Canonical cfInd_linear := Linear cfInd_is_linear.
End Def.
Local Notation "''Ind[' B , A ]" := (@cfInd B A) : ring_scope.
Local Notation "''Ind[' B ]" := 'Ind[B, _] : ring_scope.

(* Specifications: *)
Lemma cfIndE (G H : {group gT}) phi x :
H \subset G -> 'Ind[G, H] phi x = #|H|%:R^-1 * (\sum_(y in G) phi (x ^ y)).
Lemma cfIndEsdprod (phi : 'CF(K)) x :
K ><| H = G -> 'Ind[G] phi x = \sum_(w in H) phi (x ^ w)%g.
Lemma cfInd_on A phi :
H \subset G -> phi \in 'CF(H, A) -> 'Ind[G] phi \in 'CF(G, class_support A G).
Lemma cfInd_id phi : 'Ind[H] phi = phi.
Lemma cfInd_normal phi : H <| G -> 'Ind[G] phi \in 'CF(G, H).
Lemma cfInd1 phi : H \subset G -> 'Ind[G] phi 1%g = #|G : H|%:R * phi 1%g.
Lemma cfInd_cfun1 : H <| G -> 'Ind[G, H] 1 = #|G : H|%:R *: '1_H.
Lemma cfnorm_Ind_cfun1 : H <| G -> '['Ind[G, H] 1] = #|G : H|%:R.
Lemma cfIndInd phi :
K \subset G -> H \subset K -> 'Ind[G] ('Ind[K] phi) = 'Ind[G] phi.
(* This is Isaacs, Lemma (5.2). *)
Lemma Frobenius_reciprocity phi psi : '[phi, 'Res[H] psi] = '['Ind[G] phi, psi].
Definition cfdot_Res_r := Frobenius_reciprocity.
Lemma cfdot_Res_l psi phi : '['Res[H] psi, phi] = '[psi, 'Ind[G] phi].
End Induced.