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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: Finite_sets_facts.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.
-
-Section Finite_sets_facts.
-Variable U: Type.
-
-Lemma finite_cardinal :
- (X: (Ensemble U)) (Finite U X) -> (EX n:nat |(cardinal U X n)).
-Proof.
-NewInduction 1 as [|A _ [n H]].
-Exists O; Auto with sets.
-Exists (S n); Auto with sets.
-Qed.
-
-Lemma cardinal_finite:
- (X: (Ensemble U)) (n: nat) (cardinal U X n) -> (Finite U X).
-Proof.
-NewInduction 1; Auto with sets.
-Qed.
-
-Theorem Add_preserves_Finite:
- (X: (Ensemble U)) (x: U) (Finite U X) -> (Finite U (Add U X x)).
-Proof.
-Intros X x H'.
-Elim (classic (In U X x)); Intro H'0; Auto with sets.
-Rewrite (Non_disjoint_union U X x); Auto with sets.
-Qed.
-Hints Resolve Add_preserves_Finite.
-
-Theorem Singleton_is_finite: (x: U) (Finite U (Singleton U x)).
-Proof.
-Intro x; Rewrite <- (Empty_set_zero U (Singleton U x)).
-Change (Finite U (Add U (Empty_set U) x)); Auto with sets.
-Qed.
-Hints Resolve Singleton_is_finite.
-
-Theorem Union_preserves_Finite:
- (X, Y: (Ensemble U)) (Finite U X) -> (Finite U Y) ->
- (Finite U (Union U X Y)).
-Proof.
-Intros X Y H'; Elim H'.
-Rewrite (Empty_set_zero U Y); Auto with sets.
-Intros A H'0 H'1 x H'2 H'3.
-Rewrite (Union_commutative U (Add U A x) Y).
-Rewrite <- (Union_add U Y A x).
-Rewrite (Union_commutative U Y A); Auto with sets.
-Qed.
-
-Lemma Finite_downward_closed:
- (A: (Ensemble U)) (Finite U A) ->
- (X: (Ensemble U)) (Included U X A) -> (Finite U X).
-Proof.
-Intros A H'; Elim H'; Auto with sets.
-Intros X H'0.
-Rewrite (less_than_empty U X H'0); Auto with sets.
-Intros; Elim Included_Add with U X A0 x; Auto with sets.
-NewDestruct 1 as [A' [H5 H6]].
-Rewrite H5; Auto with sets.
-Qed.
-
-Lemma Intersection_preserves_finite:
- (A: (Ensemble U)) (Finite U A) ->
- (X: (Ensemble U)) (Finite U (Intersection U X A)).
-Proof.
-Intros A H' X; Apply Finite_downward_closed with A; Auto with sets.
-Qed.
-
-Lemma cardinalO_empty:
- (X: (Ensemble U)) (cardinal U X O) -> X == (Empty_set U).
-Proof.
-Intros X H; Apply (cardinal_invert U X O); Trivial with sets.
-Qed.
-Hints Resolve cardinalO_empty.
-
-Lemma inh_card_gt_O:
- (X: (Ensemble U)) (Inhabited U X) -> (n: nat) (cardinal U X n) -> (gt n O).
-Proof.
-NewInduction 1 as [x H'].
-Intros n H'0.
-Elim (gt_O_eq n); Auto with sets.
-Intro H'1; Generalize H'; Generalize H'0.
-Rewrite <- H'1; Intro H'2.
-Rewrite (cardinalO_empty X); Auto with sets.
-Intro H'3; Elim H'3.
-Qed.
-
-Lemma card_soustr_1:
- (X: (Ensemble U)) (n: nat) (cardinal U X n) ->
- (x: U) (In U X x) -> (cardinal U (Subtract U X x) (pred n)).
-Proof.
-Intros X n H'; Elim H'.
-Intros x H'0; Elim H'0.
-Clear H' n X.
-Intros X n H' H'0 x H'1 x0 H'2.
-Elim (classic (In U X x0)).
-Intro H'4; Rewrite (add_soustr_xy U X x x0).
-Elim (classic x == x0).
-Intro H'5.
-Absurd (In U X x0); Auto with sets.
-Rewrite <- H'5; Auto with sets.
-Intro H'3; Try Assumption.
-Cut (S (pred n)) = (pred (S n)).
-Intro H'5; Rewrite <- H'5.
-Apply card_add; Auto with sets.
-Red; Intro H'6; Elim H'6.
-Intros H'7 H'8; Try Assumption.
-Elim H'1; Auto with sets.
-Unfold 2 pred; Symmetry.
-Apply S_pred with m := O.
-Change (gt n O).
-Apply inh_card_gt_O with X := X; Auto with sets.
-Apply Inhabited_intro with x := x0; Auto with sets.
-Red; Intro H'3.
-Apply H'1.
-Elim H'3; Auto with sets.
