Library Stdlib.Logic.PropExtensionalityFacts

Some facts and definitions about propositional and predicate extensionality

We investigate the relations between the following extensionality principles

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).