073a7a36fa680d6889440e6651"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.field.falgebra.html#8fcc6f073a7a36fa680d6889440e6651"><span class="id" title="notation">FalgType</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.field.falgebra.html#8fcc6f073a7a36fa680d6889440e6651"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a><a class="idref" href="mathcomp.field.falgebra.html#8fcc6f073a7a36fa680d6889440e6651"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfun_base"><span class="id" title="definition">cfun_base</span></a> <span class="id" title="var">A</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.classfun.html#Defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#7ed77f64965385480255328de15d774c"><span class="id" title="notation">::&:</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</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#classfun"><span class="id" title="record">classfun</span></a> :=<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.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#0785b929dd7e2c08474c202f37716668"><span class="id" title="notation">_xB</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">|</span></a> <span class="id" title="var">xB</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><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.classfun.html#Defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#7ed77f64965385480255328de15d774c"><span class="id" title="notation">::&:</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#c3ff8d84d4e3e273bebfcf7502deb41a"><span class="id" title="notation">]</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="classfun_on"><span class="id" title="definition">classfun_on</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#cfun_base"><span class="id" title="definition">cfun_base</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfdot"><span class="id" title="definition">cfdot</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#Defs.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#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#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.classfun.html#Defs.B"><span class="id" title="variable">B</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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/> -<span class="id" title="keyword">Definition</span> <a name="cfdotr_head"><span class="id" title="definition">cfdotr_head</span></a> <span class="id" title="var">k</span> <span class="id" title="var">psi</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.classfun.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.character.classfun.html#cfdot"><span class="id" title="definition">cfdot</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfnorm_head"><span class="id" title="definition">cfnorm_head</span></a> <span class="id" title="var">k</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.classfun.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.character.classfun.html#cfdot"><span class="id" title="definition">cfdot</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Coercion</span> <span class="id" title="var">seq_of_cfun</span> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cforder"><span class="id" title="definition">cforder</span></a> <span class="id" title="var">phi</span> := <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.ssreflect.div.html#lcmn"><span class="id" title="definition">lcmn</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">/</span></a>1%<span class="id" title="var">N</span><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">x</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#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#Defs.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.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.algnum.html#e0ef1169a0283239169c3e064b6dca4c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.field.algnum.html#e0ef1169a0283239169c3e064b6dca4c"><span class="id" title="notation">]</span></a>%<span class="id" title="var">C</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">"</span></a>''CF' ( G )" := (<a class="idref" href="mathcomp.character.classfun.html#classfun"><span class="id" title="record">classfun</span></a> <span class="id" title="var">G</span>) : <span class="id" title="var">type_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="ade764dfeb1c233c7a245d960646d8b8"><span class="id" title="notation">"</span></a>''CF' ( G )" := (@<a class="idref" href="mathcomp.algebra.vector.html#fullv"><span class="id" title="definition">fullv</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.character.classfun.html#cfun_vectType"><span class="id" title="definition">cfun_vectType</span></a> <span class="id" title="var">G</span>)) : <span class="id" title="var">vspace_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="483aeb200c980494b369f3b9c18f042e"><span class="id" title="notation">"</span></a>''1_' A" := (<a class="idref" href="mathcomp.character.classfun.html#cfun_indicator"><span class="id" title="definition">cfun_indicator</span></a> <span class="id" title="var">_</span> <span class="id" title="var">A</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">"</span></a>''CF' ( G , A )" := (<a class="idref" href="mathcomp.character.classfun.html#classfun_on"><span class="id" title="definition">classfun_on</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="bdc9a72479dc093ecf77d273ad008efe"><span class="id" title="notation">"</span></a>1" := (@<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.one"><span class="id" title="definition">GRing.one</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfun_ringType"><span class="id" title="definition">cfun_ringType</span></a> <span class="id" title="var">_</span>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">cfun_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="b24325f513e9801012e6322700f34266"><span class="id" title="notation">"</span></a>phi ^*" := (<a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a> <span class="id" title="var">phi</span>) : <span class="id" title="var">cfun_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="cfConjC_closed"><span class="id" title="abbreviation">cfConjC_closed</span></a> := (<a class="idref" href="mathcomp.character.classfun.html#cfAut_closed"><span class="id" title="definition">cfAut_closed</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>).<br/> -</div> - -<div class="doc"> - Workaround for overeager projection reduction. -</div> -<div class="code"> -<span class="id" title="keyword">Notation</span> <a name="eqcfP"><span class="id" title="abbreviation">eqcfP</span></a> := (@<a class="idref" href="mathcomp.ssreflect.eqtype.html#eqP"><span class="id" title="lemma">eqP</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfun_eqType"><span class="id" title="definition">cfun_eqType</span></a> <span class="id" title="var">_</span>) <span class="id" title="var">_</span> <span class="id" title="var">_</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="018669adc5ec173554da02f8822d051d"><span class="id" title="notation">"</span></a>#[ phi ]" := (<a class="idref" href="mathcomp.character.classfun.html#cforder"><span class="id" title="definition">cforder</span></a> <span class="id" title="var">phi</span>) : <span class="id" title="var">cfun_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">"</span></a>''[' u , v ]_ G":= (@<a class="idref" href="mathcomp.character.classfun.html#cfdot"><span class="id" title="definition">cfdot</span></a> <span class="id" title="var">_</span> <span class="id" title="var">G</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">"</span></a>''[' u , v ]" := (<a class="idref" href="mathcomp.character.classfun.html#cfdot"><span class="id" title="definition">cfdot</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">"</span></a>''[' u ]_ G" := <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><span class="id" title="var">u</span><a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">,</span></a> <span class="id" title="var">u</span><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> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">"</span></a>''[' u ]" := <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><span class="id" title="var">u</span><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <span class="id" title="var">u</span><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a> : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="cfdotr"><span class="id" title="abbreviation">cfdotr</span></a> := (<a class="idref" href="mathcomp.character.classfun.html#cfdotr_head"><span class="id" title="definition">cfdotr_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>).<br/> -<span class="id" title="keyword">Notation</span> <a name="cfnorm"><span class="id" title="abbreviation">cfnorm</span></a> := (<a class="idref" href="mathcomp.character.classfun.html#cfnorm_head"><span class="id" title="definition">cfnorm_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>).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Predicates"><span class="id" title="section">Predicates</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Predicates.gT"><span class="id" title="variable">gT</span></a> <a name="Predicates.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="Predicates.D"><span class="id" title="variable">D</span></a> : <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.classfun.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>) (<a name="Predicates.R"><span class="id" title="variable">R</span></a> : <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.classfun.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><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">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.classfun.html#Predicates.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</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.classfun.html#Predicates.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">tau</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.classfun.html#Predicates.D"><span class="id" title="variable">D</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.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.classfun.html#Predicates.R"><span class="id" title="variable">R</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">Definition</span> <a name="cfker"><span class="id" title="definition">cfker</span></a> <span class="id" title="var">phi</span> := <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.classfun.html#Predicates.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><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">y</span><a class="idref" href="mathcomp.ssreflect.fintype.html#f3be25edeb0349b0a76405eded9d0b98"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#ce8c9a990e3e773a56ef37417d3761c6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfaithful"><span class="id" title="definition">cfaithful</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.classfun.html#phi"><span class="id" title="variable">phi</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#b54f5e35cb228bba5934c852e0951c39"><span class="id" title="notation">[1]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="ortho_rec"><span class="id" title="definition">ortho_rec</span></a> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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">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="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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">psi</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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.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.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</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.character.classfun.html#S1"><span class="id" title="variable">S1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="pair_ortho_rec"><span class="id" title="definition">pair_ortho_rec</span></a> <span class="id" title="var">S</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">psi</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">S'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.character.classfun.html#ortho_rec"><span class="id" title="definition">ortho_rec</span></a> <span class="id" title="var">psi</span> <span class="id" title="var">S'</span> <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#pair_ortho_rec"><span class="id" title="definition">pair_ortho_rec</span></a> <span class="id" title="var">S'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - We exclude 0 from pairwise orthogonal sets. -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <span class="id" title="var">S</span> := <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>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.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.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#pair_ortho_rec"><span class="id" title="definition">pair_ortho_rec</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="orthonormal"><span class="id" title="definition">orthonormal</span></a> <span class="id" title="var">S</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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">psi</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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.character.classfun.html#pair_ortho_rec"><span class="id" title="definition">pair_ortho_rec</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="isometry"><span class="id" title="definition">isometry</span></a> <span class="id" title="var">tau</span> := <span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span>, <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a> <a class="idref" href="mathcomp.character.classfun.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#tau"><span class="id" title="variable">tau</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="isometry_from_to"><span class="id" title="definition">isometry_from_to</span></a> <span class="id" title="var">mCFD</span> <span class="id" title="var">tau</span> <span class="id" title="var">mCFR</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#prop_in2"><span class="id" title="definition">prop_in2</span></a> <a class="idref" href="mathcomp.character.classfun.html#mCFD"><span class="id" title="variable">mCFD</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#inPhantom"><span class="id" title="definition">inPhantom</span></a> (<a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a>))<br/> - <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#prop_in1"><span class="id" title="definition">prop_in1</span></a> <a class="idref" href="mathcomp.character.classfun.html#mCFD"><span class="id" title="variable">mCFD</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#inPhantom"><span class="id" title="definition">inPhantom</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#in_mem"><span class="id" title="definition">in_mem</span></a> (<a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="mathcomp.character.classfun.html#mCFR"><span class="id" title="variable">mCFR</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Predicates"><span class="id" title="section">Predicates</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Outside section so the nosimpl does not get "cooked" out. -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="orthogonal"><span class="id" title="definition">orthogonal</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">D</span> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#nosimpl"><span class="id" title="abbreviation">nosimpl</span></a> (@<a class="idref" href="mathcomp.character.classfun.html#ortho_rec"><span class="id" title="definition">ortho_rec</span></a> <a class="idref" href="mathcomp.character.classfun.html#gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.character.classfun.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</span></a>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">"</span></a>{ 'in' CFD , 'isometry' tau , 'to' CFR }" :=<br/> - (<a class="idref" href="mathcomp.character.classfun.html#isometry_from_to"><span class="id" title="definition">isometry_from_to</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">CFD</span>) <span class="id" title="var">tau</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">CFR</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'in' CFD , 'isometry' tau , 'to' CFR }")<br/> - : <span class="id" title="var">type_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="ClassFun"><span class="id" title="section">ClassFun</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="ClassFun.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="ClassFun.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.classfun.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">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.classfun.html#ClassFun.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">H</span> <span class="id" title="var">K</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.classfun.html#ClassFun.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> <span class="id" title="var">psi</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.classfun.html#ClassFun.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun0"><span class="id" title="lemma">cfun0</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="support_cfun"><span class="id" title="lemma">support_cfun</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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#ClassFun.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunJ"><span class="id" title="lemma">cfunJ</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_repr"><span class="id" title="lemma">cfun_repr</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_inP"><span class="id" title="lemma">cfun_inP</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.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.classfun.html#ClassFun.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="mathcomp.character.classfun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#8c08d4203604dbed63e7afa9b689d858"><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#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#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfuniE"><span class="id" title="lemma">cfuniE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_A</span></a> <a class="idref" href="mathcomp.character.classfun.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#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="support_cfuni"><span class="id" title="lemma">support_cfuni</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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.ssralg.html#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_A</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.character.classfun.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_mul_cfuni"><span class="id" title="lemma">eq_mul_cfuni</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#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.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.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.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">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_cfuni"><span class="id" title="lemma">eq_cfuni</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#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.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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.classfun.html#ClassFun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><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="cfuniG"><span class="id" title="lemma">cfuniG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><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> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun1E"><span class="id" title="lemma">cfun1E</span></a> <span class="id" title="var">g</span> : (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.classfun.html#ClassFun.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="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.ssralg.html#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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#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#ClassFun.G"><span class="id" title="variable">G</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="cfun11"><span class="id" title="lemma">cfun11</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.classfun.html#ClassFun.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>) 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="prod_cfunE"><span class="id" title="lemma">prod_cfunE</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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#ClassFun.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.classfun.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.classfun.html#ClassFun.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.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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.character.classfun.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#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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="exp_cfunE"><span class="id" title="lemma">exp_cfunE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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#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.classfun.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mul_cfuni"><span class="id" title="lemma">mul_cfuni</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <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">_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.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">_B</span></a> <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.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">)</span></a> <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.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.classfun.html#ClassFun.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="cfun_classE"><span class="id" title="lemma">cfun_classE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.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#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.classfun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</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#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.classfun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</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="cfun_on_sum"><span class="id" title="lemma">cfun_on_sum</span></a> <span class="id" title="var">A</span> :<br/> - <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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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.vector.html#86045227306b00d93321ae4ef5385bbb"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#86045227306b00d93321ae4ef5385bbb"><span class="id" title="notation">sum_</span></a><a class="idref" href="mathcomp.algebra.vector.html#86045227306b00d93321ae4ef5385bbb"><span class="id" title="notation">(</span></a><span class="id" title="var">xG</span> <a class="idref" href="mathcomp.algebra.vector.html#86045227306b00d93321ae4ef5385bbb"><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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.vector.html#86045227306b00d93321ae4ef5385bbb"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#xG"><span class="id" title="variable">xG</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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.vector.html#86045227306b00d93321ae4ef5385bbb"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.vector.html#6231d90025dd46a75d146519d384c2b5"><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">_xG</span></a><a class="idref" href="mathcomp.algebra.vector.html#6231d90025dd46a75d146519d384c2b5"><span class="id" title="notation">]></span></a>)%<span class="id" title="var">VS</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_onP"><span class="id" title="lemma">cfun_onP</span></a> <span class="id" title="var">A</span> <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> (<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.character.classfun.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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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> 0) (<a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_on0"><span class="id" title="lemma">cfun_on0</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sum_by_classes"><span class="id" title="lemma">sum_by_classes</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.character.classfun.html#ClassFun.gT"><span class="id" title="variable">gT</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#R"><span class="id" title="variable">R</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.classfun.html#ClassFun.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> <span class="id" title="keyword">∀</span> <span class="id" title="var">g</span> <span class="id" title="var">h</span>, <a class="idref" href="mathcomp.character.classfun.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.character.classfun.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.character.classfun.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.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <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">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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.classfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.character.classfun.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> <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">xG</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><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.classfun.html#ClassFun.