-Rewrite H'3; Auto with sets.
-Elim (classic x == x0).
-Intro H'3; Rewrite <- H'3.
-Cut (Subtract U (Add U X x) x) == X; Auto with sets.
-Intro H'4; Rewrite H'4; Auto with sets.
-Intros H'3 H'4; Try Assumption.
-Absurd (In U (Add U X x) x0); Auto with sets.
-Red; Intro H'5; Try Exact H'5.
-LApply (Add_inv U X x x0); Tauto.
-Qed.
-
-Lemma cardinal_is_functional:
- (X: (Ensemble U)) (c1: nat) (cardinal U X c1) ->
- (Y: (Ensemble U)) (c2: nat) (cardinal U Y c2) -> X == Y ->
- c1 = c2.
-Proof.
-Intros X c1 H'; Elim H'.
-Intros Y c2 H'0; Elim H'0; Auto with sets.
-Intros A n H'1 H'2 x H'3 H'5.
-Elim (not_Empty_Add U A x); Auto with sets.
-Clear H' c1 X.
-Intros X n H' H'0 x H'1 Y c2 H'2.
-Elim H'2.
-Intro H'3.
-Elim (not_Empty_Add U X x); Auto with sets.
-Clear H'2 c2 Y.
-Intros X0 c2 H'2 H'3 x0 H'4 H'5.
-Elim (classic (In U X0 x)).
-Intro H'6; Apply f_equal with nat.
-Apply H'0 with Y := (Subtract U (Add U X0 x0) x).
-ElimType (pred (S c2)) = c2; Auto with sets.
-Apply card_soustr_1; Auto with sets.
-Rewrite <- H'5.
-Apply Sub_Add_new; Auto with sets.
-Elim (classic x == x0).
-Intros H'6 H'7; Apply f_equal with nat.
-Apply H'0 with Y := X0; Auto with sets.
-Apply Simplify_add with x := x; Auto with sets.
-Pattern 2 x; Rewrite H'6; Auto with sets.
-Intros H'6 H'7.
-Absurd (Add U X x) == (Add U X0 x0); Auto with sets.
-Clear H'0 H' H'3 n H'5 H'4 H'2 H'1 c2.
-Red; Intro H'.
-LApply (Extension U (Add U X x) (Add U X0 x0)); Auto with sets.
-Clear H'.
-Intro H'; Red in H'.
-Elim H'; Intros H'0 H'1; Red in H'0; Clear H' H'1.
-Absurd (In U (Add U X0 x0) x); Auto with sets.
-LApply (Add_inv U X0 x0 x); [ Intuition | Apply (H'0 x); Apply Add_intro2 ].
-Qed.
-
-Lemma cardinal_Empty : (m:nat)(cardinal U (Empty_set U) m) -> O = m.
-Proof.
-Intros m Cm; Generalize (cardinal_invert U (Empty_set U) m Cm).
-Elim m; Auto with sets.
-Intros; Elim H0; Intros; Elim H1; Intros; Elim H2; Intros.
-Elim (not_Empty_Add U x x0 H3).
-Qed.
-
-Lemma cardinal_unicity :
- (X: (Ensemble U)) (n: nat) (cardinal U X n) ->
- (m: nat) (cardinal U X m) -> n = m.
-Proof.
-Intros; Apply cardinal_is_functional with X X; Auto with sets.
-Qed.
-
-Lemma card_Add_gen:
- (A: (Ensemble U))
- (x: U) (n, n': nat) (cardinal U A n) -> (cardinal U (Add U A x) n') ->
- (le n' (S n)).
-Proof.
-Intros A x n n' H'.
-Elim (classic (In U A x)).
-Intro H'0.
-Rewrite (Non_disjoint_union U A x H'0).
-Intro H'1; Cut n = n'.
-Intro E; Rewrite E; Auto with sets.
-Apply cardinal_unicity with A; Auto with sets.
-Intros H'0 H'1.
-Cut n'=(S n).
-Intro E; Rewrite E; Auto with sets.
-Apply cardinal_unicity with (Add U A x); Auto with sets.
-Qed.
-
-Lemma incl_st_card_lt:
- (X: (Ensemble U)) (c1: nat) (cardinal U X c1) ->
- (Y: (Ensemble U)) (c2: nat) (cardinal U Y c2) -> (Strict_Included U X Y) ->
- (gt c2 c1).
-Proof.
-Intros X c1 H'; Elim H'.
-Intros Y c2 H'0; Elim H'0; Auto with sets arith.
-Intro H'1.
-Elim (Strict_Included_strict U (Empty_set U)); Auto with sets arith.
-Clear H' c1 X.
-Intros X n H' H'0 x H'1 Y c2 H'2.
-Elim H'2.
-Intro H'3; Elim (not_SIncl_empty U (Add U X x)); Auto with sets arith.
-Clear H'2 c2 Y.
