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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-(** Some facts and definitions about propositional and predicate extensionality
-
-We investigate the relations between the following extensionality principles
-
-- Provable-proposition extensionality
-- Predicate extensionality
-- Propositional functional extensionality
-
-Table of contents
-
-1. Definitions
-
-2.1 Predicate extensionality <-> Proposition extensionality + Propositional functional extensionality
-
-2.2 Propositional extensionality -> Provable propositional extensionality
-*)
-
-Set Implicit Arguments.
-
-(**********************************************************************)
-(** * Definitions *)
-
-(** Propositional extensionality *)
-
-Local Notation PropositionalExtensionality :=
-  (forall A B : Prop, (A <-> B) -> A = B).
-
-(** Provable-proposition extensionality *)
-
-Local Notation ProvablePropositionExtensionality :=
-  (forall A:Prop, A -> A = True).
-
-(** Predicate extensionality *)
-
-Local Notation PredicateExtensionality :=
-  (forall (A:Type) (P Q : A -> Prop), (forall x, P x <-> Q x) -> P = Q).
-
-(** Propositional functional extensionality *)
-
-Local Notation PropositionalFunctionalExtensionality :=
-  (forall (A:Type) (P Q : A -> Prop), (forall x, P x = Q x) -> P = Q).
-
-(**********************************************************************)
-(** * Propositional and predicate extensionality                      *)
-
-(**********************************************************************)
-(** ** Predicate extensionality <-> Propositional extensionality + Propositional functional extensionality *)
-
-Lemma PredExt_imp_PropExt : PredicateExtensionality -> PropositionalExtensionality.
-Proof.
-  intros Ext A B Equiv. 
-  change A with ((fun _ => A) I).
-  now rewrite Ext with (P := fun _ : True =>A) (Q := fun _ => B).
-Qed.
-
-Lemma PredExt_imp_PropFunExt : PredicateExtensionality -> PropositionalFunctionalExtensionality.
-Proof.
-  intros Ext A P Q Eq. apply Ext. intros x. now rewrite (Eq x).
-Qed.
-
-Lemma PropExt_and_PropFunExt_imp_PredExt :
-  PropositionalExtensionality -> PropositionalFunctionalExtensionality -> PredicateExtensionality.
-Proof.
-  intros Ext FunExt A P Q Equiv.
-  apply FunExt. intros x. now apply Ext.
-Qed.
-
-Theorem PropExt_and_PropFunExt_iff_PredExt :
-  PropositionalExtensionality /\ PropositionalFunctionalExtensionality <-> PredicateExtensionality.
-Proof.
-  firstorder using PredExt_imp_PropExt, PredExt_imp_PropFunExt, PropExt_and_PropFunExt_imp_PredExt.
-Qed.
-
-(**********************************************************************)
-(** ** Propositional extensionality + Provable proposition extensionality *)
-
-Lemma PropExt_imp_ProvPropExt : PropositionalExtensionality -> ProvablePropositionExtensionality.
-Proof.
-  intros Ext A Ha; apply Ext; split; trivial.
-Qed.