X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-basic_topologies.ma;h=f2ce654bae925b1dd036d565c5aecf0a27351112;hb=6006ddfa14ae1caa5adf92119fc38a3a7c4919b3;hp=c0fb6c6a7ff64f158bb667149ca1530ec497d69f;hpb=2d7053c212c790d528e82ba37c3e927070de7ae5;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
index c0fb6c6a7..f2ce654ba 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
@@ -1,4 +1,4 @@
-(**************************************************************************)
+ (**************************************************************************)
 (*       ___                                                              *)
 (*      ||M||                                                             *)
 (*      ||A||       A project by Andrea Asperti                           *)
@@ -15,44 +15,38 @@
 include "o-algebra.ma".
 include "o-saturations.ma".
 
-record basic_topology: Type ≝
- { carrbt:> OA;
-   A: arrows1 SET (oa_P carrbt) (oa_P carrbt);
-   J: arrows1 SET (oa_P carrbt) (oa_P carrbt);
-   A_is_saturation: is_saturation ? A;
-   J_is_reduction: is_reduction ? J;
-   compatibility: ∀U,V. (A U >< J V) = (U >< J V)
+record Obasic_topology: Type2 ≝
+ { Ocarrbt:> OA;
+   oA: Ocarrbt ⇒ Ocarrbt;
+   oJ: Ocarrbt ⇒ Ocarrbt;
+   oA_is_saturation: is_o_saturation ? oA;
+   oJ_is_reduction: is_o_reduction ? oJ;
+   Ocompatibility: ∀U,V. (oA U >< oJ V) = (U >< oJ V)
  }.
 
-record continuous_relation (S,T: basic_topology) : Type ≝
- { cont_rel:> arrows1 ? S T;
+record Ocontinuous_relation (S,T: Obasic_topology) : Type2 ≝
+ { Ocont_rel:> arrows2 OA S T;
    (* reduces uses eq1, saturated uses eq!!! *)
-   reduced: ∀U. U = J ? U → cont_rel U = J ? (cont_rel U);
-   saturated: ∀U. U = A ? U → cont_rel⎻* U = A ? (cont_rel⎻* U)
+   Oreduced: ∀U. U = oJ ? U → Ocont_rel U = oJ ? (Ocont_rel U);
+   Osaturated: ∀U. U = oA ? U → Ocont_rel⎻* U = oA ? (Ocont_rel⎻* U)
  }. 
 
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
+definition Ocontinuous_relation_setoid: Obasic_topology → Obasic_topology → setoid2.
  intros (S T); constructor 1;
-  [ apply (continuous_relation S T)
+  [ apply (Ocontinuous_relation S T)
   | constructor 1;
-     [ (*apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));*)
-       apply (λr,s:continuous_relation S T.r⎻* ∘ (A S) = s⎻* ∘ (A ?));
-     | simplify; intros; apply refl1;
-     | simplify; intros; apply sym1; apply H
-     | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+     [ alias symbol "eq" = "setoid2 eq".
+       alias symbol "compose" = "category2 composition".
+       apply (λr,s:Ocontinuous_relation S T. (r⎻* ) ∘ (oA S) = (s⎻* ∘ (oA ?)));
+     | simplify; intros; apply refl2;
+     | simplify; intros; apply sym2; apply e
+     | simplify; intros; apply trans2; [2: apply e |3: apply e1; |1: skip]]]
 qed.
 
-definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows1 ? S T ≝ cont_rel.
-
-coercion cont_rel'.
+definition Ocontinuous_relation_of_Ocontinuous_relation_setoid: 
+  ∀P,Q. Ocontinuous_relation_setoid P Q → Ocontinuous_relation P Q ≝ λP,Q,c.c.
+coercion Ocontinuous_relation_of_Ocontinuous_relation_setoid.
 
-definition cont_rel'':
- ∀S,T: basic_topology.
-  continuous_relation_setoid S T → unary_morphism (oa_P (carrbt S)) (oa_P (carrbt T)).
- intros; apply rule cont_rel; apply c;
-qed.
-
-coercion cont_rel''.
 (*
 theorem continuous_relation_eq':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
@@ -101,68 +95,71 @@ theorem continuous_relation_eq_inv':
  split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
 qed.
 *)
-definition continuous_relation_comp:
+
+definition Ocontinuous_relation_comp:
  ∀o1,o2,o3.
-  continuous_relation_setoid o1 o2 →
-   continuous_relation_setoid o2 o3 →
-    continuous_relation_setoid o1 o3.
+  Ocontinuous_relation_setoid o1 o2 →
+   Ocontinuous_relation_setoid o2 o3 →
+    Ocontinuous_relation_setoid o1 o3.
  intros (o1 o2 o3 r s); constructor 1;
-  [ apply (s ∘ r)
+  [ apply (s ∘ r);
   | intros;
-    apply sym1;
+    apply sym1; 
     change in match ((s ∘ r) U) with (s (r U));
-    (*BAD*) unfold FunClass_1_OF_carr1;
-    apply (.= ((reduced : ?)\sup -1));
-     [ (*BAD*) change with (eq1 ? (r U) (J ? (r U)));
-       (* BAD U *) apply (.= (reduced ??? U ?)); [ assumption | apply refl1 ]
+    apply (.= (Oreduced : ?)\sup -1);
+     [ apply (.= (Oreduced :?)); [ assumption | apply refl1 ]
      | apply refl1]
   | intros;
-    apply sym;
+    apply sym1;
     change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
-    apply (.= (saturated : ?)\sup -1);
-     [ apply (.= (saturated : ?)); [ assumption | apply refl ]
-     | apply refl]]
+    apply (.= (Osaturated : ?)\sup -1);
+     [ apply (.= (Osaturated : ?)); [ assumption | apply refl1 ]
+     | apply refl1]]
 qed.
 