-Intros X0 c2 H'2 H'3 x0 H'4 H'5; Elim (classic (In U X0 x)).
-Intro H'6; Apply gt_n_S.
-Apply H'0 with Y := (Subtract U (Add U X0 x0) x).
-ElimType (pred (S c2)) = c2; Auto with sets arith.
-Apply card_soustr_1; Auto with sets arith.
-Apply incl_st_add_soustr; Auto with sets arith.
-Elim (classic x == x0).
-Intros H'6 H'7; Apply gt_n_S.
-Apply H'0 with Y := X0; Auto with sets arith.
-Apply sincl_add_x with x := x0.
-Rewrite <- H'6; Auto with sets arith.
-Pattern 1 x0; Rewrite <- H'6; Trivial with sets arith.
-Intros H'6 H'7; Red in H'5.
-Elim H'5; Intros H'8 H'9; Try Exact H'8; Clear H'5.
-Red in H'8.
-Generalize (H'8 x).
-Intro H'5; LApply H'5; Auto with sets arith.
-Intro H; Elim Add_inv with U X0 x0 x; Auto with sets arith.
-Intro; Absurd (In U X0 x); Auto with sets arith.
-Intro; Absurd x==x0; Auto with sets arith.
-Qed.
-
-Lemma incl_card_le:
- (X,Y: (Ensemble U)) (n,m: nat) (cardinal U X n) -> (cardinal U Y m) ->
- (Included U X Y) -> (le n m).
-Proof.
-Intros;
-Elim Included_Strict_Included with U X Y; Auto with sets arith; Intro.
-Cut (gt m n); Auto with sets arith.
-Apply incl_st_card_lt with X := X Y := Y; Auto with sets arith.
-Generalize H0; Rewrite <- H2; Intro.
-Cut n=m.
-Intro E; Rewrite E; Auto with sets arith.
-Apply cardinal_unicity with X; Auto with sets arith.
-Qed.
-
-Lemma G_aux:
- (P:(Ensemble U) ->Prop)
- ((X:(Ensemble U))
- (Finite U X) -> ((Y:(Ensemble U)) (Strict_Included U Y X) ->(P Y)) ->(P X)) ->
- (P (Empty_set U)).
-Proof.
-Intros P H'; Try Assumption.
-Apply H'; Auto with sets.
-Clear H'; Auto with sets.
-Intros Y H'; Try Assumption.
-Red in H'.
-Elim H'; Intros H'0 H'1; Try Exact H'1; Clear H'.
-LApply (less_than_empty U Y); [Intro H'3; Try Exact H'3 | Assumption].
-Elim H'1; Auto with sets.
-Qed.
-
-Hints Unfold not.
-
-Lemma Generalized_induction_on_finite_sets:
- (P:(Ensemble U) ->Prop)
- ((X:(Ensemble U))
- (Finite U X) -> ((Y:(Ensemble U)) (Strict_Included U Y X) ->(P Y)) ->(P X)) ->
- (X:(Ensemble U)) (Finite U X) ->(P X).
-Proof.
-Intros P H'0 X H'1.
-Generalize P H'0; Clear H'0 P.
-Elim H'1.
-Intros P H'0.
-Apply G_aux; Auto with sets.
-Clear H'1 X.
-Intros A H' H'0 x H'1 P H'3.
-Cut (Y:(Ensemble U)) (Included U Y (Add U A x)) ->(P Y); Auto with sets.
-Generalize H'1.
-Apply H'0.
-Intros X K H'5 L Y H'6; Apply H'3; Auto with sets.
-Apply Finite_downward_closed with A := (Add U X x); Auto with sets.
-Intros Y0 H'7.
-Elim (Strict_inclusion_is_transitive_with_inclusion U Y0 Y (Add U X x)); Auto with sets.
-Intros H'2 H'4.
-Elim (Included_Add U Y0 X x);
- [Intro H'14 |
- Intro H'14; Elim H'14; Intros A' E; Elim E; Intros H'15 H'16; Clear E H'14 |
- Idtac]; Auto with sets.
-Elim (Included_Strict_Included U Y0 X); Auto with sets.
-Intro H'9; Apply H'5 with Y := Y0; Auto with sets.
-Intro H'9; Rewrite H'9.
-Apply H'3; Auto with sets.
-Intros Y1 H'8; Elim H'8.
-Intros H'10 H'11; Apply H'5 with Y := Y1; Auto with sets.
-Elim (Included_Strict_Included U A' X); Auto with sets.
-Intro H'8; Apply H'5 with Y := A'; Auto with sets.
-Rewrite <- H'15; Auto with sets.
-Intro H'8.
-Elim H'7.
-Intros H'9 H'10; Apply H'10 Orelse Elim H'10; Try Assumption.
-Generalize H'6.
-Rewrite <- H'8.
-Rewrite <- H'15; Auto with sets.
-Qed.
-
-End Finite_sets_facts.
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