(** Earlier, the Peano-like recursor was built and proved in a way due to Conor McBride, see "The view from the left" *)
Inductive PeanoView : positive -> Type :=
| PeanoOne : PeanoView 1
| PeanoSucc : forall p, PeanoView p -> PeanoView (succ p).
Fixpoint peanoView_xO p (q:PeanoView p) : PeanoView (p~0) :=
match q in PeanoView x return PeanoView (x~0) with
| PeanoOne => PeanoSucc _ PeanoOne
| PeanoSucc _ q => PeanoSucc _ (PeanoSucc _ (peanoView_xO _ q))
end.
Fixpoint peanoView_xI p (q:PeanoView p) : PeanoView (p~1) :=
match q in PeanoView x return PeanoView (x~1) with
| PeanoOne => PeanoSucc _ (PeanoSucc _ PeanoOne)
| PeanoSucc _ q => PeanoSucc _ (PeanoSucc _ (peanoView_xI _ q))
end.
Fixpoint peanoView p : PeanoView p :=
match p return PeanoView p with
| 1 => PeanoOne
| p~0 => peanoView_xO p (peanoView p)
| p~1 => peanoView_xI p (peanoView p)
end.
Definition PeanoView_iter (P:positive->Type)
(a:P 1) (f:forall p, P p -> P (succ p)) :=
(fix iter p (q:PeanoView p) : P p :=
match q in PeanoView p return P p with
| PeanoOne => a
| PeanoSucc _ q => f _ (iter _ q)
end).

Lemma peano_equiv (P:positive->Type) (a:P 1) (f:forall p, P p -> P (succ p)) p :
PeanoView_iter P a f p (peanoView p) = peano_rect P a f p.