big mess of notation

[helm.git] / helm / software / matita / library / formal_topology / basic_topologies.ma

diff --git a/helm/software/matita/library/formal_topology/basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies.ma
index 0c153b9d3ade1d6760fc6710a2606a908eef005f..7ebec5e3f4607e968feb097b083d8c10457c606a 100644 (file)
--- a/helm/software/matita/library/formal_topology/basic_topologies.ma
+++ b/helm/software/matita/library/formal_topology/basic_topologies.ma
@@ -24,10 +24,12 @@ record basic_topology: Type1 ≝
    compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
  }.
 
+definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
+
 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 → (foo ?? cont_rel)⎻* U = _1A ? ((foo ?? cont_rel)⎻* U)
  }. 
 
 definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
@@ -46,7 +48,7 @@ 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.(foo ?? a)⎻* (A o1 X) = (foo ?? 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;]
@@ -65,7 +67,7 @@ qed.*)
 
 axiom 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.(foo ?? a)⎻* (A o1 X) = (foo ?? 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)) → 
@@ -99,20 +101,24 @@ 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 sym1; unfold foo;
        apply (.= †(minus_star_image_comp ??????));
-       apply (.= (saturated ?????)\sup -1);
+       apply (.= (saturated ?? s ((foo ?? r)⎻* U) ?)^-1);
         [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
-        | apply (.= (minus_star_image_comp ??????)\sup -1);
+        | change in ⊢ (? ? ? % ?) with (((foo ?? s)⎻* ∘ (foo ?? r)⎻* ) U);
+          apply (.= (minus_star_image_comp ??????)^-1);
           apply refl1]]
 qed.
 
@@ -139,24 +145,29 @@ 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' (minus_star_image ?? (foo ?? 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);]
+              minus_star_image o2 o3 (foo ?? b) (A o2 (minus_star_image o1 o2 (foo ?? a) (A o1 X)))
+            =_1 minus_star_image o2 o3 (foo ?? b') (A o2 (minus_star_image o1 o2 (foo ?? 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 "compose" (instance 1) = "category1 composition".
+alias symbol "compose" (instance 1) = "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;
+              minus_star_image ?? (foo ?? (b ∘ a)) (A o1 X) =_1 minus_star_image ?? (foo ?? (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 ((foo ?? b)⎻* ((foo ?? 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 ((foo ?? b')⎻* ((foo ?? 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;]