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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: FunctionalExtensionality.v 10739 2008-04-01 14:45:20Z herbelin $ i*)
(** This module states the axiom of (dependent) functional extensionality and (dependent) eta-expansion.
It introduces a tactic [extensionality] to apply the axiom of extensionality to an equality goal.
It also defines two lemmas for expansion of fixpoint defs using extensionnality and proof-irrelevance
to avoid a side condition on the functionals. *)
Require Import Coq.Program.Utils.
Require Import Coq.Program.Wf.
Require Import Coq.Program.Equality.
Set Implicit Arguments.
Unset Strict Implicit.
(** 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.
(** 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.
(** Apply [fun_extensionality], introducing variable x. *)
Tactic Notation "extensionality" ident(x) :=
match goal with
[ |- ?X = ?Y ] => apply (@fun_extensionality _ _ X Y) || apply (@fun_extensionality_dep _ _ X Y) ; intro x
end.
(** Eta expansion follows from extensionality. *)
Lemma eta_expansion_dep : forall A (B : A -> Type) (f : forall x : A, B x),
f = fun x => f x.
Proof.
intros.
extensionality x.
reflexivity.
Qed.
Lemma eta_expansion : forall A B (f : A -> B),
f = fun x => f x.
Proof.
intros ; apply eta_expansion_dep.
Qed.
(** 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. *)
Program 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).
extensionality y ; apply H.
rewrite H0 ; auto.
Qed.
(** For a function defined with Program using a measure. *)
Program 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).
extensionality y ; apply H.
rewrite H0 ; auto.
Qed.
|