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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#xG"><span class="id" title="variable">xG</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#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> <a class="idref" href="mathcomp.character.classfun.html#xG"><span class="id" title="variable">xG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_base_free"><span class="id" title="lemma">cfun_base_free</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfun_base"><span class="id" title="definition">cfun_base</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dim_cfun"><span class="id" title="lemma">dim_cfun</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">dim</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.classfun.html#ClassFun.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> <a class="idref" href="http://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.classfun.html#ClassFun.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="dim_cfun_on"><span class="id" title="lemma">dim_cfun_on</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.algebra.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">dim</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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#7ed77f64965385480255328de15d774c"><span class="id" title="notation">::&:</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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="dim_cfun_on_abelian"><span class="id" title="lemma">dim_cfun_on_abelian</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#A"><span class="id" title="variable">A</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#ClassFun.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.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#6d9094556d4642bd9374f6c3dcaee079"><span class="id" title="notation">dim</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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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.classfun.html#A"><span class="id" title="variable">A</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="cfuni_on"><span class="id" title="lemma">cfuni_on</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#a2e268216161616d418b1dd69313ec2b"><span class="id" title="notation">_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.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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mul_cfuni_on"><span class="id" title="lemma">mul_cfuni_on</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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.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">_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.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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_onE"><span class="id" title="lemma">cfun_onE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">A</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#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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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="http://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#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_onT"><span class="id" title="lemma">cfun_onT</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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.ssreflect.finset.html#d1cce020b4b43370087fd70de1477ab6"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d1cce020b4b43370087fd70de1477ab6"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d1cce020b4b43370087fd70de1477ab6"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d1cce020b4b43370087fd70de1477ab6"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_onD1"><span class="id" title="lemma">cfun_onD1</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">A</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#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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ca7f9c8131cd704a6703ad86f415c132"><span class="id" title="notation">^#</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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="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#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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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> 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="cfun_onG"><span class="id" title="lemma">cfun_onG</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#ClassFun.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.classfun.html#ClassFun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunD1E"><span class="id" title="lemma">cfunD1E</span></a> <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.classfun.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.classfun.html#ClassFun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ca7f9c8131cd704a6703ad86f415c132"><span class="id" title="notation">^#</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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="http://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#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> 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="cfunGid"><span class="id" title="lemma">cfunGid</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.classfun.html#ClassFun.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.classfun.html#ClassFun.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="http://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#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.classfun.html#ClassFun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_onS"><span class="id" title="lemma">cfun_onS</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#B"><span class="id" title="variable">B</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#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.character.classfun.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.classfun.html#ClassFun.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.classfun.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfun_complement"><span class="id" title="lemma">cfun_complement</span></a> <span class="id" title="var">A</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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#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.classfun.html#ClassFun.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.vector.html#df663072855a4e0a1a944084f6a33d9e"><span class="id" title="notation">+</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.classfun.html#ClassFun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#160b1f5d5c3e3c6d274924649d717d8d"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>%<span class="id" title="var">SET</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#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.classfun.html#ClassFun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfConjCE"><span class="id" title="lemma">cfConjCE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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="mathcomp.character.classfun.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.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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="cfConjCK"><span class="id" title="lemma">cfConjCK</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> (<span class="id" title="keyword">fun</span> <span class="id" title="var">phi</span> ⇒ <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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="cfConjC_cfun1"><span class="id" title="lemma">cfConjC_cfun1</span></a> : (1<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#b8b2ebc8e1a8b9aa935c0702efb5dccf"><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#b8b2ebc8e1a8b9aa935c0702efb5dccf"><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.classfun.html#ClassFun.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/> -</div> - -<div class="doc"> - Class function kernel and faithful class functions -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfker_is_group"><span class="id" title="lemma">cfker_is_group</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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfker_group</span> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfker_is_group"><span class="id" title="lemma">cfker_is_group</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_sub"><span class="id" title="lemma">cfker_sub</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.classfun.html#phi"><span class="id" title="variable">phi</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#ClassFun.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_norm"><span class="id" title="lemma">cfker_norm</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#ClassFun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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="cfker_normal"><span class="id" title="lemma">cfker_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.classfun.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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfkerMl"><span class="id" title="lemma">cfkerMl</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfkerMr"><span class="id" title="lemma">cfkerMr</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker1"><span class="id" title="lemma">cfker1</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_cfun0"><span class="id" title="lemma">cfker_cfun0</span></a> : @<a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.G"><span class="id" title="variable">G</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.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_add"><span class="id" title="lemma">cfker_add</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <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#phi"><span class="id" title="variable">phi</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.classfun.html#psi"><span class="id" title="variable">psi</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.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_sum"><span class="id" title="lemma">cfker_sum</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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">Phi</span> : <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#ClassFun.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.classfun.html#ClassFun.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#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.classfun.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><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#Phi"><span class="id" title="variable">Phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>)<br/> - <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.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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Phi"><span class="id" title="variable">Phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_scale"><span class="id" title="lemma">cfker_scale</span></a> <span class="id" title="var">a</span> <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.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_scale_nz"><span class="id" title="lemma">cfker_scale_nz</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#a"><span class="id" title="variable">a</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.classfun.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.classfun.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#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_opp"><span class="id" title="lemma">cfker_opp</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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.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#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_cfun1"><span class="id" title="lemma">cfker_cfun1</span></a> : @<a class="idref" href="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#ClassFun.G"><span class="id" title="variable">G</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="mathcomp.character.classfun.html#ClassFun.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_mul"><span class="id" title="lemma">cfker_mul</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <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#phi"><span class="id" title="variable">phi</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.classfun.html#psi"><span class="id" title="variable">psi</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.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_prod"><span class="id" title="lemma">cfker_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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">Phi</span> : <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#ClassFun.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.classfun.html#ClassFun.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#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.character.classfun.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0e81c215a8a995136d6989d77fd3e46b"><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#Phi"><span class="id" title="variable">Phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>)<br/> - <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.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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Phi"><span class="id" title="variable">Phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfaithfulE"><span class="id" title="lemma">cfaithfulE</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> <a class="idref" href="mathcomp.character.classfun.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="mathcomp.character.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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#b54f5e35cb228bba5934c852e0951c39"><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>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#ClassFun"><span class="id" title="section">ClassFun</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">"</span></a>''CF' ( G , A )" := (<a class="idref" href="mathcomp.character.classfun.html#classfun_on"><span class="id" title="definition">classfun_on</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span>) : <span class="id" title="var">ring_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">cfun_onT</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DotProduct"><span class="id" title="section">DotProduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="DotProduct.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="DotProduct.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.classfun.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">M</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.classfun.html#DotProduct.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> <span class="id" title="var">psi</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.classfun.html#DotProduct.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">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</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.classfun.html#DotProduct.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="cfdotE"><span class="id" title="lemma">cfdotE</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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#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.classfun.html#DotProduct.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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="cfdotElr"><span class="id" title="lemma">cfdotElr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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#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.classfun.html#DotProduct.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.classfun.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#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.classfun.html#A"><span class="id" title="variable">A</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#B"><span class="id" title="variable">B</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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="cfdotEl"><span class="id" title="lemma">cfdotEl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#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.classfun.html#A"><span class="id" title="variable">A</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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="cfdotEr"><span class="id" title="lemma">cfdotEr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#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.classfun.html#A"><span class="id" title="variable">A</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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_complement"><span class="id" title="lemma">cfdot_complement</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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#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.classfun.html#DotProduct.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.classfun.html#DotProduct.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#160b1f5d5c3e3c6d274924649d717d8d"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormE"><span class="id" title="lemma">cfnormE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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="http://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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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<a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_cfdotl"><span class="id" title="lemma">eq_cfdotl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi1</span> <span class="id" title="var">phi2</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi1"><span class="id" title="variable">phi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi2"><span class="id" title="variable">phi2</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.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#phi1"><span class="id" title="variable">phi1</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#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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi2"><span class="id" title="variable">phi2</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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_cfuni"><span class="id" title="lemma">cfdot_cfuni</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#8c3a7a3fdf9f8133b80af549f7126572"><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">_A</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#483aeb200c980494b369f3b9c18f042e"><span class="id" title="notation">'1</span></a><a class="idref" href="mathcomp.character.classfun.html#483aeb200c980494b369f3b9c18f042e"><span class="id" title="notation">_B</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.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#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#69c431a9c94f6f30a655bd7ddb59037b"><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.classfun.html#DotProduct.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/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm1"><span class="id" title="lemma">cfnorm1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">'[</span></a>1<a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><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> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdotrE"><span class="id" title="lemma">cfdotrE</span></a> <span class="id" title="var">psi</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfdotr"><span class="id" title="abbreviation">cfdotr</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.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#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>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdotr_is_linear"><span class="id" title="lemma">cfdotr_is_linear</span></a> <span class="id" title="var">xi</span> : <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.classfun.html#cfdotr"><span class="id" title="abbreviation">cfdotr</span></a> <a class="idref" href="mathcomp.character.classfun.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.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.classfun.html#DotProduct.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.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#32d8c90f413029fb5c0e82f0559cd7ef"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#32d8c90f413029fb5c0e82f0559cd7ef"><span class="id" title="notation">o</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfdotr_additive</span> <span class="id" title="var">xi</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.classfun.html#cfdotr_is_linear"><span class="id" title="lemma">cfdotr_is_linear</span></a> <a class="idref" href="mathcomp.character.classfun.html#xi"><span class="id" title="variable">xi</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfdotr_linear</span> <span class="id" title="var">xi</span> := <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.classfun.html#cfdotr_is_linear"><span class="id" title="lemma">cfdotr_is_linear</span></a> <a class="idref" href="mathcomp.character.classfun.html#xi"><span class="id" title="variable">xi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot0l"><span class="id" title="lemma">cfdot0l</span></a> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a>0<a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotNl"><span class="id" title="lemma">cfdotNl</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><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.classfun.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#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="http://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#8d0566c961139ec21811f52ef0c317db"><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#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#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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotDl"><span class="id" title="lemma">cfdotDl</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.classfun.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="http://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.classfun.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#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.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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#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.classfun.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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotBl"><span class="id" title="lemma">cfdotBl</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.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="http://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.classfun.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#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.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><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#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.classfun.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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotMnl"><span class="id" title="lemma">cfdotMnl</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</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#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="http://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.classfun.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#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.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.character.classfun.html#n"><span class="id" title="variable">n</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdot_suml"><span class="id" title="lemma">cfdot_suml</span></a> <span class="id" title="var">xi</span> <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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#DotProduct.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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">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.classfun.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="http://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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">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.classfun.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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotZl"><span class="id" title="lemma">cfdotZl</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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#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="http://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#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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdotC"><span class="id" title="lemma">cfdotC</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><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#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.classfun.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.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">)^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_cfdotr"><span class="id" title="lemma">eq_cfdotr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi1</span> <span class="id" title="var">psi2</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#DotProduct.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi1"><span class="id" title="variable">psi1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi2"><span class="id" title="variable">psi2</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.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#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#psi1"><span class="id" title="variable">psi1</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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#psi2"><span class="id" title="variable">psi2</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="cfdotBr"><span class="id" title="lemma">cfdotBr</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><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#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.classfun.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>.