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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \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: Image.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ 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: (x: U) (In ? X x) -> (y: V) y == (f x) -> (In ? (Im X f) y).
Lemma Im_def:
(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.
Hints Resolve Im_def.
Lemma Im_add:
(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; Intros x0 H'.
Elim H'; Intros.
Rewrite H0.
Elim Add_inv with U X x x1; Auto with sets.
NewDestruct 1; Auto with sets.
Elim Add_inv with V (Im X f) (f x) x0; Auto with sets.
NewDestruct 1 as [x0 H y H0].
Rewrite H0; Auto with sets.
NewDestruct 1; Auto with sets.
Qed.
Lemma image_empty: (f: U -> V) (Im (Empty_set U) f) == (Empty_set V).
Proof.
Intro f; Try Assumption.
Apply Extensionality_Ensembles.
Split; Auto with sets.
Red.
Intros x H'; Elim H'.
Intros x0 H'0; Elim H'0; Auto with sets.
Qed.
Hints Resolve image_empty.
Lemma finite_image:
(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.
Hints Resolve finite_image.
Lemma Im_inv:
(X: (Ensemble U)) (f: U -> V) (y: V) (In ? (Im X f) y) ->
(exT ? [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] (x, y: U) (f x) == (f y) -> x == y.
Lemma not_injective_elim:
(f: U -> V) ~ (injective f) ->
(EXT x | (EXT y | (f x) == (f y) /\ ~ x == y)).
Proof.
Unfold injective; Intros f H.
Cut (EXT x | ~ ((y: U) (f x) == (f y) -> x == y)).
2: Apply not_all_ex_not with P:=[x:U](y: U) (f x) == (f y) -> x == y;
Trivial with sets.
NewDestruct 1 as [x C]; Exists x.
Cut (EXT y | ~((f x)==(f y)->x==y)).
2: Apply not_all_ex_not with P:=[y:U](f x)==(f y)->x==y; Trivial with sets.
NewDestruct 1 as [y D]; Exists y.
Apply imply_to_and; Trivial with sets.
Qed.
Lemma cardinal_Im_intro:
(A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal ? A n) ->
(EX 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:
(A: (Ensemble U)) (f: U -> V) (injective f) ->
(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:
(A: (Ensemble U)) (f: U -> V) (n: nat) (injective f) -> (cardinal ? A n) ->
(n': nat) (cardinal ? (Im A f) n') -> n' = n.
Proof.
NewInduction 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; Intro; Apply H'2.
Apply In_Image_elim with f; Trivial with sets.
Qed.
Lemma cardinal_decreases:
(A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal U A n) ->
(n': nat) (cardinal V (Im A f) n') -> (le n' n).
Proof.
NewInduction 1 as [|A n H'0 H'1 x H'2]; Auto with sets.
Rewrite (image_empty f); Intros.
Cut n' = O.
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:
(A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal U A n) ->
(n': nat) (cardinal V (Im A f) n') -> (lt n' n) -> ~ (injective f).
Proof.
Unfold not; Intros A f n CAn n' CIfn' ltn'n I.
Cut n' = n.
Intro E; Generalize ltn'n; Rewrite E; Exact (lt_n_n n).
Apply injective_preserves_cardinal with A := A f := f n := n; Trivial with sets.
Qed.
Lemma Pigeonhole_principle:
(A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal ? A n) ->
(n': nat) (cardinal ? (Im A f) n') -> (lt n' n) ->
(EXT x | (EXT 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.
Hints Resolve Im_def image_empty finite_image : sets v62.
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