X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-concrete_spaces.ma;h=6a9d7794db701429f657746dc1b41dc4922aad58;hb=02ce2fd650a68d04fff678441cca9086c8310005;hp=80ee4649fee1f149370a456e5c81d28f9375d26e;hpb=8b7ad9b29b3448b72e476ba077e2d0faad86c058;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
index 80ee4649f..6a9d7794d 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
@@ -17,78 +17,69 @@ include "o-saturations.ma".
 
 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
 notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
-interpretation "Universal image ⊩⎻*" 'box x = (fun_1 _ _ (or_f_minus_star _ _) (rel x)).
+interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (rel x)).
  
 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
 notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
-interpretation "Existential image ⊩" 'diamond x = (fun_1 _ _ (or_f _ _) (rel x)).
+interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (rel x)).
 
 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
-interpretation "Universal pre-image ⊩*" 'rest x = (fun_1 _ _ (or_f_star _ _) (rel x)).
+interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (rel x)).
 
 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
-interpretation "Existential pre-image ⊩⎻" 'ext x = (fun_1 _ _ (or_f_minus _ _) (rel x)).
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (rel x)).
 
-lemma hint : ∀p,q.arrows1 OA p q → ORelation_setoid p q.
-intros; assumption;
-qed.
-
-coercion hint nocomposites.
-
-definition A : ∀b:BP. unary_morphism (oa_P (form b)) (oa_P (form b)).
+definition A : ∀b:BP. unary_morphism1 (form b) (form b).
 intros; constructor 1;
  [ apply (λx.□_b (Ext⎽b x));
- | do 2 unfold uncurry_arrows; intros; apply  (†(†H));]
+ | do 2 unfold FunClass_1_OF_Type_OF_setoid21; intros; apply  (†(†e));]
 qed.
 
-lemma xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx nocomposites.
-
-lemma down_p : ∀S,I:SET.∀u:S⇒S.∀c:arrows1 SET I S.∀a:I.∀a':I.a=a'→u (c a)=u (c a').
-intros; unfold uncurry_arrows; apply (†(†H));
+lemma down_p : ∀S:SET1.∀I:SET.∀u:S⇒S.∀c:arrows2 SET1 I S.∀a:I.∀a':I.a=a'→u (c a)=u (c a').
+intros; apply (†(†e));
 qed.
 
-alias symbol "eq" = "setoid eq".
-alias symbol "and" = "o-algebra binary meet".
-record concrete_space : Type ≝
+record concrete_space : Type2 ≝
  { bp:> BP;
    (*distr : is_distributive (form bp);*)
-   downarrow: unary_morphism (oa_P (form bp)) (oa_P (form bp));
+   downarrow: unary_morphism1 (form bp) (form bp);
    downarrow_is_sat: is_saturation ? downarrow;
    converges: ∀q1,q2.
      (Ext⎽bp q1 ∧ (Ext⎽bp q2)) = (Ext⎽bp ((downarrow q1) ∧ (downarrow q2)));
    all_covered: Ext⎽bp (oa_one (form bp)) = oa_one (concr bp);
-   il2: ∀I:SET.∀p:arrows1 SET I (oa_P (form bp)).
+   il2: ∀I:SET.∀p:arrows2 SET1 I (form bp).
      downarrow (∨ { x ∈ I | downarrow (p x) | down_p ???? }) =
      ∨ { x ∈ I | downarrow (p x) | down_p ???? };
    il1: ∀q.downarrow (A ? q) = A ? q
  }.
 
 interpretation "o-concrete space downarrow" 'downarrow x = 
-  (fun_1 __ (downarrow _) x).
+  (fun11 __ (downarrow _) x).
 
 definition bp': concrete_space → basic_pair ≝ λc.bp c.
 coercion bp'.
 
-lemma setoid_OF_OA : OA → setoid.
-intros; apply (oa_P o);
+definition bp'': concrete_space → objs2 BP.
+ intro; apply (bp' c);
 qed.
-
-coercion setoid_OF_OA.
+coercion bp''.
 
 definition binary_downarrow : 
   ∀C:concrete_space.binary_morphism1 (form C) (form C) (form C).
 intros; constructor 1;
-[ intros; apply (↓ c ∧ ↓ c1);
-| intros; apply ((†H)‡(†H1));]
+[ intros; apply (↓ t ∧ ↓ t1);
+| intros;
+  alias symbol "prop2" = "prop21".
+  alias symbol "prop1" = "prop11".
+  apply ((†e)‡(†e1));]
 qed.
 
-interpretation "concrete_space binary ↓" 'fintersects a b = (fun1 _ _ _ (binary_downarrow _) a b).
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (binary_downarrow _) a b).
 
