Renaming.

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / subsets.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
index 17128ae6a97942fe7831a06baede3ce05443b1dc..4152f48529a54ea91d8ef461b6dcfd8b45144169 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -12,13 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "logic/cprop_connectives.ma".
 include "categories.ma".
 
-record powerset_carrier (A: SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
+record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
 
-definition subseteq_operator: ∀A: SET. powerset_carrier A → powerset_carrier A → CProp2 ≝
- λA:SET.λU,V.∀a:A. mem_operator ? U a → mem_operator ? V a.
+definition subseteq_operator: ∀A: SET. powerset_carrier A → powerset_carrier A → CProp0 ≝
+ λA:objs1 SET.λU,V.∀a:A. mem_operator ? U a → mem_operator ? V a.
 
 theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
  intros 6; intros 2;
@@ -74,6 +73,8 @@ qed.
 
 interpretation "subseteq" 'subseteq U V = (fun21 ___ (subseteq _) U V).
 
+
+
 theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
  intros 4; assumption.
 qed.
@@ -85,13 +86,14 @@ qed.
 definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
  intros;
  constructor 1;
-  [ apply (λA.λU,V:Ω \sup A.exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V))
+  (* Se metto x al posto di ? ottengo una universe inconsistency *)
+  [ apply (λA:objs1 SET.λU,V:Ω \sup A.(exT2 ? (λx:A.?(*x*) ∈ U) (λx:A.?(*x*) ∈ V) : CProp0))
   | intros;
-    constructor 1; intro; cases H; exists; [1,4: apply w]
-     [ apply (. #‡e); assumption
-     | apply (. #‡e1); assumption
-     | apply (. #‡(e \sup -1)); assumption;
-     | apply (. #‡(e1 \sup -1)); assumption]]
+    constructor 1; intro; cases e2; exists; [1,4: apply w]
+     [ apply (. #‡e^-1); assumption
+     | apply (. #‡e1^-1); assumption
+     | apply (. #‡e); assumption;
+     | apply (. #‡e1); assumption]]
 qed.
 
 interpretation "overlaps" 'overlaps U V = (fun21 ___ (overlaps _) U V).
@@ -104,8 +106,8 @@ definition intersects:
     intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1;
   | intros;
     split; intros 2; simplify in f ⊢ %;
-    [ apply (. (#‡e)‡(#‡e1)); assumption
-    | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+    [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+    | apply (. (#‡e)‡(#‡e1)); assumption]]
 qed.
 
 interpretation "intersects" 'intersects U V = (fun21 ___ (intersects _) U V).
@@ -118,15 +120,16 @@ definition union:
     intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1
   | intros;
     split; intros 2; simplify in f ⊢ %;
-    [ apply (. (#‡e)‡(#‡e1)); assumption
-    | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+    [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+    | apply (. (#‡e)‡(#‡e1)); assumption]]
 qed.
 
 interpretation "union" 'union U V = (fun21 ___ (union _) U V).
 
+(* qua non riesco a mettere set *)
 definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
  intros; constructor 1;
-  [ apply (λa:A.{b | eq ? a b}); unfold setoid1_of_setoid; simplify;
+  [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
     intros; simplify;
     split; intro;
     apply (.= e1);
@@ -136,4 +139,29 @@ definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
      [ apply a |4: apply a'] try assumption; apply sym; assumption]
 qed.
 
-interpretation "singleton" 'singl a = (fun11 __ (singleton _) a).
\ No newline at end of file
+interpretation "singleton" 'singl a = (fun11 __ (singleton _) a).
+
+definition big_intersects:
+ ∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
+ intros; constructor 1;
+  [ intro; whd; whd in I;
+    apply ({x | ∀i:I. x ∈ c i});
+    simplify; intros; split; intros; [ apply (. (e^-1‡#)); | apply (. e‡#); ]
+    apply f;
+  | intros; split; intros 2; simplify in f ⊢ %; intro;
+     [ apply (. (#‡(e i)^-1)); apply f;
+     | apply (. (#‡e i)); apply f]]
+qed.
+
+definition big_union:
+ ∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
+ intros; constructor 1;
+  [ intro; whd; whd in A; whd in I;
+    apply ({x | ∃i:I. x ∈ c i });
+    simplify; intros; split; intros; cases e1; clear e1; exists; [1,3:apply w]
+    [ apply (. (e^-1‡#)); | apply (. (e‡#)); ]
+    apply x;
+  | intros; split; intros 2; simplify in f ⊢ %; cases f; clear f; exists; [1,3:apply w]
+     [ apply (. (#‡(e w)^-1)); apply x;
+     | apply (. (#‡e w)); apply x]]
+qed.