Proof that \Omega \sup A is an overlap algebra, up to universe inconsistency :-(

[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 7eb2d1e9423bc88f1026ccf38c1a3ac88193b8f7..5803e6906cf1afeb711ca78cd5ead5ae5bd37a69 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -12,7 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "logic/cprop_connectives.ma".
 include "categories.ma".
 
 record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
@@ -140,4 +139,65 @@ 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 ∈ t i});
+    simplify; intros; split; intros; [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+    apply H;
+  | intros; split; intros 2; simplify in f ⊢ %; intro;
+     [ apply (. (#‡(e i))); apply f;
+     | apply (. (#‡(e i)\sup -1)); 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 I;
+    apply ({x | ∃i:I. x ∈ t i});
+    simplify; intros; split; intros; cases H; clear H; exists; [1,3:apply w]
+    [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+    apply x;
+  | intros; split; intros 2; simplify in f ⊢ %; cases f; clear f; exists; [1,3:apply w]
+     [ apply (. (#‡(e w))); apply x;
+     | apply (. (#‡(e w)\sup -1)); apply x]]
+qed.
+
+(* incluso prima non funziona piu' nulla *)
+include "o-algebra.ma".
+
+
+axiom daemon: False.
+definition SUBSETS: SET → OAlgebra.
+ intro A; constructor 1;
+  [ apply (Ω \sup A);
+  | apply subseteq;
+  | apply overlaps;
+  | apply big_intersects;
+  | apply big_union;
+  | apply ({x | True});
+    simplify; intros; apply (refl1 ? (eq1 CPROP));
+  | apply ({x | False});
+    simplify; intros; apply (refl1 ? (eq1 CPROP));
+  | intros; whd; intros; assumption
+  | intros; whd; split; assumption
+  | intros; apply transitive_subseteq_operator; [2: apply f; | skip | assumption]
+  | intros; cases f; exists [apply w] assumption
+  | intros; intros 2; apply (f ? f1 i);
+  | intros; intros 2; apply f; exists; [apply i] assumption;
+  | intros 3; cases f;
+  | intros 3; constructor 1;
+  | intros; cases f; exists; [apply w]
+     [ assumption
+     | whd; intros; cases i; simplify; assumption]
+  | intros; split; intro;
+     [ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
+     | cases H; cases x; exists; [apply w1] [assumption | exists; [apply w] assumption]]
+  | intros; intros 2; cases (H (singleton ? a) ?);
+     [ exists; [apply a] [assumption | change with (a = a); apply refl1;]
+     | change in x1 with (a = w); change with (mem A a q); apply (. (x1 \sup -1‡#));
+       assumption]]
+qed.
\ No newline at end of file