1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
|
(************************************************************************)
(* 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: Classical_sets.v 8642 2006-03-17 10:09:02Z notin $ i*)
Require Export Ensembles.
Require Export Constructive_sets.
Require Export Classical_Type.
(* Hints Unfold not . *)
Section Ensembles_classical.
Variable U : Type.
Lemma not_included_empty_Inhabited :
forall A:Ensemble U, ~ Included U A (Empty_set U) -> Inhabited U A.
Proof.
intros A NI.
elim (not_all_ex_not U (fun x:U => ~ In U A x)).
intros x H; apply Inhabited_intro with x.
apply NNPP; auto with sets.
red in |- *; intro.
apply NI; red in |- *.
intros x H'; elim (H x); trivial with sets.
Qed.
Hint Resolve not_included_empty_Inhabited.
Lemma not_empty_Inhabited :
forall A:Ensemble U, A <> Empty_set U -> Inhabited U A.
Proof.
intros; apply not_included_empty_Inhabited.
red in |- *; auto with sets.
Qed.
Lemma Inhabited_Setminus :
forall X Y:Ensemble U,
Included U X Y -> ~ Included U Y X -> Inhabited U (Setminus U Y X).
Proof.
intros X Y I NI.
elim (not_all_ex_not U (fun x:U => In U Y x -> In U X x) NI).
intros x YX.
apply Inhabited_intro with x.
apply Setminus_intro.
apply not_imply_elim with (In U X x); trivial with sets.
auto with sets.
Qed.
Hint Resolve Inhabited_Setminus.
Lemma Strict_super_set_contains_new_element :
forall X Y:Ensemble U,
Included U X Y -> X <> Y -> Inhabited U (Setminus U Y X).
Proof.
auto 7 with sets.
Qed.
Hint Resolve Strict_super_set_contains_new_element.
Lemma Subtract_intro :
forall (A:Ensemble U) (x y:U), In U A y -> x <> y -> In U (Subtract U A x) y.
Proof.
unfold Subtract at 1 in |- *; auto with sets.
Qed.
Hint Resolve Subtract_intro.
Lemma Subtract_inv :
forall (A:Ensemble U) (x y:U), In U (Subtract U A x) y -> In U A y /\ x <> y.
Proof.
intros A x y H'; elim H'; auto with sets.
Qed.
Lemma Included_Strict_Included :
forall X Y:Ensemble U, Included U X Y -> Strict_Included U X Y \/ X = Y.
Proof.
intros X Y H'; try assumption.
elim (classic (X = Y)); auto with sets.
Qed.
Lemma Strict_Included_inv :
forall X Y:Ensemble U,
Strict_Included U X Y -> Included U X Y /\ Inhabited U (Setminus U Y X).
Proof.
intros X Y H'; red in H'.
split; [ tauto | idtac ].
elim H'; intros H'0 H'1; try exact H'1; clear H'.
apply Strict_super_set_contains_new_element; auto with sets.
Qed.
Lemma not_SIncl_empty :
forall X:Ensemble U, ~ Strict_Included U X (Empty_set U).
Proof.
intro X; red in |- *; intro H'; try exact H'.
lapply (Strict_Included_inv X (Empty_set U)); auto with sets.
intro H'0; elim H'0; intros H'1 H'2; elim H'2; clear H'0.
intros x H'0; elim H'0.
intro H'3; elim H'3.
Qed.
Lemma Complement_Complement :
forall A:Ensemble U, Complement U (Complement U A) = A.
Proof.
unfold Complement in |- *; intros; apply Extensionality_Ensembles;
auto with sets.
red in |- *; split; auto with sets.
red in |- *; intros; apply NNPP; auto with sets.
Qed.
End Ensembles_classical.
Hint Resolve Strict_super_set_contains_new_element Subtract_intro
not_SIncl_empty: sets v62.
|
