1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "formal_topology/basic_topologies.ma".
16 include "formal_topology/o-basic_topologies.ma".
17 include "formal_topology/relations_to_o-algebra.ma".
19 definition o_basic_topology_of_basic_topology: basic_topology → Obasic_topology.
20 intros (b); constructor 1;
21 [ apply (POW' b) | apply (A b) | apply (J b);
22 | apply (A_is_saturation b) | apply (J_is_reduction b) | apply (compatibility b) ]
25 definition o_continuous_relation_of_continuous_relation:
26 ∀BT1,BT2.continuous_relation BT1 BT2 →
27 Ocontinuous_relation (o_basic_topology_of_basic_topology BT1) (o_basic_topology_of_basic_topology BT2).
28 intros (BT1 BT2 c); constructor 1;
29 [ apply (orelation_of_relation ?? c) | apply (reduced ?? c) | apply (saturated ?? c) ]
34 lemma o_continuous_relation_of_continuous_relation_morphism :
35 ∀S,T:category2_of_category1 BTop.
36 unary_morphism2 (arrows2 (category2_of_category1 BTop) S T)
37 (arrows2 OBTop (o_basic_topology_of_basic_topology S) (o_basic_topology_of_basic_topology T)).
40 [ apply (o_continuous_relation_of_continuous_relation S T);
41 | cases daemon (*apply (o_relation_pair_of_relation_pair_is_morphism S T)*)]
44 definition BTop_to_OBTop: carr3 (arrows3 CAT2 (category2_of_category1 BTop) OBTop).
46 [ apply o_basic_topology_of_basic_topology;
47 | intros; apply o_continuous_relation_of_continuous_relation_morphism;
48 | cases daemon (*apply o_relation_topology_of_relation_topology_morphism_respects_id*);
49 | cases daemon (*apply o_relation_topology_of_relation_topology_morphism_respects_comp*);]
53 alias symbol "eq" (instance 2) = "setoid1 eq".
54 alias symbol "eq" (instance 1) = "setoid2 eq".
55 theorem BTop_to_OBTop_faithful:
56 ∀S,T.∀f,g:arrows2 (category2_of_category1 BTop) S T.
57 map_arrows2 ?? BTop_to_OBTop ?? f = map_arrows2 ?? BTop_to_OBTop ?? g → f=g.
58 intros; change with (∀b.A ? (ext ?? f b) = A ? (ext ?? g b));
60 apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
62 apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
67 include "formal_topology/notation.ma".
69 theorem BTop_to_OBTop_full:
70 ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BTop_to_OBTop S T g = f).
72 cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
76 | apply hide; intros; lapply (Oreduced ?? f ? e);
77 cases Hg; lapply (e3 U) as K; apply (.= K);
78 apply (.= Hletin); apply rule (†(K^-1));
79 | apply hide; intros; lapply (Osaturated ?? f ? e);
80 cases Hg; lapply (e1 U) as K; apply (.= K);
81 apply (.= Hletin); apply rule (†(K^-1));
83 | simplify; unfold BTop_to_OBTop; simplify;
84 unfold o_continuous_relation_of_continuous_relation_morphism; simplify;
85 cases Hg; whd; simplify; intro;