some notation for map_arrows2

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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_topologies_to_o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_topologies_to_o-basic_topologies.ma
index 282519dc7db6166cb5529ff50311a749fe44ec5c..bea3207b6021c35904f57f255a79a01f249ba6d5 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_topologies_to_o-basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_topologies_to_o-basic_topologies.ma
@@ -16,24 +16,72 @@ include "basic_topologies.ma".
 include "o-basic_topologies.ma".
 include "relations_to_o-algebra.ma".
 
-(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_basic_topology: cic:/matita/formal_topology/basic_topologies/basic_topology.ind#xpointer(1/1) → basic_topology.
- intro;
+definition o_basic_topology_of_basic_topology: basic_topology → Obasic_topology.
+ intros (b); constructor 1;
+  [ apply (POW' b) | apply (A b) | apply (J b);
+  | apply (A_is_saturation b) | apply (J_is_reduction b) | apply (compatibility b) ]
+qed.
+
+definition o_continuous_relation_of_continuous_relation:
+ ∀BT1,BT2.continuous_relation BT1 BT2 →
+  Ocontinuous_relation (o_basic_topology_of_basic_topology BT1) (o_basic_topology_of_basic_topology BT2).
+ intros (BT1 BT2 c); constructor 1;
+  [ apply (orelation_of_relation ?? c) | apply (reduced ?? c) | apply (saturated ?? c) ]
+qed.
+
+axiom daemon: False.
+
+lemma o_continuous_relation_of_continuous_relation_morphism :
+  ∀S,T:category2_of_category1 BTop.
+  unary_morphism2 (arrows2 (category2_of_category1 BTop) S T)
+   (arrows2 OBTop (o_basic_topology_of_basic_topology S) (o_basic_topology_of_basic_topology T)).
+intros (S T);
+   constructor 1;
+     [ apply (o_continuous_relation_of_continuous_relation 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).
  constructor 1;
-  [ apply (SUBSETS (carrbt b));
-  | apply (A b);
-  | apply (J b);
-  | apply (A_is_saturation b);
-  | apply (J_is_reduction b);
-  | apply (compatibility b); ]
+  [ apply o_basic_topology_of_basic_topology;
+  | intros; apply o_continuous_relation_of_continuous_relation_morphism;
+  | cases daemon (*apply o_relation_topology_of_relation_topology_morphism_respects_id*);
+  | 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;
 qed.
+*)
 
-definition o_relation_pair_of_relation_pair:
- ∀S,T.cic:/matita/formal_topology/basic_topologies/continuous_relation.ind#xpointer(1/1) S T →
-  continuous_relation (o_basic_topology_of_basic_topology S) (o_basic_topology_of_basic_topology T).
+include "notation.ma".
+
+theorem BTop_to_OBTop_full: 
+   ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BTop_to_OBTop S T g = f).
  intros;
- constructor 1;
-  [ apply (orelation_of_relation ?? (cont_rel ?? c));
-  | apply (reduced ?? c);
-  | apply (saturated ?? c); ]
+ cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
+ exists[
+   constructor 1;
+    [ apply g
+    | apply hide; intros; lapply (Oreduced ?? f ? e);
+      cases Hg; lapply (e3 U) as K; apply (.= K);
+      apply (.= Hletin); apply rule (†(K^-1));
+    | apply hide; intros; lapply (Osaturated ?? f ? e);
+      cases Hg; lapply (e1 U) as K; apply (.= K);
+      apply (.= Hletin); apply rule (†(K^-1));
+    ]
+ | simplify; unfold BTop_to_OBTop; simplify;
+   unfold o_continuous_relation_of_continuous_relation_morphism; simplify;
+   cases Hg; whd; simplify; intro; 
 qed.
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