(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
(******************************************************************************)
(* This file provides basic notions of virtual character theory:              *)
(*       'Z[S, A] == collective predicate for the phi that are Z-linear       *)
(*                   combinations of elements of S : seq 'CF(G) and have      *)
(*                   support in A : {set gT}.                                 *)
(*          'Z[S] == collective predicate for the Z-linear combinations of    *)
(*                   elements of S.                                           *)
(*      'Z[irr G] == the collective predicate for virtual characters.         *)
(*         dirr G == the collective predicate for normal virtual characters,  *)
(*                   i.e., virtual characters of norm 1:                      *)
(*               mu \in dirr G <=> m \in 'Z[irr G] and '[mu] = 1              *)
(*                             <=> mu or - mu \in irr G.                      *)
(* --> othonormal subsets of 'Z[irr G] are contained in dirr G.               *)
(*        dIirr G == an index type for normal virtual characters.             *)
(*         dchi i == the normal virtual character of index i.                 *)
(*       of_irr i == the (unique) irreducible constituent of dchi i:          *)
(*                   dchi i = 'chi_(of_irr i) or - 'chi_(of_irr i).           *)
(*        ndirr i == the index of - dchi i.                                   *)
(*        dirr1 G == the normal virtual character index of 1 : 'CF(G), the    *)
(*                   principal character.                                     *)
(* dirr_dIirr j f == the index i (or dirr1 G if it does not exist) such that  *)
(*                   dchi i = f j.                                            *)
(* dirr_constt phi == the normal virtual character constituents of phi:       *)
(*                    i \in dirr_constt phi <=> [dchi i, phi] > 0.            *)
(*  to_dirr phi i == the normal virtual character constituent of phi with an  *)
(*                   irreducible constituent i, when i \in irr_constt phi.    *)
(******************************************************************************)
Section Basics.
Variables (gT : finGroupType) (B : {set gT}) (S : seq 'CF(B)) (A : {set gT}).
Definition Zchar : pred_class :=
[pred phi in 'CF(B, A) | dec_Cint_span (in_tuple S) phi].
Fact Zchar_key : pred_key Zchar. Proof. by []. Qed.
Canonical Zchar_keyed := KeyedPred Zchar_key.

(* Specifications: *)
Lemma cfun0_zchar : 0 \in Zchar.
Fact Zchar_zmod : zmod_closed Zchar.