Section FixPoint.
Variable f : (N.t -> A) -> N.t -> A.
Variable f_wd : Proper ((N.eq==>Aeq)==>N.eq==>Aeq) f.
(* We need an assumption saying that for every n, the step function (f h n) calls h only on the segment [0 ... n - 1]. This means that if h1 and h2 coincide on values < n, then (f h1 n) coincides with (f h2 n) *)
Hypothesis step_good :
forall (n : N.t) (h1 h2 : N.t -> A),
(forall m : N.t, m < n -> Aeq (h1 m) (h2 m)) -> Aeq (f h1 n) (f h2 n).
(** NB: without the [step_good] hypothesis, we have proved that [strong_rec a f 0] is [f (fun _ => a) 0]. Now we can prove that the first argument of [f] is arbitrary in this case... *)

Theorem strong_rec_0_any : forall (a : A)(any : N.t->A),
Aeq (strong_rec a f 0) (f any 0).