X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-concrete_spaces.ma;h=0bd8715a67550692dbc7fbdc702e8ecbdff15f86;hb=e39b9a73fa95490d29237e31cfd3ff7f6aa07e3d;hp=135dae3c981e0ae95c48d4c9544b5a059c65451b;hpb=62e9e8296d172d6497f9ad29bad402fbad2014c3;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 135dae3c9..0bd8715a6 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
@@ -15,34 +15,86 @@
 include "o-basic_pairs.ma".
 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)).
+ 
+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)).
+
+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)).
+
+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)).
+
+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)).
+intros; constructor 1;
+ [ apply (λx.□_b (Ext⎽b x));
+ | do 2 unfold uncurry_arrows; intros; apply  (†(†H));]
+qed.
+
 lemma xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx.
+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; change in c with (I ⇒ S);
+apply (†(†H));
+qed.
 
+alias symbol "eq" = "setoid eq".
+alias symbol "and" = "o-algebra binary meet".
 record concrete_space : Type ≝
  { bp:> BP;
-   downarrow: form bp → oa_P (form bp);
+   (*distr : is_distributive (form bp);*)
+   downarrow: unary_morphism (oa_P (form bp)) (oa_P (form bp));
    downarrow_is_sat: is_saturation ? downarrow;
-   converges: ∀q1,q2:form bp.
-     or_f_minus ?? (⊩) q1 ∧ or_f_minus ?? (⊩) q2 = 
-     or_f_minus ?? (⊩) ((downarrow q1) ∧ (downarrow q2));
-   all_covered: (*⨍^-_bp*) or_f_minus ?? (⊩) (oa_one (form bp)) = oa_one (concr bp);
-   il2: ∀I:setoid.∀p:ums I (oa_P (form bp)).
-     downarrow (oa_join ? I (mk_unary_morphism ?? (λi:I. downarrow (p i)) ?)) =
-     oa_join ? I (mk_unary_morphism ?? (λi:I. downarrow (p i)) ?)
+   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)).
+     downarrow (∨ { x ∈ I | downarrow (p x) | down_p ???? }) =
+     ∨ { x ∈ I | downarrow (p x) | down_p ???? };
+   il1: ∀q.downarrow (A ? q) = A ? q
  }.
 
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
+interpretation "o-concrete space downarrow" 'downarrow x = 
+  (fun_1 __ (downarrow _) x).
 
+definition bp': concrete_space → basic_pair ≝ λc.bp c.
 coercion bp'.
 
+lemma setoid_OF_OA : OA → setoid.
+intros; apply (oa_P o);
+qed.
+
+coercion setoid_OF_OA.
+
+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));]
+qed.
+
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun1 _ _ _ (binary_downarrow _) a b).
+
 record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
  { rp:> arrows1 ? CS1 CS2;
    respects_converges:
-    ∀b,c.
-     extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
-     BPextS CS1 ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+    ∀b,c. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
    respects_all_covered:
-    extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
+     eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (oa_one (form CS2))))
+           (Ext⎽CS1 (oa_one (form CS1)))
  }.
 
 definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
@@ -76,29 +128,22 @@ definition convergent_relation_space_composition:
  intros; constructor 1;
      [ intros; whd in c c1 ⊢ %;
        constructor 1;
-        [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
+        [ apply (c1 ∘ c);
         | intros;
-          change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
-          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
-            with (c1 \sub \f ∘ c \sub \f);
-          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
-            with (c1 \sub \f ∘ c \sub \f);
-          apply (.= (extS_com ??????));
-          apply (.= (†(respects_converges ?????)));
-          apply (.= (respects_converges ?????));
-          apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
-          apply refl1;
-        | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
-          apply (.= (extS_com ??????));
-          apply (.= (†(respects_all_covered ???)));
-          apply (.= respects_all_covered ???);
-          apply refl1]
+          change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
+          unfold uncurry_arrows;
+          alias symbol "trans" = "trans1".
+          apply (.= († (respects_converges : ?)));
+          apply (respects_converges ?? c (c1\sub\f⎻ b) (c1\sub\f⎻ c2));
+        | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
+          unfold uncurry_arrows;
+          apply (.= (†(respects_all_covered :?)));
+          apply rule (respects_all_covered ?? c);]
      | intros;
-       change with (b ∘ a = b' ∘ a');
+       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));
-       apply refl1]
+       apply ( (H‡H1));]
 qed.
 
 definition CSPA: category1.
@@ -107,25 +152,16 @@ definition CSPA: category1.
   | apply convergent_relation_space_setoid
   | intro; constructor 1;
      [ apply id1
-     | intros;
-       unfold id; simplify;
-       apply (.= (equalset_extS_id_X_X ??));
-       apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
-                    (equalset_extS_id_X_X ??)\sup -1)));
-       apply refl1;
-     | apply (.= (equalset_extS_id_X_X ??));
-       apply refl1]
+     | intros; apply refl1;
+     | apply refl1]
   | apply convergent_relation_space_composition
   | intros; simplify;
     change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
-    apply (.= ASSOC1);
-    apply refl1
+    apply ASSOC1;
   | intros; simplify;
     change with (a ∘ id1 ? o1 = a);
-    apply (.= id_neutral_right1 ????);
-    apply refl1
+    apply (id_neutral_right1 : ?);
   | intros; simplify;
     change with (id1 ? o2 ∘ a = a);
-    apply (.= id_neutral_left1 ????);
-    apply refl1]
+    apply (id_neutral_left1 : ?);]
 qed.