+(**************************************************************************)
+(* ___ *)
+(* ||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.
+*)