<br/> - <span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfun_dot_additive</span> <span class="id" title="var">xi</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.classfun.html#cfdotBr"><span class="id" title="lemma">cfdotBr</span></a> <a class="idref" href="mathcomp.character.classfun.html#xi"><span class="id" title="variable">xi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot0r"><span class="id" title="lemma">cfdot0r</span></a> <span class="id" title="var">xi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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> 0<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/> -<span class="id" title="keyword">Lemma</span> <a name="cfdotNr"><span class="id" title="lemma">cfdotNr</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.algebra.ssralg.html#8d0566c961139ec21811f52ef0c317db"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#8d0566c961139ec21811f52ef0c317db"><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#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.classfun.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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotDr"><span class="id" title="lemma">cfdotDr</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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#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.classfun.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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotMnr"><span class="id" title="lemma">cfdotMnr</span></a> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.classfun.html#n"><span class="id" title="variable">n</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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.algebra.ssralg.html#e9001f602764f7896bb1eb34bf606a23"><span class="id" title="notation">*+</span></a> <a class="idref" href="mathcomp.character.classfun.html#n"><span class="id" title="variable">n</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdot_sumr"><span class="id" title="lemma">cfdot_sumr</span></a> <span class="id" title="var">xi</span> <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.classfun.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#DotProduct.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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">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.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.classfun.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.classfun.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">]</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfdotZr"><span class="id" title="lemma">cfdotZr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">xi</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.classfun.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="http://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#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><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.classfun.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.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_cfAut"><span class="id" title="lemma">cfdot_cfAut</span></a> (<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>) <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.ssreflect.fintype.html#image"><span class="id" title="abbreviation">image</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#DotProduct.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#3d6621e6eef40dcc7dc9a612222d0b4e"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#3d6621e6eef40dcc7dc9a612222d0b4e"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#3d6621e6eef40dcc7dc9a612222d0b4e"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#3d6621e6eef40dcc7dc9a612222d0b4e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#3d6621e6eef40dcc7dc9a612222d0b4e"><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> <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#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.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.classfun.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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_conjC"><span class="id" title="lemma">cfdot_conjC</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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="http://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.classfun.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#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.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_conjCl"><span class="id" title="lemma">cfdot_conjCl</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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#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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_conjCr"><span class="id" title="lemma">cfdot_conjCr</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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="http://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.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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#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.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_ge0"><span class="id" title="lemma">cfnorm_ge0</span></a> <span class="id" title="var">phi</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><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.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_eq0"><span class="id" title="lemma">cfnorm_eq0</span></a> <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.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.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.classfun.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="cfnorm_gt0"><span class="id" title="lemma">cfnorm_gt0</span></a> <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.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#1f96a77ded31d6d5fa0c8fe9a087652a"><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.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sqrt_cfnorm_ge0"><span class="id" title="lemma">sqrt_cfnorm_ge0</span></a> <span class="id" title="var">phi</span> : 0 <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sqrt_cfnorm_eq0"><span class="id" title="lemma">sqrt_cfnorm_eq0</span></a> <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.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.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.classfun.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="sqrt_cfnorm_gt0"><span class="id" title="lemma">sqrt_cfnorm_gt0</span></a> <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.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#1f96a77ded31d6d5fa0c8fe9a087652a"><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.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormZ"><span class="id" title="lemma">cfnormZ</span></a> <span class="id" title="var">a</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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="http://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.classfun.html#a"><span class="id" title="variable">a</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 <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">_G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormN"><span class="id" title="lemma">cfnormN</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><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.classfun.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="http://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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_sign"><span class="id" title="lemma">cfnorm_sign</span></a> <span class="id" title="var">n</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><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.classfun.html#n"><span class="id" title="variable">n</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#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="http://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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormD"><span class="id" title="lemma">cfnormD</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">d</span> := <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</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> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.classfun.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormB"><span class="id" title="lemma">cfnormB</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">d</span> := <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#d"><span class="id" title="variable">d</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.classfun.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormDd"><span class="id" title="lemma">cfnormDd</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</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> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnormBd"><span class="id" title="lemma">cfnormBd</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_conjC"><span class="id" title="lemma">cfnorm_conjC</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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="http://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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfCauchySchwarz"><span class="id" title="lemma">cfCauchySchwarz</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><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#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#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.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 <a class="idref" href="mathcomp.algebra.ssrnum.html#8f3b4963db7e39cb42e593806a8ca50e"><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.classfun.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.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.classfun.html#psi"><span class="id" title="variable">psi</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfCauchySchwarz_sqrt"><span class="id" title="lemma">cfCauchySchwarz_sqrt</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><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#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#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.algebra.ssrnum.html#55297ec87c6b3f98c14c99daeafb55d3"><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#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cf_triangle_lerif"><span class="id" title="lemma">cf_triangle_lerif</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</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#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.sqrtC"><span class="id" title="abbreviation">sqrtC</span></a> <a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">]</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.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.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>0 <a class="idref" href="mathcomp.algebra.ssrnum.html#cb42ec59ad57b25928e1718b4e69e031"><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.ssreflect.tuple.html#44d42eed9f717544480811f482f3c650"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#44d42eed9f717544480811f482f3c650"><span class="id" title="notation">tuple</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.ssreflect.tuple.html#44d42eed9f717544480811f482f3c650"><span class="id" title="notation">]</span></a> 0 <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_cons"><span class="id" title="lemma">orthogonal_cons</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a>) <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthoP"><span class="id" title="lemma">orthoP</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.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</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#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#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="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonalP"><span class="id" title="lemma">orthogonalP</span></a> <span class="id" title="var">S</span> <span class="id" title="var">R</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.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span>, <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthoPl"><span class="id" title="lemma">orthoPl</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">S</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.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.classfun.html#S"><span class="id" title="variable">S</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">psi</span>, <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_sym"><span class="id" title="lemma">orthogonal_sym</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#symmetric"><span class="id" title="definition">symmetric</span></a> (@<a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#DotProduct.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthoPr"><span class="id" title="lemma">orthoPr</span></a> <span class="id" title="var">S</span> <span class="id" title="var">psi</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.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.classfun.html#S"><span class="id" title="variable">S</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">phi</span>, <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_orthogonal"><span class="id" title="lemma">eq_orthogonal</span></a> <span class="id" title="var">R1</span> <span class="id" title="var">R2</span> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#R1"><span class="id" title="variable">R1</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.character.classfun.html#R2"><span class="id" title="variable">R2</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#S1"><span class="id" title="variable">S1</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.character.classfun.html#S2"><span class="id" title="variable">S2</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#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="http://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#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_catl"><span class="id" title="lemma">orthogonal_catl</span></a> <span class="id" title="var">R1</span> <span class="id" title="var">R2</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> (<a class="idref" href="mathcomp.character.classfun.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.character.classfun.html#R2"><span class="id" title="variable">R2</span></a>) <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R1"><span class="id" title="variable">R1</span></a> <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R2"><span class="id" title="variable">R2</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_catr"><span class="id" title="lemma">orthogonal_catr</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> (<a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</span></a>) <a class="idref" href="http://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#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</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#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="span_orthogonal"><span class="id" title="lemma">span_orthogonal</span></a> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> <span class="id" title="var">phi1</span> <span class="id" title="var">phi2</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</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#phi1"><span class="id" title="variable">phi1</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.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</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.classfun.html#phi2"><span class="id" title="variable">phi2</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.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</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="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi1"><span class="id" title="variable">phi1</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#phi2"><span class="id" title="variable">phi2</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">Lemma</span> <a name="orthogonal_split"><span class="id" title="lemma">orthogonal_split</span></a> <span class="id" title="var">S</span> <span class="id" title="keyword">beta</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#2d3f7aca3c5e595bced87000c0854440"><span class="id" title="notation">{</span></a><span class="id" title="var">X</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#2d3f7aca3c5e595bced87000c0854440"><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.classfun.html#DotProduct.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#2d3f7aca3c5e595bced87000c0854440"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.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.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#2d3f7aca3c5e595bced87000c0854440"><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#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">Y</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.character.classfun.html#beta"><span class="id" title="variable">beta</span></a> <a class="idref" href="http://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#X"><span class="id" title="variable">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#c7f78cf1f6a5e4f664654f7d671ca752"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.character.classfun.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#d7e433f5d2fe56f5b712860a9ff2a681"><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#X"><span class="id" title="variable">X</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#Y"><span class="id" title="variable">Y</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.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#Y"><span class="id" title="variable">Y</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><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><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#2d3f7aca3c5e595bced87000c0854440"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_orthogonal"><span class="id" title="lemma">map_orthogonal</span></a> <span class="id" title="var">M</span> (<span class="id" title="var">nu</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.classfun.html#DotProduct.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.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.classfun.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) <span class="id" title="var">S</span> <span class="id" title="var">R</span> (<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.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.classfun.html#DotProduct.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="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><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#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.classfun.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#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#nu"><span class="id" title="variable">nu</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.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#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.classfun.html#S"><span class="id" title="variable">S</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.classfun.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#ca592708f529c7c7ee5f3dbd6cf93463"><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#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.classfun.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#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.character.classfun.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#ca592708f529c7c7ee5f3dbd6cf93463"><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#orthogonal"><span class="id" title="definition">orthogonal</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.character.classfun.html#nu"><span class="id" title="variable">nu</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.character.classfun.html#nu"><span class="id" title="variable">nu</span></a> <a class="idref" href="mathcomp.character.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_oppr"><span class="id" title="lemma">orthogonal_oppr</span></a> <span class="id" title="var">S</span> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.character.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_oppl"><span class="id" title="lemma">orthogonal_oppl</span></a> <span class="id" title="var">S</span> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.character.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pairwise_orthogonalP"><span class="id" title="lemma">pairwise_orthogonalP</span></a> <span class="id" title="var">S</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.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (0 <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>)<br/> - <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#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.classfun.html#S"><span class="id" title="variable">S</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> <span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span>, <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>)<br/> - (<a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pairwise_orthogonal_cat"><span class="id" title="lemma">pairwise_orthogonal_cat</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> (<a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><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.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</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.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</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">Lemma</span> <a name="eq_pairwise_orthogonal"><span class="id" title="lemma">eq_pairwise_orthogonal</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_pairwise_orthogonal"><span class="id" title="lemma">sub_pairwise_orthogonal</span></a> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> :<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.classfun.html#S1"><span class="id" title="variable">S1</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.classfun.html#S2"><span class="id" title="variable">S2</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="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.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</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#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</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#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthogonal_free"><span class="id" title="lemma">orthogonal_free</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_pairwise_orthogonal"><span class="id" title="lemma">filter_pairwise_orthogonal</span></a> <span class="id" title="var">S</span> <span class="id" title="var">p</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.character.classfun.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormal_not0"><span class="id" title="lemma">orthonormal_not0</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.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> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormalE"><span class="id" title="lemma">orthonormalE</span></a> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</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">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="mathcomp.character.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.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.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormal_orthogonal"><span class="id" title="lemma">orthonormal_orthogonal</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormal_cat"><span class="id" title="lemma">orthonormal_cat</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> (<a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.character.classfun.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#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#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</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.classfun.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.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</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.classfun.html#orthogonal"><span class="id" title="definition">orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</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">Lemma</span> <a name="eq_orthonormal"><span class="id" title="lemma">eq_orthonormal</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</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.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormal_free"><span class="id" title="lemma">orthonormal_free</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormalP"><span class="id" title="lemma">orthonormalP</span></a> <span class="id" title="var">S</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.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.character.classfun.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#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#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.classfun.html#S"><span class="id" title="variable">S</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> <span class="id" title="keyword">∀</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span>, <a class="idref" href="mathcomp.character.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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#6411ed08724033ae48d2865f0380d533"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>)<br/> - (<a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_orthonormal"><span class="id" title="lemma">sub_orthonormal</span></a> <span class="id" title="var">S1</span> <span class="id" title="var">S2</span> :<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.classfun.html#S1"><span class="id" title="variable">S1</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.classfun.html#S2"><span class="id" title="variable">S2</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="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.