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
133
134
135
136
137
138
139
|
(***********************************************************************)
(* 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 *)
(***********************************************************************)
(* Finite sets library.
* Authors: Pierre Letouzey and Jean-Christophe Filliâtre
* Institution: LRI, CNRS UMR 8623 - Université Paris Sud
* 91405 Orsay, France *)
(* $Id: FSetToFiniteSet.v 8876 2006-05-30 13:43:15Z letouzey $ *)
Require Import Ensembles Finite_sets.
Require Import FSetInterface FSetProperties OrderedTypeEx.
(** * Going from [FSets] with usual equality
to the old [Ensembles] and [Finite_sets] theory. *)
Module S_to_Finite_set (U:UsualOrderedType)(M:S with Module E := U).
Module MP:= Properties(M).
Import M MP FM Ensembles Finite_sets.
Definition mkEns : M.t -> Ensemble M.elt :=
fun s x => M.In x s.
Notation " !! " := mkEns.
Lemma In_In : forall s x, M.In x s <-> In _ (!!s) x.
Proof.
unfold In; compute; auto.
Qed.
Lemma Subset_Included : forall s s', s[<=]s' <-> Included _ (!!s) (!!s').
Proof.
unfold Subset, Included, In, mkEns; intuition.
Qed.
Notation " a === b " := (Same_set M.elt a b) (at level 70, no associativity).
Lemma Equal_Same_set : forall s s', s[=]s' <-> !!s === !!s'.
Proof.
intros.
rewrite double_inclusion.
unfold Subset, Included, Same_set, In, mkEns; intuition.
Qed.
Lemma empty_Empty_Set : !!M.empty === Empty_set _.
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1.
Qed.
Lemma Empty_Empty_set : forall s, Empty s -> !!s === Empty_set _.
Proof.
unfold Same_set, Included, mkEns, In.
split; intros.
destruct(H x H0).
inversion H0.
Qed.
Lemma singleton_Singleton : forall x, !!(M.singleton x) === Singleton _ x .
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1; try constructor; auto.
Qed.
Lemma union_Union : forall s s', !!(union s s') === Union _ (!!s) (!!s').
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1; [ constructor 1 | constructor 2 | | ]; auto.
Qed.
Lemma inter_Intersection : forall s s', !!(inter s s') === Intersection _ (!!s) (!!s').
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1; try constructor; auto.
Qed.
Lemma add_Add : forall x s, !!(add x s) === Add _ (!!s) x.
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1; unfold E.eq; auto with sets.
inversion H0.
constructor 2; constructor.
constructor 1; auto.
Qed.
Lemma Add_Add : forall x s s', MP.Add x s s' -> !!s' === Add _ (!!s) x.
Proof.
unfold Same_set, Included, mkEns, In.
split; intros.
red in H; rewrite H in H0.
destruct H0.
inversion H0.
constructor 2; constructor.
constructor 1; auto.
red in H; rewrite H; unfold E.eq in *.
inversion H0; auto.
inversion H1; auto.
Qed.
Lemma remove_Subtract : forall x s, !!(remove x s) === Subtract _ (!!s) x.
Proof.
unfold Same_set, Included, mkEns, In.
split; intro; set_iff; inversion 1; unfold E.eq in *; auto with sets.
split; auto.
swap H1.
inversion H2; auto.
Qed.
Lemma mkEns_Finite : forall s, Finite _ (!!s).
Proof.
intro s; pattern s; apply set_induction; clear s; intros.
intros; replace (!!s) with (Empty_set elt); auto with sets.
symmetry; apply Extensionality_Ensembles.
apply Empty_Empty_set; auto.
replace (!!s') with (Add _ (!!s) x).
constructor 2; auto.
symmetry; apply Extensionality_Ensembles.
apply Add_Add; auto.
Qed.
Lemma mkEns_cardinal : forall s, cardinal _ (!!s) (M.cardinal s).
Proof.
intro s; pattern s; apply set_induction; clear s; intros.
intros; replace (!!s) with (Empty_set elt); auto with sets.
rewrite cardinal_1; auto with sets.
symmetry; apply Extensionality_Ensembles.
apply Empty_Empty_set; auto.
replace (!!s') with (Add _ (!!s) x).
rewrite (cardinal_2 H0 H1); auto with sets.
symmetry; apply Extensionality_Ensembles.
apply Add_Add; auto.
Qed.
End S_to_Finite_set.
|