Definition eqb_list {A: Type} (eqb_A: A -> A -> bool): list A -> list A -> bool :=
  fix eqb_list (l1 l2: list A): bool :=
    match l1, l2 with
    | nil, nil => true
    | a1 :: l1, a2 :: l2 => eqb_A a1 a2 && eqb_list l1 l2
    | _, _ => false
    end.

(* Specifications: *)
Lemma eqb_list_spec: forall {A: Type} (eqb_A: A -> A -> bool), (forall a1 a2, eqb_A a1 a2 = true <-> a1 = a2) -> (forall l1 l2, eqb_list eqb_A l1 l2 = true <-> l1 = l2).
Lemma nat_ind2_Type: forall P : nat -> Type, ((forall n, (forall j:nat, (j<n )%nat -> P j) -> P n):Type) -> (forall n, P n).
Lemma nat_ind2: forall P : nat -> Prop, (forall n, (forall j:nat, (j<n )%nat -> P j) -> P n) -> (forall n, P n).
Lemma equiv_e2 : forall A B: Prop, A=B -> B -> A.