(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* 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.          *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Delimit Scope cfun_scope with CF.
Reserved Notation "''CF' ( G , A )" (at level 8, format "''CF' ( G ,  A )").
Reserved Notation "''CF' ( G )" (at level 8, format "''CF' ( G )").
Reserved Notation "''1_' G" (at level 8, G at level 2, format "''1_' G").
Reserved Notation "''Res[' H , G ]" (at level 8, only parsing).
Reserved Notation "''Res[' H ]" (at level 8, format "''Res[' H ]").
Reserved Notation "''Res'" (at level 8, only parsing).
Reserved Notation "''Ind[' G , H ]" (at level 8, only parsing).
Reserved Notation "''Ind[' G ]" (at level 8, format "''Ind[' G ]").
Reserved Notation "''Ind'" (at level 8, only parsing).
Reserved Notation "'[ phi , psi ]_ G" (at level 2, only parsing).
Reserved Notation "'[ phi , psi ]"
(at level 2, format "'[hv' ''[' phi , '/ '  psi ] ']'").
Reserved Notation "'[ phi ]_ G" (at level 2, only parsing).
Reserved Notation "'[ phi ]" (at level 2, format "''[' phi ]").
Reserved Notation "phi ^u" (at level 3, format "phi ^u").
Section AlgC.
(* Arithmetic properties of group orders in the characteristic 0 field algC. *)
Variable (gT : finGroupType).
Implicit Types (G : {group gT}) (B : {set gT}).

Lemma neq0CG G : (#|G|)%:R != 0 :> algC. Proof. exact: natrG_neq0. Qed.