More (ugly) work.

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / o-basic_topologies.ma

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 a958da4258b31de8376571f0eafbbe52ffa6b7fe..7f4270c79a6bb7fd100c91cfb4a5a88a7c8d8bd5 100644 (file)
--- 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)
  }. 
@@ -51,14 +52,11 @@ 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 +73,11 @@ 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 +99,7 @@ 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 +109,18 @@ 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.
@@ -131,21 +129,11 @@ 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;
+     | 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;
@@ -167,18 +155,19 @@ definition BTop: category1.
            apply ((Hcut X) \sup -1)]
        clear Hcut; generalize in match x; clear x;
        apply (continuous_relation_eq_inv');
-       apply Hcut1;]
+       apply Hcut1;*)]
   | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
-    apply (.= †(ASSOC1‡#));
-    apply refl1
+    (*apply (.= †(ASSOC1‡#));
+    apply refl1*)
   | intros; simplify; intro; unfold continuous_relation_comp; simplify;
-    apply (.= †((id_neutral_right1 ????)‡#));
-    apply refl1
+    (*apply (.= †((id_neutral_right1 ????)‡#));
+    apply refl1*)
   | intros; simplify; intro; simplify;
     apply (.= †((id_neutral_left1 ????)‡#));
     apply refl1]
 qed.
 
+(*
 (*CSC: unused! *)
 (* this proof is more logic-oriented than set/lattice oriented *)
 theorem continuous_relation_eqS:
@@ -203,3 +192,4 @@ theorem continuous_relation_eqS:
   [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
  apply Hcut2; assumption.
 qed.
+*)
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