Logic library: Adding a characterization of excluded-middle in term of

choice of a representative in a partition of bool.

Also move a result about propositional extensionality from
ClassicalFacts.v to PropExtensionalityFacts.v, generalizing it by
symmetry.

Also spotting typos (thanks to Théo).

1 files changed, 109 insertions, 0 deletions

diff --git a/theories/Logic/PropExtensionalityFacts.v b/theories/Logic/PropExtensionalityFacts.v
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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
+
+- 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.
+
+(**********************************************************************)
+(** * 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).
+
+(**********************************************************************)
+(** * 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 and provable proposition extensionality *)
+
+Lemma PropExt_imp_ProvPropExt : PropositionalExtensionality -> ProvablePropositionExtensionality.
+Proof.
+  intros Ext A Ha; apply Ext; split; trivial.
+Qed.
+
+(**********************************************************************)
+(** ** Propositional extensionality and refutable proposition extensionality *)
+
+Lemma PropExt_imp_RefutPropExt : PropositionalExtensionality -> RefutablePropositionExtensionality.
+Proof.
+  intros Ext A Ha; apply Ext; split; easy.
+Qed.