Section FixPoint.
Variable P : A -> Type.
Variable F : forall x:A, (forall y:A, R y x -> P y) -> P x.
(** Proof that [well_founded_induction] satisfies the fixpoint equation. It requires an extra property of the functional *)
Variable EQ : forall a, P a -> P a -> Prop.
Variable Equiv: forall a: A, Equivalence (@EQ a).
Hypothesis
F_ext :
forall (x:A) (f g:forall y:A, R y x -> P y),
(forall (y:A) (p:R y x), EQ (f y p) (g y p)) -> EQ (F f) (F g).

Lemma Fix_eq : forall x:A, EQ (Fix Rwf P F x) (F (fun (y:A) (p:R y x) => Fix Rwf P F y)).