Definition positive_rec : positive -> A :=
  Fix Plt_wf (fun _ => A) iter.

(* Specifications: *)
Lemma unroll_positive_rec: forall x, positive_rec x = iter x (fun y _ => positive_rec y).
Lemma positive_rec_base: positive_rec 1%positive = v1.
Lemma positive_rec_succ: forall x, positive_rec (Pos.succ x) = f x (positive_rec x).
Lemma positive_Peano_ind: forall (P: positive -> Prop), P xH -> (forall x, P x -> P (Pos.succ x)) -> forall x, P x.
Lemma unroll_positive_rec: forall x, positive_rec x = iter x (fun y _ => positive_rec y).
Lemma positive_rec_base: positive_rec 1%positive = v1.
Lemma positive_rec_succ: forall x, positive_rec (Pos.succ x) = f x (positive_rec x).
Lemma positive_Peano_ind: forall (P: positive -> Prop), P xH -> (forall x, P x -> P (Pos.succ x)) -> forall x, P x.