X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Fbasic_topologies.ma;h=0177afb63e9cea4035b289b1d8024dce1e795c32;hb=070b44c9c2344967ca8c4531909614a0d4da2fbe;hp=0c153b9d3ade1d6760fc6710a2606a908eef005f;hpb=1ed4fe0f28d3b0b915387330cd722bfb80fb1063;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies.ma
index 0c153b9d3..0177afb63 100644
--- a/helm/software/matita/library/formal_topology/basic_topologies.ma
+++ b/helm/software/matita/library/formal_topology/basic_topologies.ma
@@ -25,9 +25,9 @@ record basic_topology: Type1 ≝
  }.
 
 record continuous_relation (S,T: basic_topology) : Type1 ≝
- { cont_rel:> arrows1 ? S T;
-   reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
-   saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
+ { cont_rel:> S ⇒_\r1 T;
+   reduced: ∀U. U =_1 J ? U → image_coercion ?? cont_rel U =_1 J ? (image_coercion ?? cont_rel U);
+   saturated: ∀U. U =_1 A ? U → (cont_rel)⎻* U = _1A ? ((cont_rel)⎻* U)
  }. 
 
 definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
@@ -46,7 +46,7 @@ coercion continuos_relation_of_continuous_relation_setoid.
 
 axiom continuous_relation_eq':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
-  a = a' → ∀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).
 (*
  intros; split; intro; unfold minus_star_image; simplify; intros;
   [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
@@ -63,35 +63,34 @@ axiom continuous_relation_eq':
      [ apply I | assumption ]]
 qed.*)
 
-axiom continuous_relation_eq_inv':
+lemma 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'.
-(* intros 6;
+  (∀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)) → 
+  (∀X.a⎻* (A o1 X) = a'⎻* (A o1 X)) → 
    ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
-  [2: clear b H a' a; intros;
+  [2: clear b f 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 *)
-       cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
+       cut (∀V:Ω^o2.V ⊆ a⎻* (A ? (extS ?? a V)));
         [2: intro; intros 2; unfold minus_star_image; simplify; intros;
             apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
        clear Hletin;
-       cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
-        [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
+       cut (∀V:Ω^o2.V ⊆ a'⎻* (A ? (extS ?? a V)));
+        [2: intro; apply (. #‡(f ?)^-1); apply Hcut] clear f Hcut;
        (* second half of the fundamental adjunction here! to be taken out too *)
-      intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
+      intro; lapply (Hcut1 {(V)}); clear Hcut1;
       unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
       whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
       apply (if ?? (A_is_saturation ???));
       intros 2 (x H); lapply (Hletin V ? x ?);
-       [ apply refl | cases H; assumption; ]
+       [ apply refl | unfold foo; apply H; ]
       change with (x ∈ A ? (ext ?? a V));
-      apply (. #‡(†(extS_singleton ????)));
+      apply (. #‡(†(extS_singleton ????)^-1));
       assumption;]
- split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
+ split; apply Hcut; [2: assumption | intro; apply sym1; apply f]
 qed.
-*)
 
 definition continuous_relation_comp:
  ∀o1,o2,o3.
@@ -99,20 +98,24 @@ definition continuous_relation_comp:
    continuous_relation_setoid o2 o3 →
     continuous_relation_setoid o1 o3.
  intros (o1 o2 o3 r s); constructor 1;
-  [ apply (s ∘ r)
+  [ alias symbol "compose" (instance 1) = "category1 composition".
+apply (s ∘ r)
   | intros;
-    apply sym1;
+    apply sym1;  
+    (*change in ⊢ (? ? ? (? ? ? ? %) ?) with (image_coercion ?? (s ∘ r) U);*)
     apply (.= †(image_comp ??????));
-    apply (.= (reduced ?????)\sup -1);
+    apply (.= (reduced ?? s (image_coercion ?? r U) ?)^-1); 
      [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
-     | apply (.= (image_comp ??????)\sup -1);
+     | change in ⊢ (? ? ? % ?) with ((image_coercion ?? s ∘ image_coercion ?? r) U);
+       apply (.= (image_comp ??????)^-1);
        apply refl1]
      | intros;
        apply sym1;
-       apply (.= †(minus_star_image_comp ??????));
-       apply (.= (saturated ?????)\sup -1);
+       apply (.= †(minus_star_image_comp ??? s r ?));
+       apply (.= (saturated ?? s ((r)⎻* U) ?)^-1);
         [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
-        | apply (.= (minus_star_image_comp ??????)\sup -1);
+        | change in ⊢ (? ? ? % ?) with ((s⎻* ∘ r⎻* ) U);
+          apply (.= (minus_star_image_comp ??????)^-1);
           apply refl1]]
 qed.
 
@@ -139,24 +142,28 @@ definition BTop: category1.
      | intros; simplify; intro x; simplify;
        lapply depth=0 (continuous_relation_eq' ???? e) as H';
        lapply depth=0 (continuous_relation_eq' ???? e1) as H1';
-       letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
+       letin K ≝ (λX.H1' ((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);]
+              (b)⎻* (A o2 ((a)⎻* (A o1 X)))
+            =_1 (b')⎻* (A o2 ((a')⎻* (A o1 X))));
+        [2: intro; apply sym1; 
+            apply (.= (†(†((H' X)^-1)))); apply sym1; apply (K X);]
        clear K H' H1';
-alias symbol "compose" (instance 1) = "category1 composition".
+alias symbol "powerset" (instance 5) = "powerset low".
+alias symbol "compose" (instance 2) = "category1 composition".
 cut (∀X:Ω^o1.
-              minus_star_image ?? (b ∘ a) (A o1 X) =_1 minus_star_image ?? (b'∘a') (A o1 X));
-        [2: intro;
+              ((b ∘ a))⎻* (A o1 X) =_1 ((b'∘a'))⎻* (A o1 X));
+        [2: intro; unfold foo;
             apply (.= (minus_star_image_comp ??????));
-            apply (.= #‡(saturated ?????));
-             [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
+            change in ⊢ (? ? ? % ?) with ((b)⎻* ((a)⎻* (A o1 X)));
+            apply (.= †(saturated ?????));
+             [ apply ((saturation_idempotent ????)^-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)]
+            change in ⊢ (? ? ? % ?) with ((b')⎻* ((a')⎻* (A o1 X)));
+            apply (.= †(saturated ?????));
+             [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
+           apply ((Hcut X)^-1)]
        clear Hcut; generalize in match x; clear x;
        apply (continuous_relation_eq_inv');
        apply Hcut1;]