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).