From a869d74f414ba786c66b8eb7450ff6343ff12ebd Mon Sep 17 00:00:00 2001
From: msozeau
Date: Tue, 16 Dec 2008 12:57:26 +0000
Subject: Move FunctionalExtensionality to Logic/ (someone please check that
 the doc is ok). Rework the .v files in Program accordingly, adding some
 documentation and proper headers. Integrate the development of an elimination
 principle for measured functions in Program/Wf by Eelis van der Weegen.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11686 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Program/FunctionalExtensionality.v | 123 ----------------------------
 1 file changed, 123 deletions(-)
 delete mode 100644 theories/Program/FunctionalExtensionality.v

(limited to 'theories/Program/FunctionalExtensionality.v')

diff --git a/theories/Program/FunctionalExtensionality.v b/theories/Program/FunctionalExtensionality.v
deleted file mode 100644
index 8bd2dfb905..0000000000
--- a/theories/Program/FunctionalExtensionality.v
+++ /dev/null
@@ -1,123 +0,0 @@
-(* -*- 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 $Id$ 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 extensionality. *)
-
-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 extensionality 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.
-
-(** Tactics to fold/unfold definitions based on [Fix_measure_sub]. *)
-
-Ltac fold_sub f :=
-  match goal with
-    | [ |- ?T ] => 
-      match T with
-        appcontext C [ @Fix_measure_sub _ _ _ _ ?arg ] => 
-        let app := context C [ f arg ] in
-         change app
-      end
-  end.
-
-Ltac unfold_sub f fargs := 
-  set (call:=fargs) ; unfold f in call ; unfold call ; clear call ; 
-  rewrite fix_sub_measure_eq_ext ; repeat fold_sub fargs ; simpl proj1_sig.
-- 
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