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</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#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S2"><span class="id" title="variable">S2</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#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.character.classfun.html#S1"><span class="id" title="variable">S1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="orthonormal2P"><span class="id" title="lemma">orthonormal2P</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><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#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#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.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><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.classfun.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="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.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><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.classfun.html#psi"><span class="id" title="variable">psi</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<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a><br/> - (<a class="idref" href="mathcomp.character.classfun.html#orthonormal"><span class="id" title="definition">orthonormal</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="conjC_pair_orthogonal"><span class="id" title="lemma">conjC_pair_orthogonal</span></a> <span class="id" title="var">S</span> <span class="id" title="var">chi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#cfConjC_closed"><span class="id" title="abbreviation">cfConjC_closed</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfReal"><span class="id" title="definition">cfReal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.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.classfun.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="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> (<a class="idref" href="mathcomp.character.classfun.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.character.classfun.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>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_real_conjC"><span class="id" title="lemma">cfdot_real_conjC</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfReal"><span class="id" title="definition">cfReal</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#8c3a7a3fdf9f8133b80af549f7126572"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><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.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.classfun.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#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.algebra.ssrnum.html#651a776cde67abfb8f7231c08f29d878"><span class="id" title="notation">^*</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="extend_cfConjC_subset"><span class="id" title="lemma">extend_cfConjC_subset</span></a> <span class="id" title="var">S</span> <span class="id" title="var">X</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#cfConjC_closed"><span class="id" title="abbreviation">cfConjC_closed</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfReal"><span class="id" title="definition">cfReal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.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#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> <a class="idref" href="mathcomp.character.classfun.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#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.character.classfun.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#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfConjC_subset"><span class="id" title="definition">cfConjC_subset</span></a> <a class="idref" href="mathcomp.character.classfun.html#X"><span class="id" title="variable">X</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#cfConjC_subset"><span class="id" title="definition">cfConjC_subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#b24325f513e9801012e6322700f34266"><span class="id" title="notation">^*</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#X"><span class="id" title="variable">X</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">]</span></a>%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Note: other isometry lemmas, and the dot product lemmas for orthogonal - and orthonormal sequences are in vcharacter, because we need the 'Z[S] - notation for the isometry domains. Alternatively, this could be moved to - cfun. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#DotProduct"><span class="id" title="section">DotProduct</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="CfunOrder"><span class="id" title="section">CfunOrder</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="CfunOrder.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="CfunOrder.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.classfun.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>) (<a name="CfunOrder.phi"><span class="id" title="variable">phi</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.classfun.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dvdn_cforderP"><span class="id" title="lemma">dvdn_cforderP</span></a> <span class="id" title="var">n</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.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.classfun.html#CfunOrder.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>, <a class="idref" href="mathcomp.character.classfun.html#CfunOrder.phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#n"><span class="id" title="variable">n</span></a> <a 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.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><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.classfun.html#CfunOrder.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> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.character.classfun.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="dvdn_cforder"><span class="id" title="lemma">dvdn_cforder</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#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.classfun.html#CfunOrder.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> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.character.classfun.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://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#CfunOrder.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.classfun.html#n"><span class="id" title="variable">n</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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="exp_cforder"><span class="id" title="lemma">exp_cforder</span></a> : <a class="idref" href="mathcomp.character.classfun.html#CfunOrder.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.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#CfunOrder.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> <a 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">End</span> <a class="idref" href="mathcomp.character.classfun.html#CfunOrder"><span class="id" title="section">CfunOrder</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphOrder"><span class="id" title="section">MorphOrder</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MorphOrder.aT"><span class="id" title="variable">aT</span></a> <a name="MorphOrder.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="MorphOrder.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.classfun.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="MorphOrder.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.classfun.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="keyword">Variable</span> <a name="MorphOrder.f"><span class="id" title="variable">f</span></a> : <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.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.classfun.html#MorphOrder.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.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.classfun.html#MorphOrder.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</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="cforder_rmorph"><span class="id" title="lemma">cforder_rmorph</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.classfun.html#MorphOrder.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.classfun.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">Lemma</span> <a name="cforder_inj_rmorph"><span class="id" title="lemma">cforder_inj_rmorph</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#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphOrder.f"><span class="id" title="variable">f</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#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#MorphOrder.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://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.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#MorphOrder"><span class="id" title="section">MorphOrder</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="BuildIsometries"><span class="id" title="section">BuildIsometries</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="BuildIsometries.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="BuildIsometries.L"><span class="id" title="variable">L</span></a> <a name="BuildIsometries.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.classfun.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">psi</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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><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">U</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.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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">W</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.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.classfun.html#BuildIsometries.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="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_iso_to"><span class="id" title="lemma">sub_iso_to</span></a> <span class="id" title="var">U1</span> <span class="id" title="var">U2</span> <span class="id" title="var">W1</span> <span class="id" title="var">W2</span> <span class="id" title="var">tau</span> :<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.classfun.html#U2"><span class="id" title="variable">U2</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.classfun.html#U1"><span class="id" title="variable">U1</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="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#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.classfun.html#W1"><span class="id" title="variable">W1</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.classfun.html#W2"><span class="id" title="variable">W2</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="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#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#U1"><span class="id" title="variable">U1</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.character.classfun.html#W1"><span class="id" title="variable">W1</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><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#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#U2"><span class="id" title="variable">U2</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.character.classfun.html#W2"><span class="id" title="variable">W2</span></a><a class="idref" href="mathcomp.character.classfun.html#9e012b901a34756891cc773ca063f3a7"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isometry_of_free"><span class="id" title="lemma">isometry_of_free</span></a> <span class="id" title="var">S</span> <span class="id" title="var">f</span> :<br/> - <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> <a class="idref" href="mathcomp.character.classfun.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> <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.classfun.html#S"><span class="id" title="variable">S</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.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</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.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.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">tau</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.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><span class="id" title="notation">linear</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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</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.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.classfun.html#BuildIsometries.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="mathcomp.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><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#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.classfun.html#S"><span class="id" title="variable">S</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="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</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.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#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.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</span> <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.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</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.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isometry_of_cfnorm"><span class="id" title="lemma">isometry_of_cfnorm</span></a> <span class="id" title="var">S</span> <span class="id" title="var">tauS</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#pairwise_orthogonal"><span class="id" title="definition">pairwise_orthogonal</span></a> <a class="idref" href="mathcomp.character.classfun.html#tauS"><span class="id" title="variable">tauS</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.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfnorm"><span class="id" title="abbreviation">cfnorm</span></a> <a class="idref" href="mathcomp.character.classfun.html#tauS"><span class="id" title="variable">tauS</span></a> <a class="idref" href="http://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.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfnorm"><span class="id" title="abbreviation">cfnorm</span></a> <a class="idref" href="mathcomp.character.classfun.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.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">tau</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.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><span class="id" title="notation">linear</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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</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.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.classfun.html#BuildIsometries.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="mathcomp.algebra.ssralg.html#b9a9030f88e15d1a3aacd4e8ec9a2391"><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="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#tauS"><span class="id" title="variable">tauS</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#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.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.algebra.vector.html#fb707feae4acc20b3f4404c2e515b2a1"><span class="id" title="notation">>></span></a>%<span class="id" title="var">VS</span> <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.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</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.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isometry_raddf_inj"><span class="id" title="lemma">isometry_raddf_inj</span></a> <span class="id" title="var">U</span> (<span class="id" title="var">tau</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#b15d1bebaaff5b5ed693647b6d36f348"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b15d1bebaaff5b5ed693647b6d36f348"><span class="id" title="notation">additive</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.classfun.html#BuildIsometries.L"><span class="id" title="variable">L</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.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.classfun.html#BuildIsometries.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="mathcomp.algebra.ssralg.html#b15d1bebaaff5b5ed693647b6d36f348"><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#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.classfun.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#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#tau"><span class="id" title="variable">tau</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.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.classfun.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#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">u</span> <span class="id" title="var">v</span>, <a class="idref" href="mathcomp.character.classfun.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#51dc792c356ca1a71a3094b50d6bb2fb"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.character.classfun.html#v"><span class="id" title="variable">v</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#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#b4f176550f5b849a7fbba2ee164934d3"><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.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.classfun.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#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.classfun.html#tau"><span class="id" title="variable">tau</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="opp_isometry"><span class="id" title="lemma">opp_isometry</span></a> : @<a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#BuildIsometries.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#BuildIsometries.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">-%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#a8ac36d488c8d5cdcfec5adcde894e5f"><span class="id" title="notation">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#BuildIsometries"><span class="id" title="section">BuildIsometries</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">Variables</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.A"><span class="id" title="variable">A</span></a> <a name="Restrict.B"><span class="id" title="variable">B</span></a> : <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.classfun.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>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfRes_subproof"><span class="id" title="lemma">cfRes_subproof</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.classfun.html#Restrict.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/> - <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.character.classfun.html#H"><span class="id" title="abbreviation">H</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.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="abbreviation">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#G"><span class="id" title="abbreviation">G</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.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</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.classfun.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.classfun.html#H"><span class="id" title="abbreviation">H</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="cfRes"><span class="id" title="definition">cfRes</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 1 (<a class="idref" href="mathcomp.character.classfun.html#cfRes_subproof"><span class="id" title="lemma">cfRes_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfResE"><span class="id" title="lemma">cfResE</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#Restrict.A"><span class="id" title="variable">A</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#Restrict.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#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#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.classfun.html#Restrict.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes1"><span class="id" title="lemma">cfRes1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_is_linear"><span class="id" title="lemma">cfRes_is_linear</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.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfRes_additive</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.classfun.html#cfRes_is_linear"><span class="id" title="lemma">cfRes_is_linear</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfRes_linear</span> := <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.classfun.html#cfRes_is_linear"><span class="id" title="lemma">cfRes_is_linear</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_cfun1"><span class="id" title="lemma">cfRes_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</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> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_is_multiplicative"><span class="id" title="lemma">cfRes_is_multiplicative</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.multiplicative"><span class="id" title="abbreviation">multiplicative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfRes_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.AddRMorphism"><span class="id" title="abbreviation">AddRMorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfRes_is_multiplicative"><span class="id" title="lemma">cfRes_is_multiplicative</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfRes_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Restrict"><span class="id" title="section">Restrict</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">"</span></a>''Res[' H , G ]" := (@<a class="idref" href="mathcomp.character.classfun.html#cfRes"><span class="id" title="definition">cfRes</span></a> <span class="id" title="var">_</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">"</span></a>''Res[' H ]" := <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><span class="id" title="var">H</span><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a> : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">"</span></a>''Res'" := <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><span class="id" title="var">_</span><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a> (<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="MoreRestrict"><span class="id" title="section">MoreRestrict</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MoreRestrict.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="MoreRestrict.G"><span class="id" title="variable">G</span></a> <a name="MoreRestrict.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.classfun.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">A</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.classfun.html#MoreRestrict.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.classfun.html#MoreRestrict.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="cfResEout"><span class="id" title="lemma">cfResEout</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#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#MoreRestrict.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#MoreRestrict.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#b3ebd0deddd84fd60e149cb5ef719351"><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#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.classfun.html#MoreRestrict.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.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfResRes"><span class="id" title="lemma">cfResRes</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#MoreRestrict.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.classfun.html#MoreRestrict.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#MoreRestrict.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#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.classfun.html#A"><span class="id" title="variable">A</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">'</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.classfun.html#MoreRestrict.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.classfun.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#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.classfun.html#A"><span class="id" title="variable">A</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#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_id"><span class="id" title="lemma">cfRes_id</span></a> <span class="id" title="var">A</span> <span class="id" title="var">psi</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.classfun.html#A"><span class="id" title="variable">A</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#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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_cfker_Res"><span class="id" title="lemma">sub_cfker_Res</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#MoreRestrict.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.classfun.html#A"><span class="id" title="variable">A</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.classfun.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.classfun.html#A"><span class="id" title="variable">A</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.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.classfun.html#MoreRestrict.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.classfun.html#MoreRestrict.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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_cfker_Res"><span class="id" title="lemma">eq_cfker_Res</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#MoreRestrict.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.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</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.classfun.html#MoreRestrict.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.classfun.html#MoreRestrict.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.classfun.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#MoreRestrict.H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_sub_ker"><span class="id" title="lemma">cfRes_sub_ker</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#MoreRestrict.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.classfun.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.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.classfun.html#MoreRestrict.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.classfun.html#MoreRestrict.