X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fsubsets.ma;h=95d0284dc05ccc5397688f29c3cd26fefc60ae68;hb=1470ff47df1349333c6b721a1c162cc7dfc6806f;hp=17128ae6a97942fe7831a06baede3ce05443b1dc;hpb=3d7b244a79a1c57d3355deb2f9a70764cde077b9;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
index 17128ae6a..95d0284dc 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -12,13 +12,15 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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: A ⇒_1 CPROP }.
+interpretation "powerset low" 'powerset A = (powerset_carrier A).
+notation "hvbox(a break ∈. b)" non associative with precedence 45 for @{ 'mem_low $a $b }.
+interpretation "memlow" 'mem_low a S = (fun11 ?? (mem_operator ? S) a).
 
-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: objs1 SET. Ω^A → Ω^A → CProp0 ≝
+ λA:objs1 SET.λU,V.∀a:A. a ∈. U → a ∈. V.
 
 theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
  intros 6; intros 2;
@@ -42,9 +44,9 @@ qed.
 interpretation "powerset" 'powerset A = (powerset_setoid1 A).
 
 interpretation "subset construction" 'subset \eta.x =
- (mk_powerset_carrier _ (mk_unary_morphism1 _ CPROP x _)).
+ (mk_powerset_carrier ? (mk_unary_morphism1 ? CPROP x ?)).
 
-definition mem: ∀A. binary_morphism1 A (Ω \sup A) CPROP.
+definition mem: ∀A. A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
   [ apply (λx,S. mem_operator ? S x)
@@ -57,9 +59,9 @@ definition mem: ∀A. binary_morphism1 A (Ω \sup A) CPROP.
      | apply s1; assumption]]
 qed.
 
-interpretation "mem" 'mem a S = (fun21 ___ (mem _) a S).
+interpretation "mem" 'mem a S = (fun21 ??? (mem ?) a S).
 
-definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition subseteq: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
   [ apply (λU,V. subseteq_operator ? U V)
@@ -72,61 +74,60 @@ definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
       apply (transitive_subseteq_operator ???? s s4) ]]
 qed.
 
-interpretation "subseteq" 'subseteq U V = (fun21 ___ (subseteq _) U V).
+interpretation "subseteq" 'subseteq U V = (fun21 ??? (subseteq ?) U V).
 
-theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
+theorem subseteq_refl: ∀A.∀S:Ω^A.S ⊆ S.
  intros 4; assumption.
 qed.
 
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
+theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω^A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
  intros; apply transitive_subseteq_operator; [apply S2] assumption.
 qed.
 
-definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition overlaps: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
-  [ apply (λA.λU,V:Ω \sup A.exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V))
+  [ apply (λA:objs1 SET.λU,V:Ω^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).
+interpretation "overlaps" 'overlaps U V = (fun21 ??? (overlaps ?) U V).
 
-definition intersects:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition intersects: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
  intros;
  constructor 1;
   [ apply rule (λU,V. {x | x ∈ U ∧ x ∈ V });
     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).
+interpretation "intersects" 'intersects U V = (fun21 ??? (intersects ?) U V).
 
-definition union:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition union: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
  intros;
  constructor 1;
   [ apply (λU,V. {x | x ∈ U ∨ x ∈ V });
     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).
+interpretation "union" 'union U V = (fun21 ??? (union ?) U V).
 
-definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
+(* qua non riesco a mettere set *)
+definition singleton: ∀A:setoid. A ⇒_1 Ω^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 +137,45 @@ 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).
+notation > "{ term 19 a : S }" with precedence 90 for @{fun11 ?? (singleton $S) $a}.
+
+definition big_intersects: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^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.(I ⇒_2 Ω^A) ⇒_2 Ω^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.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (\emsp) term 90 p)" 
+non associative with precedence 50 for @{ 'bigcup $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋃) \below (ident i ∈  I) break term 90 p)" 
+non associative with precedence 50 for @{ 'bigcup_mk (λ${ident i}:$I.$p) }.
+notation > "hovbox(⋃ f)" non associative with precedence 60 for @{ 'bigcup $f }.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (\emsp) term 90 p)" 
+non associative with precedence 50 for @{ 'bigcap $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (⋂) \below (ident i ∈  I) break term 90 p)" 
+non associative with precedence 50 for @{ 'bigcap_mk (λ${ident i}:$I.$p) }.
+notation > "hovbox(⋂ f)" non associative with precedence 60 for @{ 'bigcap $f }.
+
+interpretation "bigcup" 'bigcup f = (fun12 ?? (big_union ??) f).
+interpretation "bigcap" 'bigcap f = (fun12 ?? (big_intersects ??) f).
+interpretation "bigcup mk" 'bigcup_mk f = (fun12 ?? (big_union ??) (mk_unary_morphism2 ?? f ?)).
+interpretation "bigcap mk" 'bigcap_mk f = (fun12 ?? (big_intersects ??) (mk_unary_morphism2 ?? f ?)).