End Inertia.
Arguments inertia {gT B%g} phi%CF.
Arguments cfclass {gT A%g} phi%CF B%g.
Arguments conjg_Iirr_inj {gT H} y [i1 i2] : rename.
Notation "''I[' phi ] " := (inertia phi) : group_scope.
Notation "''I[' phi ] " := (inertia_group phi) : Group_scope.
Notation "''I_' G [ phi ] " := (G%g :&: 'I[phi]) : group_scope.
Notation "''I_' G [ phi ] " := (G :&: 'I[phi])%G : Group_scope.
Notation "phi ^: G" := (cfclass phi G) : cfun_scope.
Section ConjRestrict.
Variables (gT : finGroupType) (G H K : {group gT}).

(* Specifications: *)
Lemma cfConjgRes_norm phi y :
y \in 'N(K) -> y \in 'N(H) -> ('Res[K, H] phi ^ y)%CF = 'Res (phi ^ y)%CF.
Lemma cfConjgRes phi y :
H <| G -> K <| G -> y \in G -> ('Res[K, H] phi ^ y)%CF = 'Res (phi ^ y)%CF.
Lemma sub_inertia_Res phi :
G \subset 'N(K) -> 'I_G[phi] \subset 'I_G['Res[K, H] phi].
Lemma cfConjgInd_norm phi y :
y \in 'N(K) -> y \in 'N(H) -> ('Ind[H, K] phi ^ y)%CF = 'Ind (phi ^ y)%CF.
Lemma cfConjgInd phi y :
H <| G -> K <| G -> y \in G -> ('Ind[H, K] phi ^ y)%CF = 'Ind (phi ^ y)%CF.
Lemma sub_inertia_Ind phi :
G \subset 'N(H) -> 'I_G[phi] \subset 'I_G['Ind[H, K] phi].