Definition peano_rec (P:positive->Set) := peano_rect P.
(** Peano induction *)
Definition peano_ind (P:positive->Prop) := peano_rect P.
(** Peano case analysis *)

Theorem peano_case :
forall P:positive -> Prop,
P 1 -> (forall n:positive, P (succ n)) -> forall p:positive, P p.