update in groud_2 and models

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

diff --git a/helm/software/matita/library/formal_topology/basic_topologies_to_o-basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies_to_o-basic_topologies.ma
index 88f9e2393b158219e764cff479c8e6aff91ed6fc..58b75fb680ececa60755fcb2e77b7105b9245254 100644 (file)
--- a/helm/software/matita/library/formal_topology/basic_topologies_to_o-basic_topologies.ma
+++ b/helm/software/matita/library/formal_topology/basic_topologies_to_o-basic_topologies.ma
@@ -41,7 +41,7 @@ intros (S T);
      | cases daemon (*apply (o_relation_pair_of_relation_pair_is_morphism S T)*)]
 qed.
 
-definition BTop_to_OBTop: carr3 (arrows3 CAT2 (category2_of_category1 BTop) OBTop).
+definition BTop_to_OBTop: carr3 ((category2_of_category1 BTop) ⇒_\c3 OBTop).
  constructor 1;
   [ apply o_basic_topology_of_basic_topology;
   | intros; apply o_continuous_relation_of_continuous_relation_morphism;
@@ -49,39 +49,43 @@ definition BTop_to_OBTop: carr3 (arrows3 CAT2 (category2_of_category1 BTop) OBTo
   | cases daemon (*apply o_relation_topology_of_relation_topology_morphism_respects_comp*);]
 qed.
 
-(*
-alias symbol "eq" (instance 2) = "setoid1 eq".
-alias symbol "eq" (instance 1) = "setoid2 eq".
-theorem BTop_to_OBTop_faithful:
- ∀S,T.∀f,g:arrows2 (category2_of_category1 BTop) S T.
-  map_arrows2 ?? BTop_to_OBTop ?? f = map_arrows2 ?? BTop_to_OBTop ?? g → f=g.
- intros; change with (∀b.A ? (ext ?? f b) = A ? (ext ?? g b));
- apply (POW_faithful);
- apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
- apply sym2;
- apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
- apply sym2;
- apply e;
+theorem BTop_to_OBTop_faithful: faithful2 ?? BTop_to_OBTop.
+ intros 5; apply (continuous_relation_eq_inv' o1 o2 f g); apply e;
 qed.
 
 include "formal_topology/notation.ma".
 
-theorem BTop_to_OBTop_full: 
-   ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BTop_to_OBTop S T g = f).
- intros;
+theorem BTop_to_OBTop_full: full2 ?? BTop_to_OBTop.
+ intros 3 (S T);
  cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
- exists[
+ (* cases Hg; *)
+ exists [
    constructor 1;
     [ apply g
-    | apply hide; intros; lapply (Oreduced ?? f ? e);
+    | unfold image_coercion; cases daemon (*apply hide; intros; lapply (Oreduced ?? f ? e); unfold image_coercion;
       cases Hg; lapply (e3 U) as K; apply (.= K);
-      apply (.= Hletin); apply rule (†(K^-1));
-    | apply hide; intros; lapply (Osaturated ?? f ? e);
+      apply (.= Hletin); apply rule (†(K^-1)); *)
+    | cases daemon (* apply hide; intros; lapply (Osaturated ?? f ? e);
       cases Hg; lapply (e1 U) as K; apply (.= K);
-      apply (.= Hletin); apply rule (†(K^-1));
+      apply (.= Hletin); apply rule (†(K^-1)); *)
     ]
  | simplify; unfold BTop_to_OBTop; simplify;
+   cases Hg; unfold o_continuous_relation_of_continuous_relation_morphism;
+   simplify;
+   change with ((orelation_of_relation ?? g)⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
+                f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
+
+   
+   change with (g⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
+                f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
+   apply sym2; whd in T;
+   intro;
+   apply trans2; [2: apply sym2; [2: apply Hg;
+   
+   whd in ⊢ (?(??%%)???);
+    apply (.= Hg^-1);
    unfold o_continuous_relation_of_continuous_relation_morphism; simplify;
-   cases Hg; whd; simplify; intro; 
+   intro; simplify;
+   unfold image_coercion; cases Hg; whd; simplify; intro; simplify;
 qed.
 *)