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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** 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.
(**********************************************************************)
(** * 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).
Definition iscontr (X:Type) := {x:X| forall y, x=y}.
Definition hfiber X Y (f:X -> Y) (y:Y) := {x:X| f x=y}.
Definition isweq X Y (f:X -> Y) := forall y, iscontr (hfiber f y).
Goal forall X Y (f:X->Y), isweq f -> {g|is_inverse f g}.
Proof.
intros * Hweq.
set (g:=fun y => proj1_sig (proj1_sig (Hweq y))).
exists g. split.
- intro x. unfold g. destruct (Hweq (f x)) as ((x0,Hx0),Hxy). simpl.
set (fiber0 := exist (fun x0 => f x0 = f x) x0 Hx0) in *.
set (fiber := exist (fun x0 => f x0 = f x) x eq_refl).
change (proj1_sig fiber0 = proj1_sig fiber).
specialize Hxy with fiber.
destruct Hxy.
reflexivity.
- intro y.
unfold g.
destruct (Hweq y) as ((x,Hxy),Hyuniq). simpl. assumption.
Qed.
Goal forall X Y (f:X->Y), {g|is_inverse f g} -> isweq f.
Proof.
destruct 1 as (g,(Hgf,Hfg)).
intro y.
set (fiber := exist (fun x => f x = y) (g y) (Hfg y)).
exists fiber.
destruct y0 as (y0,Hfy0).
unfold fiber.
rewrite <- Hfy0 in *.
assert (forall y, f_equal f (Hgf y) = Hfg (f y)). admit.
rewrite <- H.
rewrite Hgf.
reflexivity.
Qed.
(** The diagonal over A and the one-one correspondence with A *)
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.
Definition delta {A} (a:A) := {|pi1 := a; pi2 := a; eq := eq_refl a |}.
Implicit Arguments pi1 [[A]].
Implicit Arguments pi2 [[A]].
Lemma diagonal_projs_same_behavior : forall A (x:Delta A), pi1 x = pi2 x.
Proof.
destruct x as (a1,a2,Heq); assumption.
Qed.
Lemma diagonal_inverse1 : forall A, is_inverse (A:=A) delta pi1.
Proof.
split; [trivial|]; destruct b as (a1,a2,[]); reflexivity.
Qed.
Lemma diagonal_inverse2 : forall A, is_inverse (A:=A) delta pi2.
Proof.
split; [trivial|]; destruct b as (a1,a2,[]); reflexivity.
Qed.
(** 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)).
(**********************************************************************)
(** * Functional extensionality <-> Equality of projections from diagonal *)
Theorem FunctExt_iff_EqDeltaProjs : FunctionalExtensionality <-> EqDeltaProjs.
Proof.
split.
- intros FunExt *; apply FunExt, diagonal_projs_same_behavior.
- intros EqProjs **; change f with (fun x => pi1 {|pi1:=f x; pi2:=g x; eq:=H x|}).
rewrite EqProjs; reflexivity.
Qed.
(**********************************************************************)
(** * Functional extensionality <-> Unicity of bijection inverse *)
Lemma FunctExt_UniqInverse : FunctionalExtensionality -> UniqueInverse.
Proof.
intros FunExt * (Hg1f,Hfg1) (Hg2f,Hfg2).
apply FunExt. intros; congruence.
Qed.
Lemma UniqInverse_EqDeltaProjs : UniqueInverse -> EqDeltaProjs.
Proof.
intros UniqInv *.
apply UniqInv with delta; [apply diagonal_inverse1 | apply diagonal_inverse2].
Qed.
Theorem FunctExt_iff_UniqInverse : FunctionalExtensionality <-> UniqueInverse.
Proof.
split.
- apply FunctExt_UniqInverse.
- intro; apply FunctExt_iff_EqDeltaProjs, UniqInverse_EqDeltaProjs; trivial.
Qed.
(**********************************************************************)
(** * Functional extensionality <-> Bijectivity of bijective composition *)
Lemma FunctExt_BijComp : FunctionalExtensionality -> BijectivityBijectiveComp.
Proof.
intros FunExt * (Hgf,Hfg). split; unfold action.
- intros h; apply FunExt; intro b; rewrite Hfg; reflexivity.
- intros h; apply FunExt; intro a; rewrite Hgf; reflexivity.
Qed.
Lemma BijComp_FunctExt : BijectivityBijectiveComp -> FunctionalExtensionality.
Proof.
intros BijComp.
apply FunctExt_iff_UniqInverse. intros * H1 H2.
destruct BijComp with (C:=A) (1:=H2) as (Hg2f,_).
destruct BijComp with (C:=A) (1:=H1) as (_,Hfg1).
rewrite <- (Hg2f g1).
change g1 with (action g1 (fun x => x)).
rewrite -> (Hfg1 (fun x => x)).
reflexivity.
Qed.
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