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(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)
(****************************************************************************)
(*                                                                          *)
(*                         Naive Set Theory in Coq                          *)
(*                                                                          *)
(*                     INRIA                        INRIA                   *)
(*              Rocquencourt                        Sophia-Antipolis        *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*									    *)
(*			         Gilles Kahn 				    *)
(*				 Gerard Huet				    *)
(*									    *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
(* to the Newton Institute for providing an exceptional work environment    *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
(****************************************************************************)

(*i $Id$ i*)

Require Export Finite_sets.
Require Export Constructive_sets.
Require Export Classical_Type.
Require Export Classical_sets.
Require Export Powerset.
Require Export Powerset_facts.
Require Export Powerset_Classical_facts.
Require Export Gt.
Require Export Lt.
Require Export Le.
Require Export Finite_sets_facts.

Section Image.
Variables U V : Type.

Inductive Im (X:Ensemble U) (f:U -> V) : Ensemble V :=
    Im_intro : forall x:U, In _ X x -> forall y:V, y = f x -> In _ (Im X f) y.

Lemma Im_def :
 forall (X:Ensemble U) (f:U -> V) (x:U), In _ X x -> In _ (Im X f) (f x).
Proof.
intros X f x H'; try assumption.
apply Im_intro with (x := x); auto with sets.
Qed.
Hint Resolve Im_def.

Lemma Im_add :
 forall (X:Ensemble U) (x:U) (f:U -> V),
   Im (Add _ X x) f = Add _ (Im X f) (f x).
Proof.
intros X x f.
apply Extensionality_Ensembles.
split; red in |- *; intros x0 H'.
elim H'; intros.
rewrite H0.
elim Add_inv with U X x x1; auto with sets.
destruct 1; auto with sets.
elim Add_inv with V (Im X f) (f x) x0; auto with sets.
destruct 1 as [x0 H y H0].
rewrite H0; auto with sets.
destruct 1; auto with sets.
Qed.

Lemma image_empty : forall f:U -> V, Im (Empty_set U) f = Empty_set V.
Proof.
intro f; try assumption.
apply Extensionality_Ensembles.
split; auto with sets.
red in |- *.
intros x H'; elim H'.
intros x0 H'0; elim H'0; auto with sets.
Qed.
Hint Resolve image_empty.

Lemma finite_image :
 forall (X:Ensemble U) (f:U -> V), Finite _ X -> Finite _ (Im X f).
Proof.
intros X f H'; elim H'.
rewrite (image_empty f); auto with sets.
intros A H'0 H'1 x H'2; clear H' X.
rewrite (Im_add A x f); auto with sets.
apply Add_preserves_Finite; auto with sets.
Qed.
Hint Resolve finite_image.

Lemma Im_inv :
 forall (X:Ensemble U) (f:U -> V) (y:V),
   In _ (Im X f) y ->  exists x : U, In _ X x /\ f x = y.
Proof.
intros X f y H'; elim H'.
intros x H'0 y0 H'1; rewrite H'1.
exists x; auto with sets.
Qed.

Definition injective (f:U -> V) := forall x y:U, f x = f y -> x = y.

Lemma not_injective_elim :
 forall f:U -> V,
   ~ injective f ->  exists x : _, (exists y : _, f x = f y /\ x <> y).
Proof.
unfold injective in |- *; intros f H.
cut (exists x : _, ~ (forall y:U, f x = f y -> x = y)).
2: apply not_all_ex_not with (P := fun x:U => forall y:U, f x = f y -> x = y);
    trivial with sets.
destruct 1 as [x C]; exists x.
cut (exists y : _, ~ (f x = f y -> x = y)).
2: apply not_all_ex_not with (P := fun y:U => f x = f y -> x = y);
    trivial with sets.
destruct 1 as [y D]; exists y.
apply imply_to_and; trivial with sets.
Qed.

Lemma cardinal_Im_intro :
 forall (A:Ensemble U) (f:U -> V) (n:nat),
   cardinal _ A n ->  exists p : nat, cardinal _ (Im A f) p.
Proof.
intros.
apply finite_cardinal; apply finite_image.
apply cardinal_finite with n; trivial with sets.
Qed.

Lemma In_Image_elim :
 forall (A:Ensemble U) (f:U -> V),
   injective f -> forall x:U, In _ (Im A f) (f x) -> In _ A x.
Proof.
intros.
elim Im_inv with A f (f x); trivial with sets.
intros z C; elim C; intros InAz E.
elim (H z x E); trivial with sets.
Qed.

Lemma injective_preserves_cardinal :
 forall (A:Ensemble U) (f:U -> V) (n:nat),
   injective f ->
   cardinal _ A n -> forall n':nat, cardinal _ (Im A f) n' -> n' = n.
Proof.
induction 2 as [| A n H'0 H'1 x H'2]; auto with sets.
rewrite (image_empty f).
intros n' CE.
apply cardinal_unicity with V (Empty_set V); auto with sets.
intro n'.
rewrite (Im_add A x f).
intro H'3.
elim cardinal_Im_intro with A f n; trivial with sets.
intros i CI.
lapply (H'1 i); trivial with sets.
cut (~ In _ (Im A f) (f x)).
intros H0 H1.
apply cardinal_unicity with V (Add _ (Im A f) (f x)); trivial with sets.
apply card_add; auto with sets.
rewrite <- H1; trivial with sets.
red in |- *; intro; apply H'2.
apply In_Image_elim with f; trivial with sets.
Qed.

Lemma cardinal_decreases :
 forall (A:Ensemble U) (f:U -> V) (n:nat),
   cardinal U A n -> forall n':nat, cardinal V (Im A f) n' -> n' <= n.
Proof.
induction 1 as [| A n H'0 H'1 x H'2]; auto with sets.
rewrite (image_empty f); intros.
cut (n' = 0).
intro E; rewrite E; trivial with sets.
apply cardinal_unicity with V (Empty_set V); auto with sets.
intro n'.
rewrite (Im_add A x f).
elim cardinal_Im_intro with A f n; trivial with sets.
intros p C H'3.
apply le_trans with (S p).
apply card_Add_gen with V (Im A f) (f x); trivial with sets.
apply le_n_S; auto with sets.
Qed.

Theorem Pigeonhole :
 forall (A:Ensemble U) (f:U -> V) (n:nat),
   cardinal U A n ->
   forall n':nat, cardinal V (Im A f) n' -> n' < n -> ~ injective f.
Proof.
unfold not in |- *; intros A f n CAn n' CIfn' ltn'n I.
cut (n' = n).
intro E; generalize ltn'n; rewrite E; exact (lt_irrefl n).
apply injective_preserves_cardinal with (A := A) (f := f) (n := n);
 trivial with sets.
Qed.

Lemma Pigeonhole_principle :
 forall (A:Ensemble U) (f:U -> V) (n:nat),
   cardinal _ A n ->
   forall n':nat,
     cardinal _ (Im A f) n' ->
     n' < n ->  exists x : _, (exists y : _, f x = f y /\ x <> y).
Proof.
intros; apply not_injective_elim.
apply Pigeonhole with A n n'; trivial with sets.
Qed.
End Image.
Hint Resolve Im_def image_empty finite_image: sets v62.