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=90086b4d2f0e8646b973ef60925f8ccaee6db19b;hb=13114a0147a28f8c7359c9c19ee254716eb5f55a;hp=61b8f77e7582b2b0f4834537d63d1ab7909b3dc2;hpb=680b7493de237259ddb589a92be4b8bbc8de3cbf;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 61b8f77e7..90086b4d2 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
@@ -15,150 +15,120 @@
 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 = (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 = (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 = (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 = (or_f_minus _ _ (rel x)).
-
-definition A : ∀b:BP. unary_morphism (oa_P (form b)) (oa_P (form b)).
+definition A : ∀b:OBP. unary_morphism1 (Oform b) (Oform b).
 intros; constructor 1;
  [ apply (λx.□_b (Ext⎽b x));
- | do 2 unfold FunClass_1_OF_carr1; intros; apply  (†(†H));]
+ | intros; apply  (†(†e));]
 qed.
 
-lemma xxx : ∀x.carr x → carr1 (setoid1_of_setoid x). intros; assumption; qed.
-coercion xxx.
-
-definition d_p_i : 
-  ∀S,I:SET.∀d:unary_morphism S S.∀p:arrows1 SET I S.arrows1 SET I S.
-intros; constructor 1;
- [ apply (λi:I. u (c i));
- | unfold FunClass_1_OF_carr1; intros; 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 ≝
- { bp:> BP;
+record Oconcrete_space : Type2 ≝
+ { Obp:> OBP;
    (*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.
-     (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:setoid.∀p:ums I (oa_P (form bp)).
-     downarrow (oa_join ? I (d_p_i ?? downarrow p)) =
-     oa_join ? I (d_p_i ?? downarrow p);
-   il1: ∀q.downarrow (A ? q) = A ? q
+   Odownarrow: unary_morphism1 (Oform Obp) (Oform Obp);
+   Odownarrow_is_sat: is_o_saturation ? Odownarrow;
+   Oconverges: ∀q1,q2.
+     (Ext⎽Obp q1 ∧ (Ext⎽Obp q2)) = (Ext⎽Obp ((Odownarrow q1) ∧ (Odownarrow q2)));
+   Oall_covered: Ext⎽Obp (oa_one (Oform Obp)) = oa_one (Oconcr Obp);
+   Oil2: ∀I:SET.∀p:arrows2 SET1 I (Oform Obp).
+     Odownarrow (∨ { x ∈ I | Odownarrow (p x) | down_p ???? }) =
+     ∨ { x ∈ I | Odownarrow (p x) | down_p ???? };
+   Oil1: ∀q.Odownarrow (A ? q) = A ? q
  }.
 
-interpretation "o-concrete space downarrow" 'downarrow x = (fun_1 __ (downarrow _) x).
+interpretation "o-concrete space downarrow" 'downarrow x = 
+  (fun11 __ (Odownarrow _) x).
 
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
-coercion bp'.
+definition Obinary_downarrow : 
+  ∀C:Oconcrete_space.binary_morphism1 (Oform C) (Oform C) (Oform C).
+intros; constructor 1;
+[ intros; apply (↓ c ∧ ↓ c1);
+| intros;
+  alias symbol "prop2" = "prop21".
+  alias symbol "prop1" = "prop11".
+  apply ((†e)‡(†e1));]
+qed.
 
-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));
-   respects_all_covered:
-    extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
- }.
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (Obinary_downarrow _) a b).
 
-definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
- λCS1,CS2,c. rp CS1 CS2 c.
- 
-coercion rp'.
+record Oconvergent_relation_pair (CS1,CS2: Oconcrete_space) : Type2 ≝
+ { Orp:> arrows2 ? CS1 CS2;
+   Orespects_converges:
+    ∀b,c. eq1 ? (Orp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (Orp\sub\f⎻ b ↓ Orp\sub\f⎻ c));
+   Orespects_all_covered:
+     eq1 ? (Orp\sub\c⎻ (Ext⎽CS2 (oa_one (Oform CS2))))
+           (Ext⎽CS1 (oa_one (Oform CS1)))
+ }.
 
-definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
- intros;
+definition Oconvergent_relation_space_setoid: Oconcrete_space → Oconcrete_space → setoid2.
+ intros (c c1);
  constructor 1;
-  [ apply (convergent_relation_pair c c1)
+  [ apply (Oconvergent_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 (c2 c3);
+       apply (Orelation_pair_equality c c1 c2 c3);
+     | 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.
-
-coercion rp''.
-
-definition convergent_relation_space_composition:
- ∀o1,o2,o3: concrete_space.
-  binary_morphism1
-   (convergent_relation_space_setoid o1 o2)
-   (convergent_relation_space_setoid o2 o3)
-   (convergent_relation_space_setoid o1 o3).
+definition Oconvergent_relation_space_of_Oconvergent_relation_space_setoid: 
+  ∀CS1,CS2.carr2 (Oconvergent_relation_space_setoid CS1 CS2) → 
+    Oconvergent_relation_pair CS1 CS2  ≝ λP,Q,c.c.
+coercion Oconvergent_relation_space_of_Oconvergent_relation_space_setoid.
+
+definition Oconvergent_relation_space_composition:
+ ∀o1,o2,o3: Oconcrete_space.
+  binary_morphism2
+   (Oconvergent_relation_space_setoid o1 o2)
+   (Oconvergent_relation_space_setoid o2 o3)
+   (Oconvergent_relation_space_setoid o1 o3).
  intros; constructor 1;
-     [ intros; whd in c c1 ⊢ %;
+     [ intros; whd in t t1 ⊢ %;
        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))));
+          alias symbol "trans" = "trans1".
+          apply (.= († (Orespects_converges : ?)));
+          apply (Orespects_converges ?? c (c1\sub\f⎻ b) (c1\sub\f⎻ c2));
+        | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (Oform o3)))));
+          apply (.= (†(Orespects_all_covered :?)));
+          apply rule (Orespects_all_covered ?? c);]
      | 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));
-       apply refl1]
+       change with (b ∘ a = b' ∘ a'); 
+       change in e with (Orp ?? a = Orp ?? a');
+       change in e1 with (Orp ?? b = Orp ?? b');
+       apply (e‡e1);]
 qed.
 
-definition CSPA: category1.
+definition OCSPA: category2.
  constructor 1;
-  [ apply concrete_space
-  | apply convergent_relation_space_setoid
+  [ apply Oconcrete_space
+  | apply Oconvergent_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]
-  | apply convergent_relation_space_composition
-  | intros; simplify;
+     [ apply id2
+     | intros; apply refl1;
+     | apply refl1]
+  | apply Oconvergent_relation_space_composition
+  | intros; simplify; whd in a12 a23 a34;
     change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
-    apply (.= ASSOC1);
-    apply refl1
+    apply rule ASSOC;
   | intros; simplify;
-    change with (a ∘ id1 ? o1 = a);
-    apply (.= id_neutral_right1 ????);
-    apply refl1
+    change with (a ∘ id2 OBP o1 = a);
+    apply (id_neutral_right2 : ?);
   | intros; simplify;
-    change with (id1 ? o2 ∘ a = a);
-    apply (.= id_neutral_left1 ????);
-    apply refl1]
+    change with (id2 ? o2 ∘ a = a);
+    apply (id_neutral_left2 : ?);]
 qed.
+
+definition Oconcrete_space_of_OCSPA : objs2 OCSPA → Oconcrete_space ≝ λx.x.
+coercion Oconcrete_space_of_OCSPA.
+
+definition Oconvergent_relation_space_setoid_of_arrows2_OCSPA :
+ ∀P,Q. arrows2 OCSPA P Q → Oconvergent_relation_space_setoid P Q ≝ λP,Q,x.x.
+coercion Oconvergent_relation_space_setoid_of_arrows2_OCSPA.
+