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=21cb6b92e615f08280e9f032718593b4a6a1d3a5;hp=57ec577aec9d1248c0f70e521dc62329a4dafbeb;hpb=05669a3646cbf8219bf772718b4b2c5bfd1ddd82;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 57ec577ae..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
@@ -15,134 +15,120 @@
 include "o-basic_pairs.ma".
 include "o-saturations.ma".
 
-definition A : ∀b:BP. unary_morphism1 (form b) (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_Type_OF_setoid21; intros; apply  (†(†e));]
+ | intros; apply  (†(†e));]
 qed.
 
 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.
 
-record concrete_space : Type2 ≝
- { bp:> BP;
+record Oconcrete_space : Type2 ≝
+ { Obp:> OBP;
    (*distr : is_distributive (form bp);*)
-   downarrow: unary_morphism1 (form bp) (form bp);
-   downarrow_is_sat: is_o_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: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
+   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 = 
-  (fun11 __ (downarrow _) x).
+  (fun11 __ (Odownarrow _) x).
 
-definition bp': concrete_space → basic_pair ≝ λc.bp c.
-coercion bp'.
-
-definition bp'': concrete_space → objs2 BP.
- intro; apply (bp' c);
-qed.
-coercion bp''.
-
-definition binary_downarrow : 
-  ∀C:concrete_space.binary_morphism1 (form C) (form C) (form C).
+definition Obinary_downarrow : 
+  ∀C:Oconcrete_space.binary_morphism1 (Oform C) (Oform C) (Oform C).
 intros; constructor 1;
-[ intros; apply (↓ t ∧ ↓ t1);
+[ intros; apply (↓ c ∧ ↓ c1);
 | intros;
   alias symbol "prop2" = "prop21".
   alias symbol "prop1" = "prop11".
   apply ((†e)‡(†e1));]
 qed.
 
-interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (binary_downarrow _) a b).
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun21 _ _ _ (Obinary_downarrow _) a b).
 
-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:
-     eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (oa_one (form CS2))))
-           (Ext⎽CS1 (oa_one (form CS1)))
+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 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 → setoid2.
- 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 (c2 c3);
+       apply (Orelation_pair_equality c c1 c2 c3);
      | intros 1; apply refl2;
      | intros 2; apply sym2; 
      | intros 3; apply trans2]]
 qed.
 
+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 rp'': ∀CS1,CS2.carr2 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
-coercion rp''.
-
-definition rp''': ∀CS1,CS2.Type_OF_setoid2 (convergent_relation_space_setoid CS1 CS2) → arrows2 BP CS1 CS2 ≝
- λCS1,CS2,c.rp ?? c.
-coercion rp'''.
-
-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.
+definition Oconvergent_relation_space_composition:
+ ∀o1,o2,o3: Oconcrete_space.
   binary_morphism2
-   (convergent_relation_space_setoid o1 o2)
-   (convergent_relation_space_setoid o2 o3)
-   (convergent_relation_space_setoid o1 o3).
+   (Oconvergent_relation_space_setoid o1 o2)
+   (Oconvergent_relation_space_setoid o2 o3)
+   (Oconvergent_relation_space_setoid o1 o3).
  intros; constructor 1;
      [ intros; whd in t t1 ⊢ %;
        constructor 1;
-        [ apply (t1 ∘ t);
+        [ apply (c1 ∘ c);
         | intros;
-          change in ⊢ (? ? ? % ?) with (t\sub\c⎻ (t1\sub\c⎻ (Ext⎽o3 (b↓c))));
-          unfold FunClass_1_OF_Type_OF_setoid21;
+          change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
           alias symbol "trans" = "trans1".
-          apply (.= († (respects_converges : ?)));
-          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 rule (respects_all_covered ?? t);]
+          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 e with (rp'' ?? a = rp'' ?? a');
-       change in e1 with (rp'' ?? b = rp ?? b');
+       change in e with (Orp ?? a = Orp ?? a');
+       change in e1 with (Orp ?? b = Orp ?? b');
        apply (e‡e1);]
 qed.
 
-definition CSPA: category2.
+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 id2
      | intros; apply refl1;
      | apply refl1]
-  | apply convergent_relation_space_composition
+  | apply Oconvergent_relation_space_composition
   | intros; simplify; whd in a12 a23 a34;
     change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
     apply rule ASSOC;
   | intros; simplify;
-    change with (a ∘ id2 ? o1 = a);
+    change with (a ∘ id2 OBP o1 = a);
     apply (id_neutral_right2 : ?);
   | intros; simplify;
     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.
+