Definition peano_ind (P : N -> Prop) := peano_rect P.Definition peano_rec (P : N -> Set) := peano_rect P.

(* Specifications: *)
Theorem peano_rec_base P a f : peano_rec P a f 0 = a.
Theorem peano_rec_succ P a f n : peano_rec P a f (succ n) = f n (peano_rec P a f n).
Theorem bi_induction : forall A : N -> Prop, Proper (Logic.eq==>iff) A -> A 0 -> (forall n, A n <-> A (succ n)) -> forall n : N, A n.