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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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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Set Implicit Arguments.

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Definition is_inverse A B f g := (forall a:A, g (f a) = a) /\ (forall b:B, f (g b) = b).

#[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.
forall (A : Type) (x : Delta A), pi1 x = pi2 x
Proof.
forall (A : Type) (x : Delta A), pi1 x = pi2 x
  destruct x as (a1,a2,Heq); assumption.
Qed.

Lemma diagonal_inverse1 : forall A, is_inverse (A:=A) delta pi1.
forall A : Type, is_inverse delta pi1
Proof.
forall A : Type, is_inverse delta pi1
  split; [trivial|]; destruct b as (a1,a2,[]); reflexivity.
Qed.

Lemma diagonal_inverse2 : forall A, is_inverse (A:=A) delta pi2.
forall A : Type, is_inverse delta pi2
Proof.
forall A : Type, is_inverse delta pi2
  split; [trivial|]; destruct b as (a1,a2,[]); reflexivity.
Qed.

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

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

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

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

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

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Theorem FunctExt_iff_EqDeltaProjs : FunctionalExtensionality <-> EqDeltaProjs.
FunctionalExtensionality <-> EqDeltaProjs
Proof.
FunctionalExtensionality <-> EqDeltaProjs
  split.
FunctionalExtensionality -> EqDeltaProjs
EqDeltaProjs -> FunctionalExtensionality
  -
FunctionalExtensionality -> EqDeltaProjs intros FunExt *; apply FunExt, diagonal_projs_same_behavior. 
  -
EqDeltaProjs -> FunctionalExtensionality intros EqProjs **; change f with (fun x => pi1 {|pi1:=f x; pi2:=g x; eq:=H x|}).
EqProjs:EqDeltaProjs
A, B:Type
f, g:A -> B
H:forall x : A, f x = g x
(fun x : A =>
 pi1 {| pi1 := f x; pi2 := g x; eq := H x |}) = g
    rewrite EqProjs; reflexivity.
Qed.

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Lemma FunctExt_UniqInverse : FunctionalExtensionality -> UniqueInverse.
FunctionalExtensionality -> UniqueInverse
Proof.
FunctionalExtensionality -> UniqueInverse
  intros FunExt * (Hg1f,Hfg1) (Hg2f,Hfg2).
FunExt:FunctionalExtensionality
A, B:Type
f:A -> B
g1, g2:B -> A
Hg1f:forall a : A, g1 (f a) = a
Hfg1:forall b : B, f (g1 b) = b
Hg2f:forall a : A, g2 (f a) = a
Hfg2:forall b : B, f (g2 b) = b
g1 = g2
  apply FunExt.
FunExt:FunctionalExtensionality
A, B:Type
f:A -> B
g1, g2:B -> A
Hg1f:forall a : A, g1 (f a) = a
Hfg1:forall b : B, f (g1 b) = b
Hg2f:forall a : A, g2 (f a) = a
Hfg2:forall b : B, f (g2 b) = b
forall x : B, g1 x = g2 x intros; congruence.
Qed.

Lemma UniqInverse_EqDeltaProjs : UniqueInverse -> EqDeltaProjs.
UniqueInverse -> EqDeltaProjs
Proof.
UniqueInverse -> EqDeltaProjs
  intros UniqInv *.
UniqInv:UniqueInverse
A:Type
pi1 = pi2
  apply UniqInv with delta; [apply diagonal_inverse1 | apply diagonal_inverse2].
Qed.

Theorem FunctExt_iff_UniqInverse : FunctionalExtensionality <-> UniqueInverse.
FunctionalExtensionality <-> UniqueInverse
Proof.
FunctionalExtensionality <-> UniqueInverse
  split.
FunctionalExtensionality -> UniqueInverse
UniqueInverse -> FunctionalExtensionality 
  -
FunctionalExtensionality -> UniqueInverse apply FunctExt_UniqInverse.
  -
UniqueInverse -> FunctionalExtensionality intro; apply FunctExt_iff_EqDeltaProjs, UniqInverse_EqDeltaProjs; trivial.
Qed.

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Lemma FunctExt_BijComp : FunctionalExtensionality -> BijectivityBijectiveComp.
FunctionalExtensionality -> BijectivityBijectiveComp
Proof.
FunctionalExtensionality -> BijectivityBijectiveComp
  intros FunExt * (Hgf,Hfg).
FunExt:FunctionalExtensionality
A, B, C:Type
f:A -> B
g:B -> A
Hgf:forall a : A, g (f a) = a
Hfg:forall b : B, f (g b) = b
is_inverse (action f) (action g) split; unfold action.
FunExt:FunctionalExtensionality
A, B, C:Type
f:A -> B
g:B -> A
Hgf:forall a : A, g (f a) = a
Hfg:forall b : B, f (g b) = b
forall a : B -> C, (fun x : B => a (f (g x))) = a
FunExt:FunctionalExtensionality
A, B, C:Type
f:A -> B
g:B -> A
Hgf:forall a : A, g (f a) = a
Hfg:forall b : B, f (g b) = b
forall b : A -> C, (fun x : A => b (g (f x))) = b
  -
FunExt:FunctionalExtensionality
A, B, C:Type
f:A -> B
g:B -> A
Hgf:forall a : A, g (f a) = a
Hfg:forall b : B, f (g b) = b
forall a : B -> C, (fun x : B => a (f (g x))) = a intros h; apply FunExt; intro b; rewrite Hfg; reflexivity.
  -
FunExt:FunctionalExtensionality
A, B, C:Type
f:A -> B
g:B -> A
Hgf:forall a : A, g (f a) = a
Hfg:forall b : B, f (g b) = b
forall b : A -> C, (fun x : A => b (g (f x))) = b intros h; apply FunExt; intro a; rewrite Hgf; reflexivity.
Qed.

Lemma BijComp_FunctExt : BijectivityBijectiveComp -> FunctionalExtensionality.
BijectivityBijectiveComp -> FunctionalExtensionality
Proof.
BijectivityBijectiveComp -> FunctionalExtensionality
  intros BijComp.
BijComp:BijectivityBijectiveComp
FunctionalExtensionality
  apply FunctExt_iff_UniqInverse.
BijComp:BijectivityBijectiveComp
UniqueInverse intros * H1 H2.
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
g1 = g2
  destruct BijComp with (C:=A) (1:=H2) as (Hg2f,_).
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
Hg2f:forall a : B -> A, action g2 (action f a) = a
g1 = g2
  destruct BijComp with (C:=A) (1:=H1) as (_,Hfg1).
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
Hg2f:forall a : B -> A, action g2 (action f a) = a
Hfg1:forall b : A -> A, action f (action g1 b) = b
g1 = g2
  rewrite <- (Hg2f g1).
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
Hg2f:forall a : B -> A, action g2 (action f a) = a
Hfg1:forall b : A -> A, action f (action g1 b) = b
action g2 (action f g1) = g2
  change g1 with (action g1 (fun x => x)).
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
Hg2f:forall a : B -> A, action g2 (action f a) = a
Hfg1:forall b : A -> A, action f (action g1 b) = b
action g2 (action f (action g1 (fun x : A => x))) = g2
  rewrite -> (Hfg1 (fun x => x)).
BijComp:BijectivityBijectiveComp
A, B:Type
f:A -> B
g1, g2:B -> A
H1:is_inverse f g1
H2:is_inverse f g2
Hg2f:forall a : B -> A, action g2 (action f a) = a
Hfg1:forall b : A -> A, action f (action g1 b) = b
action g2 (fun x : A => x) = g2
  reflexivity.
Qed.