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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.
-
-