the decentralization of core notation continues ...

[helm.git] / matita / matita / lib / basics / sets.ma

diff --git a/matita/matita/lib/basics/sets.ma b/matita/matita/lib/basics/sets.ma
index 622025d790a65afa6269b5626b99e4d2ca329617..8e40b666bfdee2eb318478d2899a3bece809619f 100644 (file)
--- a/matita/matita/lib/basics/sets.ma
+++ b/matita/matita/lib/basics/sets.ma
@@ -10,6 +10,8 @@
       V_______________________________________________________________ *)
 
 include "basics/logic.ma".
+include "basics/core_notation/singl_1.ma".
+include "basics/core_notation/subseteq_2.ma".
 
 (**** a subset of A is just an object of type A→Prop ****)
 
@@ -27,50 +29,93 @@ interpretation "union" 'union a b = (union ? a b).
 definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a ∧ Q a.
 interpretation "intersection" 'intersects a b = (intersection ? a b).
 
+definition complement ≝ λU:Type[0].λA:U → Prop.λw.¬ A w.
+interpretation "complement" 'not a = (complement ? a).
+
+definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w ∧ ¬ B w.
+interpretation "substraction" 'minus a b = (substraction ? a b).
+
 definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
-interpretation "subset" 'subseteq a b = (intersection ? a b).
+interpretation "subset" 'subseteq a b = (subset ? a b).
 
 (* extensional equality *)
 definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a ↔ Q a.
-notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
+(* ≐ is typed as \doteq *)
+notation "A ≐ B" non associative with precedence 45 for @{'eqP $A $B}.
 interpretation "extensional equality" 'eqP a b = (eqP ? a b).
 
 lemma eqP_sym: ∀U.∀A,B:U →Prop. 
-  A =1 B → B =1 A.
+  A ≐ B → B ≐ A.
 #U #A #B #eqAB #a @iff_sym @eqAB qed.
  
 lemma eqP_trans: ∀U.∀A,B,C:U →Prop. 
-  A =1 B → B =1 C → A =1 C.
+  A ≐ B → B ≐ C → A ≐ C.
 #U #A #B #C #eqAB #eqBC #a @iff_trans // qed.
 
 lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. 
-  A =1 C  → A ∪ B =1 C ∪ B.
+  A ≐ C  → A ∪ B ≐ C ∪ B.
 #U #A #B #C #eqAB #a @iff_or_r @eqAB qed.
   
 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop. 
-  B =1 C  → A ∪ B =1 A ∪ C.
+  B ≐ C  → A ∪ B ≐ A ∪ C.
 #U #A #B #C #eqBC #a @iff_or_l @eqBC qed.
   
+lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop. 
+  A ≐ C  → A ∩ B ≐ C ∩ B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+  
+lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop. 
+  B ≐ C  → A ∩ B ≐ A ∩ C.
+#U #A #B #C #eqBC #a @iff_and_l @eqBC qed.
+
+lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop. 
+  A ≐ C  → A - B ≐ C - B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+  
+lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop. 
+  B ≐ C  → A - B ≐ A - C.
+#U #A #B #C #eqBC #a @iff_and_l /2/ qed.
+
 (* set equalities *)
+lemma union_empty_r: ∀U.∀A:U→Prop. 
+  A ∪ ∅ ≐ A.
+#U #A #w % [* // normalize #abs @False_ind /2/ | /2/]
+qed.
+
 lemma union_comm : ∀U.∀A,B:U →Prop. 
-  A ∪ B =1 B ∪ A.
+  A ∪ B ≐ B ∪ A.
 #U #A #B #a % * /2/ qed. 
 
 lemma union_assoc: ∀U.∀A,B,C:U → Prop. 
-  A ∪ B ∪ C =1 A ∪ (B ∪ C).
+  A ∪ B ∪ C ≐ A ∪ (B ∪ C).
 #S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
 qed.   
 
 lemma cap_comm : ∀U.∀A,B:U →Prop. 
-  A ∩ B =1 B ∩ A.
+  A ∩ B ≐ B ∩ A.
 #U #A #B #a % * /2/ qed. 
 
 lemma union_idemp: ∀U.∀A:U →Prop. 
-  A ∪ A =1 A.
+  A ∪ A ≐ A.
 #U #A #a % [* // | /2/] qed. 
 
 lemma cap_idemp: ∀U.∀A:U →Prop. 
-  A ∩ A =1 A.
+  A ∩ A ≐ A.
 #U #A #a % [* // | /2/] qed. 
 
+(*distributivities *)
 
+lemma distribute_intersect : ∀U.∀A,B,C:U→Prop. 
+  (A ∪ B) ∩ C ≐ (A ∩ C) ∪ (B ∩ C).
+#U #A #B #C #w % [* * /3/ | * * /3/] 
+qed.
+  
+lemma distribute_substract : ∀U.∀A,B,C:U→Prop. 
+  (A ∪ B) - C ≐ (A - C) ∪ (B - C).
+#U #A #B #C #w % [* * /3/ | * * /3/] 
+qed.
+
+(* substraction *)
+lemma substract_def:∀U.∀A,B:U→Prop. A-B ≐ A ∩ ¬B.
+#U #A #B #w normalize /2/
+qed.