# Library Coq.Logic.ExtensionalityFacts

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
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.

# 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} _.

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.

Functional extensionality

Local Notation FunctionalExtensionality :=
(forall A B (f g : A -> B), (forall x, f x = g x) -> f = g).

Equality of projections from diagonal

Local Notation EqDeltaProjs := (forall A, pi1 = pi2 :> (Delta A -> A)).

Unicity of bijection inverse

Local Notation UniqueInverse := (forall A B (f:A->B) g1 g2, is_inverse f g1 -> is_inverse f g2 -> g1 = g2).

Bijectivity of bijective composition

Definition action A B C (f:A->B) := (fun h:B->C => fun x => h (f x)).

Local Notation BijectivityBijectiveComp := (forall A B C (f:A->B) g,
is_inverse f g -> is_inverse (A:=B->C) (action f) (action g)).