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.

Lemma nat_ind2:
forall P : nat -> Prop,
(forall n, (forall j:nat, (j<n )%nat -> P j) ->  P n) ->
(forall n, P n).