Section Basics.
Variables (gT : finGroupType) (B : {set gT}) (S : seq 'CF(B)) (A : {set gT}).
Definition Zchar : {pred 'CF(B)} :=
[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.
Fact Zchar_zmod : zmod_closed Zchar.
Canonical Zchar_opprPred := OpprPred Zchar_zmod.
Canonical Zchar_addrPred := AddrPred Zchar_zmod.
Canonical Zchar_zmodPred := ZmodPred Zchar_zmod.
Fact vchar_mulr_closed : mulr_closed 'Z[irr G].
Canonical vchar_mulrPred := MulrPred vchar_mulr_closed.
Canonical vchar_smulrPred := SmulrPred vchar_mulr_closed.
Canonical vchar_semiringPred := SemiringPred vchar_mulr_closed.
Canonical vchar_subringPred := SubringPred vchar_mulr_closed.

Lemma mul_vchar A :
{in 'Z[irr G, A] &, forall phi psi, phi * psi \in 'Z[irr G, A]}.