2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/logic.ma".
14 (**** a subset of A is just an object of type A→Prop ****)
16 definition empty_set ≝ λA:Type[0].λa:A.False.
17 notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
18 interpretation "empty set" 'empty_set = (empty_set ?).
20 definition singleton ≝ λA.λx,a:A.x=a.
21 (* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
22 interpretation "singleton" 'singl x = (singleton ? x).
24 definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a ∨ Q a.
25 interpretation "union" 'union a b = (union ? a b).
27 definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a ∧ Q a.
28 interpretation "intersection" 'intersects a b = (intersection ? a b).
30 definition complement ≝ λU:Type[0].λA:U → Prop.λw.¬ A w.
31 interpretation "complement" 'not a = (complement ? a).
33 definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w ∧ ¬ B w.
34 interpretation "substraction" 'minus a b = (substraction ? a b).
36 definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
37 interpretation "subset" 'subseteq a b = (subset ? a b).
39 (* extensional equality *)
40 definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a ↔ Q a.
41 notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
42 interpretation "extensional equality" 'eqP a b = (eqP ? a b).
44 lemma eqP_sym: ∀U.∀A,B:U →Prop.
46 #U #A #B #eqAB #a @iff_sym @eqAB qed.
48 lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
49 A =1 B → B =1 C → A =1 C.
50 #U #A #B #C #eqAB #eqBC #a @iff_trans // qed.
52 lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
53 A =1 C → A ∪ B =1 C ∪ B.
54 #U #A #B #C #eqAB #a @iff_or_r @eqAB qed.
56 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
57 B =1 C → A ∪ B =1 A ∪ C.
58 #U #A #B #C #eqBC #a @iff_or_l @eqBC qed.
60 lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
61 A =1 C → A ∩ B =1 C ∩ B.
62 #U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
64 lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
65 B =1 C → A ∩ B =1 A ∩ C.
66 #U #A #B #C #eqBC #a @iff_and_l @eqBC qed.
68 lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
69 A =1 C → A - B =1 C - B.
70 #U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
72 lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
73 B =1 C → A - B =1 A - C.
74 #U #A #B #C #eqBC #a @iff_and_l /2/ qed.
77 lemma union_empty_r: ∀U.∀A:U→Prop.
79 #U #A #w % [* // normalize #abs @False_ind /2/ | /2/]
82 lemma union_comm : ∀U.∀A,B:U →Prop.
84 #U #A #B #a % * /2/ qed.
86 lemma union_assoc: ∀U.∀A,B,C:U → Prop.
87 A ∪ B ∪ C =1 A ∪ (B ∪ C).
88 #S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
91 lemma cap_comm : ∀U.∀A,B:U →Prop.
93 #U #A #B #a % * /2/ qed.
95 lemma union_idemp: ∀U.∀A:U →Prop.
97 #U #A #a % [* // | /2/] qed.
99 lemma cap_idemp: ∀U.∀A:U →Prop.
101 #U #A #a % [* // | /2/] qed.
103 (*distributivities *)
105 lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
106 (A ∪ B) ∩ C =1 (A ∩ C) ∪ (B ∩ C).
107 #U #A #B #C #w % [* * /3/ | * * /3/]
110 lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
111 (A ∪ B) - C =1 (A - C) ∪ (B - C).
112 #U #A #B #C #w % [* * /3/ | * * /3/]
116 lemma substract_def:∀U.∀A,B:U→Prop. A-B =1 A ∩ ¬B.
117 #U #A #B #w normalize /2/