Implicit Type H : {group gT}.Implicit Type M : {group gT}.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Inductive extremal_group_type :=
  ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.Definition index_extremal_group_type c :=
  match c with
  | ModularGroup => 0
  | Dihedral => 1
  | SemiDihedral => 2
  | Quaternion => 3
  | NotExtremal => 4
  end%N.

Lemma cancel_index_extremal_groups : cancel index_extremal_group_type (nth NotExtremal enum_extremal_groups).