- lambda_delta: "conversion" and "equivalence" components started

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

diff --git a/matita/matita/lib/formal_topology/basic_topologies_to_o-basic_topologies.ma b/matita/matita/lib/formal_topology/basic_topologies_to_o-basic_topologies.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "formal_topology/basic_topologies.ma".
+include "formal_topology/o-basic_topologies.ma".
+include "formal_topology/relations_to_o-algebra.ma".
+
+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 ((category2_of_category1 BTop) ⇒_\c3 OBTop).
+ constructor 1;
+  [ 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.
+
+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: full2 ?? BTop_to_OBTop.
+ intros 3 (S T);
+ cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
+ (* cases Hg; *)
+ exists [
+   constructor 1;
+    [ apply g
+    | 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)); *)
+    | cases daemon (* 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;
+   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;
+   intro; simplify;
+   unfold image_coercion; cases Hg; whd; simplify; intro; simplify;
+qed.
+*)