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 "o-basic_pairs.ma".
16 include "o-basic_topologies.ma".
18 alias symbol "eq" = "setoid1 eq".
20 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
21 definition o_basic_topology_of_o_basic_pair: BP → BTop.
25 | apply (□_t ∘ Ext⎽t);
26 | apply (◊_t ∘ Rest⎽t);
27 | intros 2; split; intro;
28 [ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
29 apply (. (#‡(lemma_10_4_a ?? (⊩) V)^-1));
30 apply f_minus_star_image_monotone;
31 apply f_minus_image_monotone;
36 | change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
37 apply (. (or_prop2 : ?) ^ -1);
39 | intros 2; split; intro;
40 [ change with (◊_t ((⊩) \sup * U) ≤ ◊_t ((⊩) \sup * V));
41 apply (. ((lemma_10_4_b ?? (⊩) U)^-1)‡#);
42 apply (f_image_monotone ?? (⊩) ? ((⊩)* V));
43 apply f_star_image_monotone;
48 | change with ((⊩) ((⊩)* V) ≤ V);
49 apply (. (or_prop1 : ?));
52 apply (.= (oa_overlap_sym' : ?));
53 change with ((◊_t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊_t ((⊩)* V))));
54 apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
55 apply (.= #‡(lemma_10_3_a : ?));
56 apply (.= (or_prop3 : ?)^-1);
57 apply (oa_overlap_sym' ? ((⊩) ((⊩)* V)) U); ]
60 definition o_continuous_relation_of_o_relation_pair:
61 ∀BP1,BP2.arrows2 BP BP1 BP2 →
62 arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
66 | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
69 change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
70 cut ((t \sub \f ∘ (⊩)) ((⊩)* U) = ((⊩) ∘ t \sub \c) ((⊩)* U)) as COM;[2:
71 cases (commute ?? t); apply (e3 ^ -1 ((⊩)* U));]
73 change in ⊢ (? ? ? % ?) with (((⊩) ∘ (⊩)* ) (((⊩) ∘ t \sub \c ∘ (⊩)* ) U));
74 apply (.= (lemma_10_3_c ?? (⊩) (t \sub \c ((⊩)* U))));
76 change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
77 change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
79 | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
82 change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
83 cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩)⎻ U)) as COM;[2:
84 cases (commute ?? t); apply (e1 ^ -1 ((⊩)⎻ U));]
86 change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘ (⊩)⎻ ) (((⊩)⎻* ∘ t \sub \c⎻* ∘ (⊩)⎻ ) U));
87 apply (.= (lemma_10_3_d ?? (⊩) (t \sub \c⎻* ((⊩)⎻ U))));
89 change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩)⎻ ) U));
90 change in e with (U=((⊩)⎻* ∘(⊩ \sub BP1)⎻ ) U);