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.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_Res"><span class="id" title="lemma">cforder_Res</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.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.classfun.html#MoreRestrict.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.classfun.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> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><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.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#MoreRestrict"><span class="id" title="section">MoreRestrict</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.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.classfun.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.classfun.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.classfun.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/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Morphim.Main"><span class="id" title="section">Main</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Morphim.Main.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.classfun.html#Morphim.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>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</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.classfun.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.classfun.html#Morphim.Main.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">Fact</span> <a name="cfMorph_subproof"><span class="id" title="lemma">cfMorph_subproof</span></a> <span class="id" title="var">phi</span> :<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.classfun.html#Morphim.Main.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><br/> - <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.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.character.classfun.html#Morphim.Main.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#Morphim.D"><span class="id" title="variable">D</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.character.classfun.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</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.classfun.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.classfun.html#Morphim.Main.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="cfMorph"><span class="id" title="definition">cfMorph</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 1 (<a class="idref" href="mathcomp.character.classfun.html#cfMorph_subproof"><span class="id" title="lemma">cfMorph_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorphE"><span class="id" title="lemma">cfMorphE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.html#Morphim.Main.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#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="mathcomp.character.classfun.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.classfun.html#Morphim.Main.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#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph1"><span class="id" title="lemma">cfMorph1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorphEout"><span class="id" title="lemma">cfMorphEout</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#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#Morphim.Main.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#Morphim.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_cfun1"><span class="id" title="lemma">cfMorph_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</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> 1.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfMorph_is_linear"><span class="id" title="lemma">cfMorph_is_linear</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMorph_additive</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.classfun.html#cfMorph_is_linear"><span class="id" title="lemma">cfMorph_is_linear</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMorph_linear</span> := <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.classfun.html#cfMorph_is_linear"><span class="id" title="lemma">cfMorph_is_linear</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfMorph_is_multiplicative"><span class="id" title="lemma">cfMorph_is_multiplicative</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.multiplicative"><span class="id" title="abbreviation">multiplicative</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMorph_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.RMorphism.Exports.AddRMorphism"><span class="id" title="abbreviation">AddRMorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph_is_multiplicative"><span class="id" title="lemma">cfMorph_is_multiplicative</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMorph_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Morphim.Main.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.character.classfun.html#Morphim.Main.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#Morphim.D"><span class="id" title="variable">D</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_inj"><span class="id" title="lemma">cfMorph_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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMorph_eq1"><span class="id" title="lemma">cfMorph_eq1</span></a> <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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_morph"><span class="id" title="lemma">cfker_morph</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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#Morphim.Main.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.character.classfun.html#Morphim.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#320f70d30c9a649ec82642b364681418"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#320f70d30c9a649ec82642b364681418"><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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#320f70d30c9a649ec82642b364681418"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_morph_im"><span class="id" title="lemma">cfker_morph_im</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</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.classfun.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#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_cfker_morph"><span class="id" title="lemma">sub_cfker_morph</span></a> <span class="id" title="var">phi</span> (<span class="id" title="var">A</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.classfun.html#Morphim.aT"><span class="id" title="variable">aT</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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#Morphim.Main.G"><span class="id" title="variable">G</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#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.classfun.html#A"><span class="id" title="variable">A</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.classfun.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.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_morphim_cfker"><span class="id" title="lemma">sub_morphim_cfker</span></a> <span class="id" title="var">phi</span> (<span class="id" title="var">A</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.classfun.html#Morphim.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</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#Morphim.Main.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.classfun.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.classfun.html#A"><span class="id" title="variable">A</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.classfun.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#A"><span class="id" title="variable">A</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_morph"><span class="id" title="lemma">cforder_morph</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://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.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#Morphim.Main"><span class="id" title="section">Main</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfResMorph"><span class="id" title="lemma">cfResMorph</span></a> (<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.classfun.html#Morphim.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">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.classfun.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.classfun.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.classfun.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#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#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#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="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">cfMorph</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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</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.classfun.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.classfun.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.classfun.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.classfun.html#Morphim"><span class="id" title="section">Morphim</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Isomorphism"><span class="id" title="section">Isomorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Isomorphism.aT"><span class="id" title="variable">aT</span></a> <a name="Isomorphism.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="Isomorphism.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.classfun.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="Isomorphism.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.classfun.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.classfun.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">Variable</span> <a name="Isomorphism.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.classfun.html#Isomorphism.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/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Isomorphism.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.classfun.html#Isomorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#Isomorphism.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#Isomorphism.f"><span class="id" title="variable">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Isomorphism.defR"><span class="id" title="variable">defR</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#isom_im"><span class="id" title="lemma">isom_im</span></a> <a class="idref" href="mathcomp.character.classfun.html#Isomorphism.isoGR"><span class="id" title="variable">isoGR</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Isomorphism.defG"><span class="id" title="variable">defG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#G1"><span class="id" title="abbreviation">G1</span></a> <a class="idref" href="http://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#Isomorphism.G"><span class="id" title="variable">G</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#isom_im"><span class="id" title="lemma">isom_im</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.classfun.html#Isomorphism.isoGR"><span class="id" title="variable">isoGR</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfIsom_key"><span class="id" title="lemma">cfIsom_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="cfIsom"><span class="id" title="definition">cfIsom</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.classfun.html#cfIsom_key"><span class="id" title="lemma">cfIsom_key</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</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.classfun.html#G1"><span class="id" title="abbreviation">G1</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><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.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.classfun.html#Isomorphism.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.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.classfun.html#Isomorphism.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>).<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfIsom_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsomE"><span class="id" title="lemma">cfIsomE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Isomorphism.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#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#Isomorphism.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom1"><span class="id" title="lemma">cfIsom1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfIsom_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfIsom_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfIsom_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfIsom_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_cfun1"><span class="id" title="lemma">cfIsom_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</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> 1. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_isom"><span class="id" title="lemma">cfker_isom</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.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.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#Isomorphism.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Isomorphism"><span class="id" title="section">Isomorphism</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="InvMorphism"><span class="id" title="section">InvMorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="InvMorphism.aT"><span class="id" title="variable">aT</span></a> <a name="InvMorphism.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="InvMorphism.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.classfun.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="InvMorphism.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.classfun.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.classfun.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">Variable</span> <a name="InvMorphism.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.classfun.html#InvMorphism.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/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="InvMorphism.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.classfun.html#InvMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.f"><span class="id" title="variable">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsomK"><span class="id" title="lemma">cfIsomK</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#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a>) (<a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</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.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsomKV"><span class="id" title="lemma">cfIsomKV</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#cfIsom"><span class="id" title="definition">cfIsom</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.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</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.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_inj"><span class="id" title="lemma">cfIsom_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.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a>). <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_eq1"><span class="id" title="lemma">cfIsom_eq1</span></a> <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.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_isom"><span class="id" title="lemma">cforder_isom</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.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism.isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://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.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#InvMorphism"><span class="id" title="section">InvMorphism</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">Variables</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>) (<a name="Coset.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.classfun.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>) (<a name="Coset.B"><span class="id" title="variable">B</span></a> : <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.classfun.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>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</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">Definition</span> <a name="cfMod"><span class="id" title="definition">cfMod</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.classfun.html#Coset.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#Coset.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.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.classfun.html#Coset.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="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="ffun_Quo"><span class="id" title="definition">ffun_Quo</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.classfun.html#Coset.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.ssreflect.finfun.html#e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">ffun</span></a> <span class="id" title="var">Hx</span> <a class="idref" href="mathcomp.ssreflect.finfun.html#e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">⇒</span></a><br/> - <a class="idref" href="mathcomp.character.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</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.classfun.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.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> <a class="idref" href="mathcomp.character.classfun.html#Hx"><span class="id" title="variable">Hx</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> 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.character.classfun.html#Hx"><span class="id" title="variable">Hx</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#Coset.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#Coset.B"><span class="id" title="variable">B</span></a>)%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.finfun.html#e4e2ffb93b77700f7a723d1db6d75bdf"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Fact</span> <a name="cfQuo_subproof"><span class="id" title="lemma">cfQuo_subproof</span></a> <span class="id" title="var">phi</span> : <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.classfun.html#Coset.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#Coset.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#ffun_Quo"><span class="id" title="definition">ffun_Quo</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="cfQuo"><span class="id" title="definition">cfQuo</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 1 (<a class="idref" href="mathcomp.character.classfun.html#cfQuo_subproof"><span class="id" title="lemma">cfQuo_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - We specialize the cfMorph lemmas to cfMod by strengthening the domain - condition G \subset 'N(H) to H <| G; the cfMorph lemmas can be used if the - stronger results are needed. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfModE"><span class="id" title="lemma">cfModE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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#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.classfun.html#Coset.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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</span></a>)%<span class="id" title="var">CF</span> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod1"><span class="id" title="lemma">cfMod1</span></a> <span class="id" title="var">phi</span> : (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>. <br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMod_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMod_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMod_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfMod_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_cfun1"><span class="id" title="lemma">cfMod_cfun1</span></a> : (1 <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</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="cfker_mod"><span class="id" title="lemma">cfker_mod</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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#Coset.B"><span class="id" title="variable">B</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Note that cfQuo is nondegenerate even when G does not normalize B. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoEnorm"><span class="id" title="lemma">cfQuoEnorm</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.classfun.html#Coset.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.classfun.html#Coset.B"><span class="id" title="variable">B</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.classfun.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.classfun.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.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">N_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ee98cf35a816a182ecdf169a5f07c7f5"><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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</span></a>)%<span class="id" title="var">CF</span> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoE"><span class="id" title="lemma">cfQuoE</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.classfun.html#Coset.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.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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#Coset.B"><span class="id" title="variable">B</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.classfun.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.classfun.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.classfun.html#Coset.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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</span></a>)%<span class="id" title="var">CF</span> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo1"><span class="id" title="lemma">cfQuo1</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.classfun.html#Coset.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="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoEout"><span class="id" title="lemma">cfQuoEout</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.classfun.html#Coset.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.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</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.classfun.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#b3ebd0deddd84fd60e149cb5ef719351"><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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</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.algebra.ssralg.html#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#862982ed16052c855fd1cdb6c8e69ea7"><span class="id" title="notation">A</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - cfQuo is only linear on the class functions that have H in their kernel. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_cfun1"><span class="id" title="lemma">cfQuo_cfun1</span></a> : (1 <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</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/> -</div> - -<div class="doc"> - Cancellation properties -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfModK"><span class="id" title="lemma">cfModK</span></a> : <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfQuo"><span class="id" title="definition">cfQuo</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoK"><span class="id" title="lemma">cfQuoK</span></a> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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">phi</span>, <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</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.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</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.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_eq1"><span class="id" title="lemma">cfMod_eq1</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#57f3294a3433a9d6d9da43648e3842d3"><span class="id" title="notation">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1)%<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.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_eq1"><span class="id" title="lemma">cfQuo_eq1</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#Coset.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Coset.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#Coset.B"><span class="id" title="variable">B</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.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#f776f4454e0f767d03ceed800abc137e"><span class="id" title="notation">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1)%<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.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Coset"><span class="id" title="section">Coset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">"</span></a>phi / H" := (<a class="idref" href="mathcomp.character.classfun.html#cfQuo"><span class="id" title="definition">cfQuo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">phi</span>) : <span class="id" title="var">cfun_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">"</span></a>phi %% H" := (@<a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H</span> <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="MoreCoset"><span class="id" title="section">MoreCoset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MoreCoset.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="MoreCoset.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.classfun.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">H</span> <span class="id" title="var">K</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.classfun.html#MoreCoset.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> : <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.classfun.html#MoreCoset.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="cfResMod"><span class="id" title="lemma">cfResMod</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</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.classfun.html#MoreCoset.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#K"><span class="id" title="variable">K</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.classfun.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#MoreCoset.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#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#MoreCoset.