Let iter (x: positive) (P: forall y, Plt y x -> A) : A :=
match peq x xH with
| left EQ => v1
| right NOTEQ => f (Pos.pred x) (P (Pos.pred x) (Ppred_Plt x NOTEQ))
end.
Definition positive_rec : positive -> A :=
Fix Plt_wf (fun _ => A) iter.

Lemma positive_Peano_ind:
forall (P: positive -> Prop),
P xH ->
(forall x, P x -> P (Pos.succ x)) ->
forall x, P x.