mpri-funcprog-project/coq/FixExtra.v

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Require Import Coq.Logic.FunctionalExtensionality.
(* This is a simplified version of the lemma [Fix_eq], which is defined in
[Coq.Init.Wf]. We use functional extensionality to remove one hypothesis.
Furthermore, we introduce the auxiliary equality [f = Fix Rwf P F] so as
to avoid duplicating the (usually large) term [F] in the right-hand side
of the conclusion. *)
Lemma Fix_eq_simplified
(A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
(P : A -> Type)
(F : forall x : A, (forall y : A, R y x -> P y) -> P x)
(f : forall x, P x) :
f = Fix Rwf P F ->
forall x : A,
f x = F x (fun (y : A) (_ : R y x) => f y).
Proof.
intros. subst. eapply Fix_eq. intros. f_equal.
eauto using functional_extensionality_dep, functional_extensionality.
Qed.