-definition BTop: category1.
+definition OBTop: category2.
  constructor 1;
-  [ apply basic_topology
-  | apply continuous_relation_setoid
+  [ apply Obasic_topology
+  | apply Ocontinuous_relation_setoid
   | intro; constructor 1;
-     [ apply id1
-     | intros; apply H;
-     | intros; apply H;]
+     [ apply id2
+     | intros; apply e;
+     | intros; apply e;]
   | intros; constructor 1;
-     [ apply continuous_relation_comp;
-     | intros; simplify; (*intro x; simplify;*)
-       change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
-       change in H with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
-       change in H1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
-       apply (.= H‡#);
-       intro x;
-        
-       change with (eq1 (oa_P (carrbt o3)) (b⎻* (a'⎻* (A o1 x))) (b'⎻*(a'⎻* (A o1 x))));
-       lapply (saturated o1 o2 a' (A o1 x):?) as X;
-         [ apply ((saturation_idempotent ?? (A_is_saturation o1) x)^-1) ]
-       change in X with (eq1 (oa_P (carrbt o2)) (a'⎻* (A o1 x)) (A o2 (a'⎻* (A o1 x)))); 
-       unfold uncurry_arrows;
-       apply (.= †X); whd in H1;
-       lapply (H1 (a'⎻* (A o1 x))) as X1;
-       change in X1 with (eq1 (oa_P (carrbt o3)) (b⎻* (A o2 (a'⎻* (A o1 x)))) (b'⎻* (A o2 (a' \sup ⎻* (A o1 x)))));
-       apply (.= X1);
-       unfold uncurry_arrows;
-       apply (†(X\sup -1));]
+     [ apply Ocontinuous_relation_comp;
+     | intros; simplify;
+       change with ((b⎻* ∘ a⎻* ) ∘ oA o1 = ((b'⎻* ∘ a'⎻* ) ∘ oA o1)); 
+       change with (b⎻* ∘ (a⎻* ∘ oA o1) = b'⎻* ∘ (a'⎻* ∘ oA o1));
+       change in e with (a⎻* ∘ oA o1 = a'⎻* ∘ oA o1);
+       change in e1 with (b⎻* ∘ oA o2 = b'⎻* ∘ oA o2);
+       apply (.= e‡#);
+       intro x;          
+       change with (eq1 ? (b⎻* (a'⎻* (oA o1 x))) (b'⎻*(a'⎻* (oA o1 x)))); 
+       apply (.= †(Osaturated o1 o2 a' (oA o1 x) ?)); [
+         apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+       apply (.= (e1 (a'⎻* (oA o1 x))));
+       change with (eq1 ? (b'⎻* (oA o2 (a'⎻* (oA o1 x)))) (b'⎻*(a'⎻* (oA o1 x))));   
+       apply (.= †(Osaturated o1 o2 a' (oA o1 x):?)^-1); [
+         apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+       apply rule #;]
   | intros; simplify;
-    change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
-    apply rule (#‡ASSOC1\sup -1);
+    change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ oA o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ oA o1));
+    apply rule (#‡ASSOC ^ -1);
   | intros; simplify;
-    change with ((a⎻* ∘ (id1 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
-    apply (#‡(id_neutral_right1 : ?));
+    change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
+    apply (#‡(id_neutral_right2 : ?));
   | intros; simplify;
-    change with (((id1 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
-    apply (#‡(id_neutral_left1 : ?));]
+    change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
+    apply (#‡(id_neutral_left2 : ?));]
 qed.
 
+definition Obasic_topology_of_OBTop: objs2 OBTop → Obasic_topology ≝ λx.x.
+coercion Obasic_topology_of_OBTop.
+
+definition Ocontinuous_relation_setoid_of_arrows2_OBTop : 
+  ∀P,Q. arrows2 OBTop P Q → Ocontinuous_relation_setoid P Q ≝ λP,Q,x.x.
+coercion Ocontinuous_relation_setoid_of_arrows2_OBTop.
+
 (*
 (*CSC: unused! *)
 (* this proof is more logic-oriented than set/lattice oriented *)
@@ -188,4 +185,4 @@ theorem continuous_relation_eqS:
   [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
  apply Hcut2; assumption.
 qed.
-*)
\ No newline at end of file
+*)