Section Def.
Variables B A : {set gT}.
Local Notation G := <<B>>.
Local Notation H := <<A>>.
(* The defalut value for the ~~ (H \subset G) case matches the one for cfRes *)
(* so that Frobenius reciprocity holds even in this degenerate case.         *)
Definition ffun_cfInd (phi : 'CF(A)) :=
[ffun x => if H \subset G then #|A|%:R^-1 * (\sum_(y in G) phi (x ^ y))
else #|G|%:R * '[phi, 1] *+ (x == 1%g)].
Fact cfInd_subproof phi : is_class_fun G (ffun_cfInd phi).
Definition cfInd phi := Cfun 1 (cfInd_subproof phi).
Canonical cfInd_additive := Additive cfInd_is_linear.
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.

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)).