-record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
- { rp:> arrows1 ? CS1 CS2;
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type2 ≝
+ { rp:> arrows2 ? CS1 CS2;
    respects_converges:
     ∀b,c. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
    respects_all_covered:
@@ -98,76 +89,76 @@ record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
 
 definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
  λCS1,CS2,c. rp CS1 CS2 c.
- 
 coercion rp'.
 
-definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
+definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid2.
  intros;
  constructor 1;
   [ apply (convergent_relation_pair c c1)
   | constructor 1;
      [ intros;
        apply (relation_pair_equality c c1 c2 c3);
-     | intros 1; apply refl1;
-     | intros 2; apply sym1; 
-     | intros 3; apply trans1]]
+     | intros 1; apply refl2;
+     | intros 2; apply sym2; 
+     | intros 3; apply trans2]]
 qed.
 
-definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
 
+definition rp'': ∀CS1,CS2.carr2 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
 coercion rp''.
 
-definition prop_1_SET : 
- ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
-intros; apply (prop_1 A B w a b H);
-qed.
+definition rp''': ∀CS1,CS2.Type_OF_setoid2 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
+coercion rp'''.
 
-interpretation "SET dagger" 'prop1 h = (prop_1_SET _ _ _ _ _ h).
+definition rp'''': ∀CS1,CS2.Type_OF_setoid2 (convergent_relation_space_setoid CS1 CS2) → carr2 (arrows2 BP CS1 CS2) ≝
+ λCS1,CS2,c.rp ?? c.
+coercion rp''''.
 
 definition convergent_relation_space_composition:
  ∀o1,o2,o3: concrete_space.
-  binary_morphism1
+  binary_morphism2
    (convergent_relation_space_setoid o1 o2)
-   (convergentin ⊢ (? (? ? ? (? ? ? (? ? ? ? ? (? ? ? (? ? ? (% ? ?))) ?)) ?) ? ? ?)_relation_space_setoid o2 o3)
+   (convergent_relation_space_setoid o2 o3)
    (convergent_relation_space_setoid o1 o3).
  intros; constructor 1;
-     [ intros; whd in c c1 ⊢ %;
+     [ intros; whd in t t1 ⊢ %;
        constructor 1;
-        [ apply (c1 ∘ c);
-        | intros;  
-          change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
+        [ apply (t1 ∘ t);
+        | intros;
+          change in ⊢ (? ? ? % ?) with (t\sub\c⎻ (t1\sub\c⎻ (Ext⎽o3 (b↓c))));
+          unfold FunClass_1_OF_Type_OF_setoid21;
           alias symbol "trans" = "trans1".
           apply (.= († (respects_converges : ?)));
-          apply (.= (respects_converges : ?));
-          apply refl1;
-        | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
+          apply (respects_converges ?? t (t1\sub\f⎻ b) (t1\sub\f⎻ c));
+        | change in ⊢ (? ? ? % ?) with (t\sub\c⎻ (t1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
+          unfold FunClass_1_OF_Type_OF_setoid21;
           apply (.= (†(respects_all_covered :?)));
-          apply (.= (respects_all_covered :?));
-          apply refl1]
+          apply rule (respects_all_covered ?? t);]
      | intros;
        change with (b ∘ a = b' ∘ a'); 
-       change in H with (rp'' ?? a = rp'' ?? a');
-       change in H1 with (rp'' ?? b = rp ?? b');
-       apply ( (H‡H1));]
+       change in e with (rp'' ?? a = rp'' ?? a');
+       change in e1 with (rp'' ?? b = rp ?? b');
+       apply (e‡e1);]
 qed.
 
-definition CSPA: category1.
+definition CSPA: category2.
  constructor 1;
   [ apply concrete_space
   | apply convergent_relation_space_setoid
   | intro; constructor 1;
-     [ apply id1
+     [ apply id2
      | intros; apply refl1;
      | apply refl1]
   | apply convergent_relation_space_composition
-  | intros; simplify;
+  | intros; simplify; whd in a12 a23 a34;
     change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
-    apply ASSOC1;
+    apply rule ASSOC;
   | intros; simplify;
-    change with (a ∘ id1 ? o1 = a);
-    apply (id_neutral_right1 : ?);
+    change with (a ∘ id2 ? o1 = a);
+    apply (id_neutral_right2 : ?);
   | intros; simplify;
-    change with (id1 ? o2 ∘ a = a);
-    apply (id_neutral_left1 : ?);]
+    change with (id2 ? o2 ∘ a = a);
+    apply (id_neutral_left2 : ?);]
 qed.
\ No newline at end of file