Section Norm1vchar.
Variables (gT : finGroupType) (G : {group gT}).
Fact dirr_key : pred_key (dirr G). Proof. by []. Qed.
Canonical dirr_keyed := KeyedPred dirr_key.
Fact dirr_oppr_closed : oppr_closed (dirr G).
Definition dIirr (B : {set gT}) := (bool * (Iirr B))%type.
Definition dirr1 (B : {set gT}) : dIirr B := (false, 0).
Definition ndirr (B : {set gT}) (i : dIirr B) : dIirr B :=
(~~ i.1, i.2).
Definition dirr_dIirr (B : {set gT}) J (f : J -> 'CF(B)) j : dIirr B :=
odflt (dirr1 B) [pick i | dchi i == f j].
Definition to_dirr  (B : {set gT}) (phi : 'CF(B)) (i : Iirr B) : dIirr B :=
('[phi, 'chi_i] < 0, i).
Definition of_irr (B : {set gT}) (i : dIirr B) : Iirr B := i.2.

Lemma cfdot_todirrE (phi: 'CF(G)) i (phi_i := dchi (to_dirr phi i)) :
'[phi, phi_i] *: phi_i = '[phi, 'chi_i] *: 'chi_i.