helm/software/matita/library/formal_topology/basic_topologies_to_o-basic_topologies.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
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  13 (**************************************************************************)
  14
  15 include "formal_topology/basic_topologies.ma".
  16 include "formal_topology/o-basic_topologies.ma".
  17 include "formal_topology/relations_to_o-algebra.ma".
  18
  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) ]
  23 qed.
  24
  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) ]
  30 qed.
  31
  32 axiom daemon: False.
  33
  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)).
  38 intros (S T);
  39    constructor 1;
  40      [ apply (o_continuous_relation_of_continuous_relation S T);
  41      | cases daemon (*apply (o_relation_pair_of_relation_pair_is_morphism S T)*)]
  42 qed.
  43
  44 definition BTop_to_OBTop: carr3 ((category2_of_category1 BTop) ⇒_\c3 OBTop).
  45  constructor 1;
  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*);]
  50 qed.
  51
  52 theorem BTop_to_OBTop_faithful: faithful2 ?? BTop_to_OBTop.
  53  intros 5; apply (continuous_relation_eq_inv' o1 o2 f g); apply e;
  54 qed.
  55
  56 include "formal_topology/notation.ma".
  57
  58 theorem BTop_to_OBTop_full: full2 ?? BTop_to_OBTop.
  59  intros 3 (S T);
  60  cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
  61  (* cases Hg; *)
  62  exists [
  63    constructor 1;
  64     [ apply g
  65     | unfold image_coercion; cases daemon (*apply hide; intros; lapply (Oreduced ?? f ? e); unfold image_coercion;
  66       cases Hg; lapply (e3 U) as K; apply (.= K);
  67       apply (.= Hletin); apply rule (†(K^-1)); *)
  68     | cases daemon (* apply hide; intros; lapply (Osaturated ?? f ? e);
  69       cases Hg; lapply (e1 U) as K; apply (.= K);
  70       apply (.= Hletin); apply rule (†(K^-1)); *)
  71     ]
  72  | simplify; unfold BTop_to_OBTop; simplify;
  73    cases Hg; unfold o_continuous_relation_of_continuous_relation_morphism;
  74    simplify;
  75    change with ((orelation_of_relation ?? g)⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
  76                 f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
  77
  78
  79    change with (g⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
  80                 f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
  81    apply sym2; whd in T;
  82    intro;
  83    apply trans2; [2: apply sym2; [2: apply Hg;
  84
  85    whd in ⊢ (?(??%%)???);
  86     apply (.= Hg^-1);
  87    unfold o_continuous_relation_of_continuous_relation_morphism; simplify;
  88    intro; simplify;
  89    unfold image_coercion; cases Hg; whd; simplify; intro; simplify;
  90 qed.
  91 *)