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=c0fb6c6a7ff64f158bb667149ca1530ec497d69f;hb=bd3b9cc44e65e316159ab56d3099949224413d66;hp=a958da4258b31de8376571f0eafbbe52ffa6b7fe;hpb=e88702452d7ff1dcec1156e4e5588eaf577103a0;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 a958da425..c0fb6c6a7 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
@@ -26,6 +26,7 @@ record basic_topology: Type ≝
 
 record continuous_relation (S,T: basic_topology) : Type ≝
  { cont_rel:> arrows1 ? 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)
  }. 
@@ -34,7 +35,8 @@ definition continuous_relation_setoid: basic_topology → basic_topology → set
  intros (S T); constructor 1;
   [ apply (continuous_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.∀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]]]
@@ -51,14 +53,11 @@ definition cont_rel'':
 qed.
 
 coercion cont_rel''.
-
+(*
 theorem continuous_relation_eq':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
   a = a' → ∀X.a⎻* (A o1 X) = a'⎻* (A o1 X).
- intros;
- lapply (prop_1_SET ??? H);
- 
-  split; intro; unfold minus_star_image; simplify; intros;
+ intros; apply oa_leq_antisym; intro; unfold minus_star_image; simplify; intros;
   [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
     lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
     cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption]
@@ -75,11 +74,11 @@ qed.
 
 theorem continuous_relation_eq_inv':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
-  (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
+  (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) → a=a'.
  intros 6;
  cut (∀a,a': continuous_relation_setoid o1 o2.
-  (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → 
-   ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
+  (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) → 
+   ∀V:(oa_P (carrbt o2)). A o1 (a'⎻ V) ≤ A o1 (a⎻ V));
   [2: clear b H a' a; intros;
       lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
        (* fundamental adjunction here! to be taken out *)
@@ -101,7 +100,7 @@ theorem continuous_relation_eq_inv':
       assumption;]
  split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
 qed.
-
+*)
 definition continuous_relation_comp:
  ∀o1,o2,o3.
   continuous_relation_setoid o1 o2 →
@@ -111,18 +110,18 @@ definition continuous_relation_comp:
   [ apply (s ∘ r)
   | intros;
     apply sym1;
-    apply (.= †(image_comp ??????));
-    apply (.= (reduced ?????)\sup -1);
-     [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
-     | apply (.= (image_comp ??????)\sup -1);
-       apply refl1]
-     | intros;
-       apply sym1;
-       apply (.= †(minus_star_image_comp ??????));
-       apply (.= (saturated ?????)\sup -1);
-        [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
-        | apply (.= (minus_star_image_comp ??????)\sup -1);
-          apply refl1]]
+    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 refl1]
+  | intros;
+    apply sym;
+    change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
+    apply (.= (saturated : ?)\sup -1);
+     [ apply (.= (saturated : ?)); [ assumption | apply refl ]
+     | apply refl]]
 qed.
 
 definition BTop: category1.
@@ -131,54 +130,40 @@ definition BTop: category1.
   | apply continuous_relation_setoid
   | intro; constructor 1;
      [ apply id1
-     | intros;
-       apply (.= (image_id ??));
-       apply sym1;
-       apply (.= †(image_id ??));
-       apply sym1;
-       assumption
-     | intros;
-       apply (.= (minus_star_image_id ??));
-       apply sym1;
-       apply (.= †(minus_star_image_id ??));
-       apply sym1;
-       assumption]
+     | intros; apply H;
+     | intros; apply H;]
   | intros; constructor 1;
      [ apply continuous_relation_comp;
-     | intros; simplify; intro x; simplify;
-       lapply depth=0 (continuous_relation_eq' ???? H) as H';
-       lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
-       letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
-       cut (∀X:Ω \sup o1.
-              minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
-            = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
-        [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
-       clear K H' H1';
-       cut (∀X:Ω \sup o1.
-              minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
-        [2: intro;
-            apply (.= (minus_star_image_comp ??????));
-            apply (.= #‡(saturated ?????));
-             [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
-            apply sym1; 
-            apply (.= (minus_star_image_comp ??????));
-            apply (.= #‡(saturated ?????));
-             [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
-           apply ((Hcut X) \sup -1)]
-       clear Hcut; generalize in match x; clear x;
-       apply (continuous_relation_eq_inv');
-       apply Hcut1;]
-  | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
-    apply (.= †(ASSOC1‡#));
-    apply refl1
-  | intros; simplify; intro; unfold continuous_relation_comp; simplify;
-    apply (.= †((id_neutral_right1 ????)‡#));
-    apply refl1
-  | intros; simplify; intro; simplify;
-    apply (.= †((id_neutral_left1 ????)‡#));
-    apply refl1]
+     | 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));]
+  | intros; simplify;
+    change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+    apply rule (#‡ASSOC1\sup -1);
+  | intros; simplify;
+    change with ((a⎻* ∘ (id1 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+    apply (#‡(id_neutral_right1 : ?));
+  | intros; simplify;
+    change with (((id1 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+    apply (#‡(id_neutral_left1 : ?));]
 qed.
 
+(*
 (*CSC: unused! *)
 (* this proof is more logic-oriented than set/lattice oriented *)
 theorem continuous_relation_eqS:
@@ -203,3 +188,4 @@ 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