(* Let's call f's second parameter its "lowers" function, since it provides it access to results for inputs with a lower measure.  In preparation of lemma similar to Fix_F_sub_rect, but for Fix_sub, we first need an extra hypothesis stating that the function body has the same result for different "lowers" functions (g and h below) as long as those produce the same results for lower inputs, regardless of the lt proofs. *)
Hypothesis equiv_lowers:
forall x0 (g h: forall x: {y: A | R y x0}, P (proj1_sig x)),
(forall x p p', g (exist (fun y: A => R y x0) x p) = h (exist (*FIXME shouldn't be needed *) (fun y => R y x0) x p')) ->
f g = f h.
(* From equiv_lowers, it follows that [Fix_F_sub A R P f x] applications do not not depend on the Acc proofs. *)

(* Specifications: *)
Lemma eq_Fix_F_sub x (a a': Acc R x):
Fix_F_sub A R P f x a =
Fix_F_sub A R P f x a'.