Scheme eq_dep_elim := Induction for eq Sort Type.Definition eqbool_dep P (h1 : P true) b :=
  match b return P b -> Prop with
  | true => fun (h2 : P true) => h1 = h2
  | false => fun (h2 : P false) => False
  end.

(* Specifications: *)
Lemma eqbool_irrelevance : forall (b : bool) (h1 h2 : b = true), h1 = h2.
Lemma eqbool_irrelevance : forall (b : bool) (h1 h2 : b = true), h1 = h2.