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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Some facts and definitions about propositional and predicate extensionality  We investigate the relations between the following extensionality principles  - Proposition extensionality - Predicate extensionality - Propositional functional extensionality - Provable-proposition extensionality - Refutable-proposition extensionality - Extensional proposition representatives - Extensional predicate representatives - Extensional propositional function representatives  Table of contents  1. Definitions  2.1 Predicate extensionality <-> Proposition extensionality + Propositional functional extensionality  2.2 Propositional extensionality -> Provable propositional extensionality  2.3 Propositional extensionality -> Refutable propositional extensionality  *)
Set Implicit Arguments.
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(** * 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).
(** Refutable-proposition extensionality *)
Local Notation RefutablePropositionExtensionality :=
(forall A:Prop, ~A -> A = False).
(** 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).
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(** * Propositional and predicate extensionality                      *)
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(** ** Predicate extensionality <-> Propositional extensionality + Propositional functional extensionality *)

(* Specifications: *)
Lemma PredExt_imp_PropExt : PredicateExtensionality -> PropositionalExtensionality.
Lemma PredExt_imp_PropFunExt : PredicateExtensionality -> PropositionalFunctionalExtensionality.
Lemma PropExt_and_PropFunExt_imp_PredExt :
PropositionalExtensionality -> PropositionalFunctionalExtensionality -> PredicateExtensionality.
Theorem PropExt_and_PropFunExt_iff_PredExt :
PropositionalExtensionality /\ PropositionalFunctionalExtensionality <-> PredicateExtensionality.