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">(</span></a><a class="idref" href="mathcomp.character.classfun.html#06cfee1f9bee2f85a584c7483fb59a63"><span class="id" title="notation">psi</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">K</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.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.classfun.html#H"><span class="id" title="variable">H</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#K"><span class="id" title="variable">K</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#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#K"><span class="id" title="variable">K</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_cfker_mod"><span class="id" title="lemma">quotient_cfker_mod</span></a> (<span class="id" title="var">A</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.classfun.html#MoreCoset.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">K</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.classfun.html#MoreCoset.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#K"><span class="id" title="variable">K</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.classfun.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#MoreCoset.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#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#K"><span class="id" title="variable">K</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#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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><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#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_cfker_mod"><span class="id" title="lemma">sub_cfker_mod</span></a> (<span class="id" title="var">A</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.classfun.html#MoreCoset.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">K</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.classfun.html#MoreCoset.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#K"><span class="id" title="variable">K</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.classfun.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#MoreCoset.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#A"><span class="id" title="variable">A</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.classfun.html#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> <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#A"><span class="id" title="variable">A</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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#A"><span class="id" title="variable">A</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#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.classfun.html#psi"><span class="id" title="variable">psi</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_quo"><span class="id" title="lemma">cfker_quo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MoreCoset.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#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.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoEker"><span class="id" title="lemma">cfQuoEker</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MoreCoset.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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><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#phi"><span class="id" title="variable">phi</span></a>)%<span class="id" title="var">CF</span> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</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#phi"><span class="id" title="variable">phi</span></a>) <a class="idref" href="mathcomp.character.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfaithful_quo"><span class="id" title="lemma">cfaithful_quo</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfaithful"><span class="id" title="definition">cfaithful</span></a> (<a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><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#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Note that there is no requirement that K be normal in H or G. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfResQuo"><span class="id" title="lemma">cfResQuo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#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.classfun.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.classfun.html#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#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.classfun.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#MoreCoset.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#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.classfun.html#H"><span class="id" title="variable">H</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#K"><span class="id" title="variable">K</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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.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.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.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#K"><span class="id" title="variable">K</span></a>)%<span class="id" title="var">CF</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuoInorm"><span class="id" title="lemma">cfQuoInorm</span></a> <span class="id" title="var">K</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#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.classfun.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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#K"><span class="id" title="variable">K</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.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">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">N_G</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">K</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">phi</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">K</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">CF</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_mod"><span class="id" title="lemma">cforder_mod</span></a> <span class="id" title="var">H</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.classfun.html#MoreCoset.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#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>) : <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MoreCoset.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#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="variable">H</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.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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">Lemma</span> <a name="cforder_quo"><span class="id" title="lemma">cforder_quo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MoreCoset.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#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.classfun.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.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="variable">H</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.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#MoreCoset"><span class="id" title="section">MoreCoset</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Product"><span class="id" title="section">Product</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Product.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="Product.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.classfun.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="cfunM_onI"><span class="id" title="lemma">cfunM_onI</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Product.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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#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.classfun.html#Product.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.classfun.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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.classfun.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#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.classfun.html#Product.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.classfun.html#A"><span class="id" title="variable">A</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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfunM_on"><span class="id" title="lemma">cfunM_on</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Product.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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#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.classfun.html#Product.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#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.classfun.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#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.classfun.html#Product.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Product"><span class="id" title="section">Product</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="SDproduct"><span class="id" title="section">SDproduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="SDproduct.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="SDproduct.G"><span class="id" title="variable">G</span></a> <a name="SDproduct.K"><span class="id" title="variable">K</span></a> <a name="SDproduct.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.classfun.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="SDproduct.defG"><span class="id" title="variable">defG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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.classfun.html#SDproduct.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.classfun.html#SDproduct.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfSdprodKey"><span class="id" title="lemma">cfSdprodKey</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/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfSdprod"><span class="id" title="definition">cfSdprod</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.classfun.html#cfSdprodKey"><span class="id" title="lemma">cfSdprodKey</span></a><br/> - (<a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfIsom"><span class="id" title="definition">cfIsom</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#tagged"><span class="id" title="definition">tagged</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_isom"><span class="id" title="lemma">sdprod_isom</span></a> <a class="idref" href="mathcomp.character.classfun.html#SDproduct.defG"><span class="id" title="variable">defG</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.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.classfun.html#SDproduct.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> <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.classfun.html#SDproduct.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">Canonical</span> <span class="id" title="var">cfSdprod_unlockable</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">unlockable</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#5ad020f08aa5d2a22e22f3b18f63fcd0"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfSdprod_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfSdprod_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfSdprod_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfSdprod_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod1"><span class="id" title="lemma">cfSdprod1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="SDproduct.nsKG"><span class="id" title="variable">nsKG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#SDproduct.G"><span class="id" title="variable">G</span></a>. <br/> -<span class="id" title="keyword">Let</span> <a name="SDproduct.sHG"><span class="id" title="variable">sHG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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#SDproduct.G"><span class="id" title="variable">G</span></a>. <br/> -<span class="id" title="keyword">Let</span> <a name="SDproduct.sKG"><span class="id" title="variable">sKG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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#SDproduct.G"><span class="id" title="variable">G</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_sdprod"><span class="id" title="lemma">cfker_sdprod</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprodEr"><span class="id" title="lemma">cfSdprodEr</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#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.classfun.html#SDproduct.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#8c08d4203604dbed63e7afa9b689d858"><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.classfun.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.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprodE"><span class="id" title="lemma">cfSdprodE</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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.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.classfun.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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprodK"><span class="id" title="lemma">cfSdprodK</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#cfSdprod"><span class="id" title="definition">cfSdprod</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.classfun.html#SDproduct.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_inj"><span class="id" title="lemma">cfSdprod_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.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_eq1"><span class="id" title="lemma">cfSdprod_eq1</span></a> <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.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfRes_sdprodK"><span class="id" title="lemma">cfRes_sdprodK</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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.classfun.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.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</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.classfun.html#SDproduct.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.classfun.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#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sdprod_cfker"><span class="id" title="lemma">sdprod_cfker</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#SDproduct.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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#cfker"><span class="id" title="definition">cfker</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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_sdprod"><span class="id" title="lemma">cforder_sdprod</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.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://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.classfun.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">End</span> <a class="idref" href="mathcomp.character.classfun.html#SDproduct"><span class="id" title="section">SDproduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="DProduct"><span class="id" title="section">DProduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="DProduct.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="DProduct.G"><span class="id" title="variable">G</span></a> <a name="DProduct.K"><span class="id" title="variable">K</span></a> <a name="DProduct.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.classfun.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="DProduct.KxH"><span class="id" title="variable">KxH</span></a> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.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.classfun.html#DProduct.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.classfun.html#DProduct.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reindex_dprod"><span class="id" title="lemma">reindex_dprod</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.classfun.html#idx"><span class="id" title="variable">idx</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.gT"><span class="id" title="variable">gT</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#R"><span class="id" title="variable">R</span></a>) :<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.classfun.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.classfun.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">g</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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.classfun.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.character.classfun.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/> - <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.classfun.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.classfun.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">k</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.K"><span class="id" title="variable">K</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">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.classfun.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.classfun.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">h</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.character.classfun.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#h"><span class="id" title="variable">h</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDprodr"><span class="id" title="definition">cfDprodr</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#dprodWsd"><span class="id" title="lemma">dprodWsd</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.KxH"><span class="id" title="variable">KxH</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDprodl"><span class="id" title="definition">cfDprodl</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfSdprod"><span class="id" title="definition">cfSdprod</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#dprodWsdC"><span class="id" title="lemma">dprodWsdC</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.KxH"><span class="id" title="variable">KxH</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="cfDprod"><span class="id" title="definition">cfDprod</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> := <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodl_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodl_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodl_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodl_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodr_additive</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodr_linear</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodr_rmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfDprodr_lrmorphism</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodl1"><span class="id" title="lemma">cfDprodl1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodr1"><span class="id" title="lemma">cfDprodr1</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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.classfun.html#psi"><span class="id" title="variable">psi</span></a> 1%<span class="id" title="var">g</span>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod1"><span class="id" title="lemma">cfDprod1</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodl_eq1"><span class="id" title="lemma">cfDprodl_eq1</span></a> <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.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#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>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfDprodr_eq1"><span class="id" title="lemma">cfDprodr_eq1</span></a> <span class="id" title="var">psi</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.classfun.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.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_cfun1r"><span class="id" title="lemma">cfDprod_cfun1r</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfDprod_cfun1l"><span class="id" title="lemma">cfDprod_cfun1l</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> 1 <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfDprod_cfun1"><span class="id" title="lemma">cfDprod_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> 1 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> 1.<br/> - <span class="id" title="keyword">Lemma</span> <a name="cfDprod_split"><span class="id" title="lemma">cfDprod_split</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1 <a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> 1 <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="DProduct.nsKG"><span class="id" title="variable">nsKG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.G"><span class="id" title="variable">G</span></a>. <br/> -<span class="id" title="keyword">Let</span> <a name="DProduct.nsHG"><span class="id" title="variable">nsHG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.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.classfun.html#DProduct.G"><span class="id" title="variable">G</span></a>. <br/> -<span class="id" title="keyword">Let</span> <a name="DProduct.cKH"><span class="id" title="variable">cKH</span></a> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.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.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#DProduct.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a>. <br/> -<span class="id" title="keyword">Let</span> <a name="DProduct.sKG"><span class="id" title="variable">sKG</span></a> := <a class="idref" href="mathcomp.fingroup.fingroup.html#normal_sub"><span class="id" title="lemma">normal_sub</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.nsKG"><span class="id" title="variable">nsKG</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="DProduct.sHG"><span class="id" title="variable">sHG</span></a> := <a class="idref" href="mathcomp.fingroup.fingroup.html#normal_sub"><span class="id" title="lemma">normal_sub</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.nsHG"><span class="id" title="variable">nsHG</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodlK"><span class="id" title="lemma">cfDprodlK</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#cfDprodl"><span class="id" title="definition">cfDprodl</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.classfun.html#DProduct.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.character.classfun.html#99c53e21af8dfe7d98f8f1da9621ebb6"><span class="id" title="notation">]</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodrK"><span class="id" title="lemma">cfDprodrK</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#cfDprodr"><span class="id" title="definition">cfDprodr</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.classfun.html#DProduct.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>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_dprodl"><span class="id" title="lemma">cfker_dprodl</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.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#DProduct.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.classfun.html#cfker"><span class="id" title="definition">cfker</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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_dprodr"><span class="id" title="lemma">cfker_dprodr</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#DProduct.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.classfun.html#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><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#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodEl"><span class="id" title="lemma">cfDprodEl</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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">k</span> <span class="id" title="var">h</span>, <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#h"><span class="id" title="variable">h</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodEr"><span class="id" title="lemma">cfDprodEr</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">k</span> <span class="id" title="var">h</span>, <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.html#h"><span class="id" title="variable">h</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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodE"><span class="id" title="lemma">cfDprodE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.character.classfun.html#DProduct.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">h</span> <span class="id" title="var">k</span>, <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.character.classfun.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.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#h"><span class="id" title="variable">h</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#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.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#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_Resl"><span class="id" title="lemma">cfDprod_Resl</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.classfun.html#DProduct.K"><span class="id" title="variable">K</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#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprod_Resr"><span class="id" title="lemma">cfDprod_Resr</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</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.classfun.html#DProduct.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.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.character.classfun.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#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodKl"><span class="id" title="lemma">cfDprodKl</span></a> (<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.classfun.html#DProduct.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>) : <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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="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="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#psi"><span class="id" title="variable">psi</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"><span class="id" title="lemma">cfDprodKr</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.classfun.html#DProduct.K"><span class="id" title="variable">K</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#phi"><span class="id" title="variable">phi</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="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.classfun.html#phi"><span class="id" title="variable">phi</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/> -</div> - -<div class="doc"> - Note that equality holds here iff either cfker phi = K and cfker psi = H, - or else phi != 0, psi != 0 and coprime #|K : cfker phi| #|H : cfker phi|. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cfker_dprod"><span class="id" title="lemma">cfker_dprod</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#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0d7ccd69af81527d9facc6293603bbef"><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#psi"><span class="id" title="variable">psi</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.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_dprod"><span class="id" title="lemma">cfdot_dprod</span></a> <span class="id" title="var">phi1</span> <span class="id" title="var">phi2</span> <span class="id" title="var">psi1</span> <span class="id" title="var">psi2</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#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi1"><span class="id" title="variable">phi1</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi1"><span class="id" title="variable">psi1</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#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi2"><span class="id" title="variable">phi2</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi2"><span class="id" title="variable">psi2</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.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi1"><span class="id" title="variable">phi1</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#phi2"><span class="id" title="variable">phi2</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.classfun.html#psi1"><span class="id" title="variable">psi1</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#psi2"><span class="id" title="variable">psi2</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="cfDprodl_iso"><span class="id" title="lemma">cfDprodl_iso</span></a> : <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodr_iso"><span class="id" title="lemma">cfDprodr_iso</span></a> : <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> <a class="idref" href="mathcomp.character.classfun.html#cfDprodr"><span class="id" title="definition">cfDprodr</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cforder_dprodl"><span class="id" title="lemma">cforder_dprodl</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.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="http://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.classfun.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">Lemma</span> <a name="cforder_dprodr"><span class="id" title="lemma">cforder_dprodr</span></a> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#018669adc5ec173554da02f8822d051d"><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.classfun.html#psi"><span class="id" title="variable">psi</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.character.classfun.html#018669adc5ec173554da02f8822d051d"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</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">End</span> <a class="idref" href="mathcomp.character.classfun.html#DProduct"><span class="id" title="section">DProduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfDprodC"><span class="id" title="lemma">cfDprodC</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.classfun.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.classfun.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.classfun.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.classfun.html#G"><span class="id" title="variable">G</span></a>) (<span class="id" title="var">HxK</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.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.classfun.html#G"><span class="id" title="variable">G</span></a>) <span class="id" title="var">chi</span> <span class="id" title="var">psi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#cfDprod"><span class="id" title="definition">cfDprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#chi"><span class="id" title="variable">chi</span></a> <a class="idref" href="mathcomp.character.classfun.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#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.classfun.html#HxK"><span class="id" title="variable">HxK</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#chi"><span class="id" title="variable">chi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Bigdproduct"><span class="id" title="section">Bigdproduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Bigdproduct.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="Bigdproduct.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="Bigdproduct.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.classfun.html#I"><span class="id" title="variable">I</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="Bigdproduct.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.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.classfun.html#Bigdproduct.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="Bigdproduct.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.classfun.html#Bigdproduct.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="Bigdproduct.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.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Bigdproduct.sAG"><span class="id" title="variable">sAG</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfBigdprodi_subproof"><span class="id" title="lemma">cfBigdprodi_subproof</span></a> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#gval"><span class="id" title="projection">gval</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">if</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> 1%<span class="id" title="var">G</span>) <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.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#30cfba5ac0056b0b6e6dae61e139d94a"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#30cfba5ac0056b0b6e6dae61e139d94a"><span class="id" title="notation">bigcup_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#30cfba5ac0056b0b6e6dae61e139d94a"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.ssreflect.finset.html#30cfba5ac0056b0b6e6dae61e139d94a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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> <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#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> <a class="idref" href="mathcomp.character.classfun.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#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#30cfba5ac0056b0b6e6dae61e139d94a"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><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#Bigdproduct.G"><span class="id" title="variable">G</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.character.classfun.html#cfDprodl"><span class="id" title="definition">cfDprodl</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi_subproof"><span class="id" title="lemma">cfBigdprodi_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</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><span class="id" title="var">_</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#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.character.classfun.html#5dde21e605844589296bf0aeffd7b476"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfBigdprodi_additive</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">additive</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">of</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.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#1f39c3338430de1e4f0dd19d42cfade9"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfBigdprodi_linear</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">linear</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">of</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.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#6190fe21ffbd3dab252b4f744e9e9c11"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfBigdprodi_rmorphism</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">rmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">of</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.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f59994a9f1c6ff43f3de0a3cea89bb6b"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfBigdprodi_lrmorphism</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">lrmorphism</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">of</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.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d17433407f88fd9a1e0740e2eddd6566"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi1"><span class="id" title="lemma">cfBigdprodi1</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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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>) : <a class="idref" href="mathcomp.character.classfun.html#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_eq1"><span class="id" title="lemma">cfBigdprodi_eq1</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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.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="http://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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodiK"><span class="id" title="lemma">cfBigdprodiK</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</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.classfun.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.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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_inj"><span class="id" title="lemma">cfBigdprodi_inj</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</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.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodEi"><span class="id" title="lemma">cfBigdprodEi</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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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><span class="id" title="keyword">∀</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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="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#cfBigdprodi"><span class="id" title="definition">cfBigdprodi</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.html#j"><span class="id" title="variable">j</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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodi_iso"><span class="id" title="lemma">cfBigdprodi_iso</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#isometry"><span class="id" title="definition">isometry</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.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> (<span class="id" title="var">phi</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">i</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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><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.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodE"><span class="id" title="lemma">cfBigdprodE</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">x</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.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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><br/> - <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ce5c039a1b6c6cda866a50fe38b40c90"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.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#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprod1"><span class="id" title="lemma">cfBigdprod1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfBigdprodK"><span class="id" title="lemma">cfBigdprodK</span></a> <span class="id" title="var">phi</span> (<span class="id" title="var">Phi</span> := <a class="idref" href="mathcomp.character.classfun.html#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>) <span class="id" title="var">i</span> (<span class="id" title="var">a</span> := <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a> 1%<span class="id" title="var">g</span> <a class="idref" href="mathcomp.algebra.ssralg.html#69c431a9c94f6f30a655bd7ddb59037b"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#Phi"><span class="id" title="variable">Phi</span></a> 1%<span class="id" title="var">g</span>) :<br/> - <a class="idref" href="mathcomp.character.classfun.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#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#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.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#a"><span class="id" title="variable">a</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#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.character.classfun.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.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.classfun.html#Bigdproduct.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_bigdprod"><span class="id" title="lemma">cfdot_bigdprod</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#d6b5e89d170c69f67e0b52af88e95c1d"><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.classfun.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#cfBigdprod"><span class="id" title="definition">cfBigdprod</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#443901d1788fc95745443c70e786b07b"><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#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">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.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#i"><span class="id" title="variable">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">End</span> <a class="idref" href="mathcomp.character.classfun.html#Bigdproduct"><span class="id" title="section">Bigdproduct</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MorphIsometry"><span class="id" title="section">MorphIsometry</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="MorphIsometry.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">D</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</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.classfun.html#MorphIsometry.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">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="cfMorph_iso"><span class="id" title="lemma">cfMorph_iso</span></a> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</span> <span class="id" title="var">D</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.classfun.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.classfun.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.classfun.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/> - <a class="idref" href="mathcomp.character.classfun.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#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="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> (<a class="idref" href="mathcomp.character.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</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.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.classfun.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.classfun.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.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.classfun.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIsom_iso"><span class="id" title="lemma">cfIsom_iso</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">G</span> (<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.classfun.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>) (<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.classfun.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.classfun.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">∀</span> <span class="id" title="var">isoG</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</span></a>, <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</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.classfun.html#isoG"><span class="id" title="variable">isoG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfMod_iso"><span class="id" title="lemma">cfMod_iso</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.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#isometry"><span class="id" title="definition">isometry</span></a> (@<a class="idref" href="mathcomp.character.classfun.html#cfMod"><span class="id" title="definition">cfMod</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfQuo_iso"><span class="id" title="lemma">cfQuo_iso</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.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#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="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">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="mathcomp.character.classfun.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.classfun.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#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#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</span></a> (@<a class="idref" href="mathcomp.character.classfun.html#cfQuo"><span class="id" title="definition">cfQuo</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.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#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_quo"><span class="id" title="lemma">cfnorm_quo</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.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#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.classfun.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.classfun.html#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="variable">H</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> <a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.character.classfun.html#c6f657b388e6d4324a06ab413a1ed00d"><span class="id" title="notation">_G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfSdprod_iso"><span class="id" title="lemma">cfSdprod_iso</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> (<span class="id" title="var">defG</span> : <a class="idref" href="mathcomp.character.classfun.html#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.classfun.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.classfun.html#G"><span class="id" title="variable">G</span></a>) : <a class="idref" href="mathcomp.character.classfun.html#isometry"><span class="id" title="definition">isometry</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.classfun.html#defG"><span class="id" title="variable">defG</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#MorphIsometry"><span class="id" title="section">MorphIsometry</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">Variable</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>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Induced.Def"><span class="id" title="section">Def</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="Induced.Def.B"><span class="id" title="variable">B</span></a> <a name="Induced.Def.A"><span class="id" title="variable">A</span></a> : <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.classfun.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>.<br/> - -<br/> -</div> - -<div class="doc"> - The defalut value for the ~~ (H \subset G) case matches the one for cfRes - so that Frobenius reciprocity holds even in this degenerate case. -</div> -<div class="code"> -<span class="id" title="keyword">Definition</span> <a name="ffun_cfInd"><span class="id" title="definition">ffun_cfInd</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.classfun.html#Induced.Def.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) :=<br/> - <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="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.classfun.html#H"><span class="id" title="abbreviation">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#G"><span class="id" title="abbreviation">G</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.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.Def.A"><span class="id" title="variable">A</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">y</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a>)<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.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="abbreviation">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#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.classfun.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> 1<a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><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.classfun.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><a class="idref" href="mathcomp.ssreflect.finfun.html#486743bb05c6aa8b9d64fd3cec29ee79"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fact</span> <a name="cfInd_subproof"><span class="id" title="lemma">cfInd_subproof</span></a> <span class="id" title="var">phi</span> : <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.character.classfun.html#G"><span class="id" title="abbreviation">G</span></a> (<a class="idref" href="mathcomp.character.classfun.html#ffun_cfInd"><span class="id" title="definition">ffun_cfInd</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="cfInd"><span class="id" title="definition">cfInd</span></a> <span class="id" title="var">phi</span> := <a class="idref" href="mathcomp.character.classfun.html#Cfun"><span class="id" title="definition">Cfun</span></a> 1 (<a class="idref" href="mathcomp.character.classfun.html#cfInd_subproof"><span class="id" title="lemma">cfInd_subproof</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_is_linear"><span class="id" title="lemma">cfInd_is_linear</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.classfun.html#cfInd"><span class="id" title="definition">cfInd</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfInd_additive</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.classfun.html#cfInd_is_linear"><span class="id" title="lemma">cfInd_is_linear</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cfInd_linear</span> := <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.classfun.html#cfInd_is_linear"><span class="id" title="lemma">cfInd_is_linear</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Induced.Def"><span class="id" title="section">Def</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndE"><span class="id" title="lemma">cfIndE</span></a> (<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.classfun.html#Induced.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> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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#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#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#H"><span class="id" title="variable">H</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">y</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#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.classfun.html#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.html#y"><span class="id" title="variable">y</span></a>)<a class="idref" href="mathcomp.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="Induced.G"><span class="id" title="variable">G</span></a> <a name="Induced.K"><span class="id" title="variable">K</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.classfun.html#Induced.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.classfun.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>) (<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.classfun.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>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndEout"><span class="id" title="lemma">cfIndEout</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#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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.classfun.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.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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#3b05480e39db306e67fadbc79d394529"><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.classfun.html#Induced.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#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.classfun.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> 1<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.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">_1</span></a>%<span class="id" title="var">G</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndEsdprod"><span class="id" title="lemma">cfIndEsdprod</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.classfun.html#Induced.K"><span class="id" title="variable">K</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.classfun.html#Induced.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.classfun.html#Induced.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.classfun.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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.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#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">w</span> <a class="idref" href="mathcomp.algebra.ssralg.html#b4ba9f64661118f4ed0bad900f98d2a2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.character.classfun.html#Induced.H"><span class="id" title="variable">H</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#phi"><span class="id" title="variable">phi</span></a> (<a class="idref" href="mathcomp.character.classfun.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#746f7e4d3218aa2699eefc064b513fc2"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.character.classfun.html#w"><span class="id" title="variable">w</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_on"><span class="id" title="lemma">cfInd_on</span></a> <span class="id" title="var">A</span> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.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.classfun.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#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.classfun.html#Induced.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.character.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Induced.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.fingroup.fingroup.html#class_support"><span class="id" title="definition">class_support</span></a> <a class="idref" href="mathcomp.character.classfun.