Great: some significant progress in fixing universe levels.

[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..6351fead91eca85d504ca36b96a93d70a4153db2 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.
@@ -85,9 +86,10 @@ 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]
+    constructor 1; intro; cases e2; exists; [1,4: apply w]
      [ apply (. #‡e); assumption
      | apply (. #‡e1); assumption
      | apply (. #‡(e \sup -1)); assumption;
@@ -124,6 +126,7 @@ 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;
@@ -136,4 +139,77 @@ 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 f;
+  | intros; split; intros 2; simplify in f ⊢ %; intro;
+     [ apply (. (#‡(e i))); apply f;
+     | apply (. (#‡(e i)\sup -1)); apply f]]
+qed.
+
+(* senza questo exT "fresco", universe inconsistency *)
+inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
+  ex_introT: ∀w:A. P w → exT A P.
+
+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:carr I. x ∈ t i*) exT (carr I) (λi. x ∈ t i)});
+    simplify; intros; split; intros; cases e1; clear e1; 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".
+
+definition SUBSETS: objs1 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;
+    (* senza questa change, universe inconsistency *)
+    whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
+    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;
+       (* senza questa change, universe inconsistency *)
+       change in ⊢ (? ? (λ_:%.?)) with (carr I);
+       exists [apply w1] exists [apply w] assumption;
+     | cases e; cases x; exists; [apply w1]
+        [ assumption
+        | (* senza questa change, universe inconsistency *)
+          whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
+          exists; [apply w] assumption]]
+  | intros; intros 2; cases (f (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.