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Require Import Coq.Program.Utils.
Require Import Coq.Program.Wf.
(** The converse of functional equality. *)
Lemma equal_f : forall A B : Type, forall (f g : A -> B),
f = g -> forall x, f x = g x.
Proof.
intros.
rewrite H.
auto.
Qed.
(** Eta expansion *)
Axiom eta_expansion_dep : forall A (B : A -> Type) (f : forall x : A, B x),
f = fun x => f x.
Lemma eta_expansion : forall A B (f : A -> B),
f = fun x => f x.
Proof.
intros ; apply eta_expansion_dep.
Qed.
(** Statements of functional equality for simple and dependent functions. *)
Axiom fun_extensionality_dep : forall A, forall B : (A -> Type),
forall (f g : forall x : A, B x),
(forall x, f x = g x) -> f = g.
Lemma fun_extensionality : forall A B (f g : A -> B),
(forall x, f x = g x) -> f = g.
Proof.
intros ; apply fun_extensionality_dep.
assumption.
Qed.
Hint Resolve fun_extensionality fun_extensionality_dep : program.
(** Unification needs help sometimes... *)
Ltac apply_ext :=
match goal with
[ |- ?x = ?y ] => apply (@fun_extensionality _ _ x y)
end.
(** The two following lemmas allow to unfold a well-founded fixpoint definition without
restriction using the functional extensionality axiom. *)
(** For a function defined with Program using a well-founded order. *)
Lemma fix_sub_eq_ext :
forall (A : Set) (R : A -> A -> Prop) (Rwf : well_founded R)
(P : A -> Set)
(F_sub : forall x : A, (forall {y : A | R y x}, P (`y)) -> P x),
forall x : A,
Fix_sub A R Rwf P F_sub x =
F_sub x (fun {y : A | R y x}=> Fix A R Rwf P F_sub (`y)).
Proof.
intros ; apply Fix_eq ; auto.
intros.
assert(f = g).
apply (fun_extensionality_dep _ _ _ _ H).
rewrite H0 ; auto.
Qed.
(** For a function defined with Program using a measure. *)
Lemma fix_sub_measure_eq_ext :
forall (A : Type) (f : A -> nat) (P : A -> Type)
(F_sub : forall x : A, (forall {y : A | f y < f x}, P (`y)) -> P x),
forall x : A,
Fix_measure_sub A f P F_sub x =
F_sub x (fun {y : A | f y < f x}=> Fix_measure_sub A f P F_sub (`y)).
Proof.
intros ; apply Fix_measure_eq ; auto.
intros.
assert(f0 = g).
apply (fun_extensionality_dep _ _ _ _ H).
rewrite H0 ; auto.
Qed.
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