Section AlgC.
(* Arithmetic properties of group orders in the characteristic 0 field algC. *)
Variable (gT : finGroupType).
Implicit Types (G : {group gT}) (B : {set gT}).
Variable B : {set gT}.
Local Notation G := <<B>>.
Record classfun : predArgType :=
Classfun {cfun_val; _ : is_class_fun G cfun_val}.
Implicit Types phi psi xi : classfun.
(* The default expansion lemma cfunE requires key = 0. *)
Fact classfun_key : unit. Proof. by []. Qed.
Definition Cfun := locked_with classfun_key (fun flag : nat => Classfun).
Canonical cfun_subType := Eval hnf in [subType for cfun_val].
Definition cfun_eqMixin := Eval hnf in [eqMixin of classfun by <:].
Canonical cfun_eqType := Eval hnf in EqType classfun cfun_eqMixin.
Definition cfun_choiceMixin := Eval hnf in [choiceMixin of classfun by <:].
Canonical cfun_choiceType := Eval hnf in ChoiceType classfun cfun_choiceMixin.
Definition fun_of_cfun phi := cfun_val phi : gT -> algC.
Coercion fun_of_cfun : classfun >-> Funclass.
Fact cfun_zero_subproof : is_class_fun G (0 : {ffun _}).
Definition cfun_comp f f0 phi := Cfun 0 (@cfun_comp_subproof f phi f0).
Definition cfun_opp := cfun_comp (oppr0 _).
Fact cfun_add_subproof phi psi : is_class_fun G [ffun x => phi x + psi x].
Definition cfun_add phi psi := Cfun 0 (cfun_add_subproof phi psi).
Fact cfun_indicator_subproof (A : {set gT}) :
is_class_fun G [ffun x => ((x \in G) && (x ^: G \subset A))%:R].
Definition cfun_indicator A := Cfun 1 (cfun_indicator_subproof A).
Local Notation "''1_' A" := (cfun_indicator A) : ring_scope.
Definition cfun_mul phi psi := Cfun 0 (cfun_mul_subproof phi psi).
Definition cfun_unit := [pred phi : classfun | [forall x in G, phi x != 0]].
Definition cfun_inv phi :=
if phi \in cfun_unit then cfun_comp (invr0 _) phi else phi.
Definition cfun_scale a := cfun_comp (mulr0 a).
Fact cfun_addA : associative cfun_add.
Fact cfun_mulD : left_distributive cfun_mul cfun_add.
Definition cfun_ringMixin :=
ComRingMixin cfun_mulA cfun_mulC cfun_mul1 cfun_mulD cfun_nz1.
Canonical cfun_ringType := RingType classfun cfun_ringMixin.
Canonical cfun_comRingType := ComRingType classfun cfun_mulC.

Lemma expS_cfunE phi n x : (phi ^+ n.+1) x = phi x ^+ n.+1.