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.character.classfun.html#Induced.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_id"><span class="id" title="lemma">cfInd_id</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_normal"><span class="id" title="lemma">cfInd_normal</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#Induced.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.classfun.html#Induced.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd1"><span class="id" title="lemma">cfInd1</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.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.classfun.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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</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.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.html#Induced.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#phi"><span class="id" title="variable">phi</span></a> 1%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfInd_cfun1"><span class="id" title="lemma">cfInd_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.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#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><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="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.html#Induced.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#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">_H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfnorm_Ind_cfun1"><span class="id" title="lemma">cfnorm_Ind_cfun1</span></a> : <a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.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#bb709747b2c5cc15f051ad5d56651c60"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#Induced.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#ac99de1882838d8c0eab088180289562"><span class="id" title="notation">]</span></a> 1<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> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.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.classfun.html#Induced.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndInd"><span class="id" title="lemma">cfIndInd</span></a> <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#Induced.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#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#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.classfun.html#Induced.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="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Isaacs, Lemma (5.2). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Frobenius_reciprocity"><span class="id" title="lemma">Frobenius_reciprocity</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span> : <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.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#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.classfun.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.classfun.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> <a class="idref" href="mathcomp.character.classfun.html#d6b5e89d170c69f67e0b52af88e95c1d"><span class="id" title="notation">'[</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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#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>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfdot_Res_r"><span class="id" title="definition">cfdot_Res_r</span></a> := <a class="idref" href="mathcomp.character.classfun.html#Frobenius_reciprocity"><span class="id" title="lemma">Frobenius_reciprocity</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfdot_Res_l"><span class="id" title="lemma">cfdot_Res_l</span></a> <span class="id" title="var">psi</span> <span class="id" title="var">phi</span> : <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.classfun.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.classfun.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.classfun.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="http://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.classfun.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.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndM"><span class="id" title="lemma">cfIndM</span></a> <span class="id" title="var">phi</span> <span class="id" title="var">psi</span>: <a class="idref" href="mathcomp.character.classfun.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.classfun.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><br/> - <a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.character.classfun.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.algebra.ssralg.html#2d0cfb150261028f4ebd2ba355623dcc"><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.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><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.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">Ind</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.character.classfun.html#Induced.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.character.classfun.html#bc3433d79d98cfd82a70b7d11c234953"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#psi"><span class="id" title="variable">psi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#Induced"><span class="id" title="section">Induced</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">"</span></a>''Ind[' G , H ]" := (@<a class="idref" href="mathcomp.character.classfun.html#cfInd"><span class="id" title="definition">cfInd</span></a> <span class="id" title="var">_</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">"</span></a>''Ind[' G ]" := <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><span class="id" title="var">G</span><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="mathcomp.character.classfun.html#c8e0ccc285497008e15e1f92f34d54bd"><span class="id" title="notation">]</span></a> : <span class="id" title="var">ring_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">"</span></a>''Ind'" := <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><span class="id" title="var">_</span><a class="idref" href="mathcomp.character.classfun.html#cd7dae4d94d0890c33d51d534038960d"><span class="id" title="notation">]</span></a> (<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="MorphInduced"><span class="id" title="section">MorphInduced</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MorphInduced.aT"><span class="id" title="variable">aT</span></a> <a name="MorphInduced.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="MorphInduced.D"><span class="id" title="variable">D</span></a> <a name="MorphInduced.G"><span class="id" title="variable">G</span></a> <a name="MorphInduced.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.classfun.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="MorphInduced.R"><span class="id" title="variable">R</span></a> <a name="MorphInduced.S"><span class="id" title="variable">S</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.classfun.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/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndMorph"><span class="id" title="lemma">cfIndMorph</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.classfun.html#MorphInduced.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.classfun.html#MorphInduced.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">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.classfun.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.classfun.html#MorphInduced.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.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#7ef99623452370540bbc44fd30b0bc94"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.character.classfun.html#f"><span class="id" title="variable">f</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#MorphInduced.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.classfun.html#MorphInduced.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#MorphInduced.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#MorphInduced.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#MorphInduced.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><br/> - <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.classfun.html#MorphInduced.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.classfun.html#cfMorph"><span class="id" title="definition">cfMorph</span></a> <a class="idref" href="mathcomp.character.classfun.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.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.classfun.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.classfun.html#MorphInduced.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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MorphInduced.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.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.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.classfun.html#MorphInduced.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>) (<a name="MorphInduced.h"><span class="id" title="variable">h</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.classfun.html#MorphInduced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.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">Hypotheses</span> (<a name="MorphInduced.isoG"><span class="id" title="variable">isoG</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.classfun.html#MorphInduced.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.g"><span class="id" title="variable">g</span></a>) (<a name="MorphInduced.isoH"><span class="id" title="variable">isoH</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.classfun.html#MorphInduced.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.h"><span class="id" title="variable">h</span></a>) (<a name="MorphInduced.eq_hg"><span class="id" title="variable">eq_hg</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.character.classfun.html#MorphInduced.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#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.h"><span class="id" title="variable">h</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.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>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="MorphInduced.sHG"><span class="id" title="variable">sHG</span></a> : <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.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#MorphInduced.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfResIsom"><span class="id" title="lemma">cfResIsom</span></a> <span class="id" title="var">phi</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.classfun.html#MorphInduced.S"><span class="id" title="variable">S</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#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.isoG"><span class="id" title="variable">isoG</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MorphInduced.isoH"><span class="id" title="variable">isoH</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.classfun.html#MorphInduced.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.classfun.html#phi"><span class="id" title="variable">phi</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfIndIsom"><span class="id" title="lemma">cfIndIsom</span></a> <span class="id" title="var">phi</span> : <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.classfun.html#MorphInduced.R"><span class="id" title="variable">R</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#cfIsom"><span class="id" title="definition">cfIsom</span></a> <a class="idref" href="mathcomp.character.classfun.html#MorphInduced.isoH"><span class="id" title="variable">isoH</span></a> <a class="idref" href="mathcomp.character.classfun.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.classfun.html#MorphInduced.isoG"><span class="id" title="variable">isoG</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.classfun.html#MorphInduced.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.classfun.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.classfun.html#MorphInduced"><span class="id" title="section">MorphInduced</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FieldAutomorphism"><span class="id" title="section">FieldAutomorphism</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="FieldAutomorphism.u"><span class="id" title="variable">u</span></a> : <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>) (<a name="FieldAutomorphism.gT"><span class="id" title="variable">gT</span></a> <a name="FieldAutomorphism.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>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="FieldAutomorphism.G"><span class="id" title="variable">G</span></a> <a name="FieldAutomorphism.K"><span class="id" title="variable">K</span></a> <a name="FieldAutomorphism.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.classfun.html#FieldAutomorphism.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="FieldAutomorphism.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.classfun.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.classfun.html#FieldAutomorphism.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>) (<a name="FieldAutomorphism.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.classfun.html#FieldAutomorphism.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="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.classfun.html#FieldAutomorphism.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">S</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</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.classfun.html#FieldAutomorphism.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="cfAutZ_nat"><span class="id" title="lemma">cfAutZ_nat</span></a> <span class="id" title="var">n</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#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> <a class="idref" href="mathcomp.algebra.ssralg.html#3b05480e39db306e67fadbc79d394529"><span class="id" title="notation">*:</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutZ_Cnat"><span class="id" title="lemma">cfAutZ_Cnat</span></a> <span class="id" title="var">z</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#z"><span class="id" title="variable">z</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="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#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#z"><span class="id" title="variable">z</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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#z"><span class="id" title="variable">z</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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutZ_Cint"><span class="id" title="lemma">cfAutZ_Cint</span></a> <span class="id" title="var">z</span> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#z"><span class="id" title="variable">z</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.Cint"><span class="id" title="definition">Cint</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#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#z"><span class="id" title="variable">z</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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#z"><span class="id" title="variable">z</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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutK"><span class="id" title="lemma">cfAutK</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#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a>) (<a class="idref" href="mathcomp.character.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> (<a class="idref" href="mathcomp.field.algC.html#algC_invaut_rmorphism"><span class="id" title="definition">algC_invaut_rmorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutVK"><span class="id" title="lemma">cfAutVK</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#cfAut"><span class="id" title="definition">cfAut</span></a> (<a class="idref" href="mathcomp.field.algC.html#algC_invaut_rmorphism"><span class="id" title="definition">algC_invaut_rmorphism</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</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.classfun.html#FieldAutomorphism.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_inj"><span class="id" title="lemma">cfAut_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.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_eq1"><span class="id" title="lemma">cfAut_eq1</span></a> <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.classfun.html#cfAut"><span class="id" title="definition">cfAut</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.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="http://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#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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="support_cfAut"><span class="id" title="lemma">support_cfAut</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</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.algebra.ssralg.html#support"><span class="id" title="abbreviation">support</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_cfAut_free"><span class="id" title="lemma">map_cfAut_free</span></a> <span class="id" title="var">S</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.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> <a class="idref" href="mathcomp.character.classfun.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> <a class="idref" href="mathcomp.algebra.vector.html#free"><span class="id" title="definition">free</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</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.classfun.html#FieldAutomorphism.u"><span class="id" title="variable">u</span></a>) <a class="idref" href="mathcomp.character.classfun.html#S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_on"><span class="id" title="lemma">cfAut_on</span></a> <span class="id" title="var">A</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.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</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.classfun.html#FieldAutomorphism.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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="http://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#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.classfun.html#FieldAutomorphism.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.classfun.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.character.classfun.html#62c6c4bf7772487a11a35ed29aee9fc7"><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>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfker_aut"><span class="id" title="lemma">cfker_aut</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.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#cfker"><span class="id" title="definition">cfker</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAut_cfuni"><span class="id" title="lemma">cfAut_cfuni</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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">_A</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <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.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">_A</span></a> <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.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.classfun.html#FieldAutomorphism.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="cforder_aut"><span class="id" title="lemma">cforder_aut</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.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">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.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.classfun.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">Lemma</span> <a name="cfAutRes"><span class="id" title="lemma">cfAutRes</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#FieldAutomorphism.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.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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">phi</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">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutMorph"><span class="id" title="lemma">cfAutMorph</span></a> (<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.classfun.html#FieldAutomorphism.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.classfun.html#FieldAutomorphism.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>) : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutIsom"><span class="id" title="lemma">cfAutIsom</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.classfun.html#FieldAutomorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.f"><span class="id" title="variable">f</span></a>) <span class="id" title="var">phi</span> :<br/> - <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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.classfun.html#isoGR"><span class="id" title="variable">isoGR</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutQuo"><span class="id" title="lemma">cfAutQuo</span></a> <span class="id" title="var">phi</span> : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="mathcomp.character.classfun.html#e2539410c9b6b2f6a5e7c9eab816ddef"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.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="cfAutMod"><span class="id" title="lemma">cfAutMod</span></a> (<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.classfun.html#FieldAutomorphism.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#FieldAutomorphism.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>) : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="mathcomp.character.classfun.html#d9227c0ae83ba0285be6deaf557252f6"><span class="id" title="notation">%%</span></a> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.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="cfAutInd"><span class="id" title="lemma">cfAutInd</span></a> (<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.classfun.html#FieldAutomorphism.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>) : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#FieldAutomorphism.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.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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#523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.character.classfun.html#523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">Ind</span></a> <a class="idref" href="mathcomp.character.classfun.html#523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#523c64af6ca44fabb3f80902002d2d06"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> : <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism.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.classfun.html#FieldAutomorphism.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.classfun.html#FieldAutomorphism.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutDprodl"><span class="id" title="lemma">cfAutDprodl</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.classfun.html#FieldAutomorphism.K"><span class="id" title="variable">K</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#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutDprodr"><span class="id" title="lemma">cfAutDprodr</span></a> (<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.classfun.html#FieldAutomorphism.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>) : <a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cfAutDprod"><span class="id" title="lemma">cfAutDprod</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.classfun.html#FieldAutomorphism.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.character.classfun.html#5112e587c59fdaca05e10d1764f09c4c"><span class="id" title="notation">)</span></a>) (<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.classfun.html#FieldAutomorphism.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.classfun.html#951fc6d5628818da9c9fe35428eee358"><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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">)^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="http://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.classfun.html#FieldAutomorphism.KxH"><span class="id" title="variable">KxH</span></a> <a class="idref" href="mathcomp.character.classfun.html#phi"><span class="id" title="variable">phi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a> <a class="idref" href="mathcomp.character.classfun.html#psi"><span class="id" title="variable">psi</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.character.classfun.html#951fc6d5628818da9c9fe35428eee358"><span class="id" title="notation">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.character.classfun.html#FieldAutomorphism"><span class="id" title="section">FieldAutomorphism</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="conj_cfRes"><span class="id" title="definition">conj_cfRes</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfAutRes"><span class="id" title="lemma">cfAutRes</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfker_conjC"><span class="id" title="definition">cfker_conjC</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfker_aut"><span class="id" title="lemma">cfker_aut</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="conj_cfQuo"><span class="id" title="definition">conj_cfQuo</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfAutQuo"><span class="id" title="lemma">cfAutQuo</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="conj_cfMod"><span class="id" title="definition">conj_cfMod</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfAutMod"><span class="id" title="lemma">cfAutMod</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="conj_cfInd"><span class="id" title="definition">conj_cfInd</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfAutInd"><span class="id" title="lemma">cfAutInd</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="cfconjC_eq1"><span class="id" title="definition">cfconjC_eq1</span></a> := <a class="idref" href="mathcomp.character.classfun.html#cfAut_eq1"><span class="id" title="lemma">cfAut_eq1</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#Num.Theory.conjC"><span class="id" title="definition">conjC</span></a>.<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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