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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) *)
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(*    //   *    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 extensionality  We investigate the relations between the following extensionality principles  - Functional extensionality - Equality of projections from diagonal - Unicity of inverse bijections - Bijectivity of bijective composition  Table of contents  1. Definitions  2. Functional extensionality <-> Equality of projections from diagonal  3. Functional extensionality <-> Unicity of inverse bijections  4. Functional extensionality <-> Bijectivity of bijective composition  *)
Set Implicit Arguments.
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(** * Definitions *)
(** Being an inverse *)
Definition is_inverse A B f g := (forall a:A, g (f a) = a) /\ (forall b:B, f (g b) = b).
(** The diagonal over A and the one-one correspondence with A *)
#[universes(template)]
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.
Definition delta {A} (a:A) := {| pi1 := a; pi2 := a; eq := eq_refl a |}.
Arguments pi1 {A} _.
Arguments pi2 {A} _.

(* Specifications: *)
Lemma diagonal_projs_same_behavior : forall A (x:Delta A), pi1 x = pi2 x.
Lemma diagonal_inverse1 : forall A, is_inverse (A:=A) delta pi1.
Lemma diagonal_inverse2 : forall A, is_inverse (A:=A) delta pi2.
Local Notation FunctionalExtensionality :=
(forall A B (f g : A -> B), (forall x, f x = g x) -> f = g).