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 "basic_pairs_to_o-basic_pairs.ma".
16 include "apply_functor.ma".
18 definition rOBP ≝ Apply (category2_of_category1 BP) OBP BP_to_OBP.
20 include "o-basic_pairs_to_o-basic_topologies.ma".
23 ∀s,t:rOBP.∀f:arrows2 OBTop (OR (ℱ_2 s)) (OR (ℱ_2 t)).
24 exT22 ? (λg:arrows2 rOBP s t.
25 map_arrows2 ?? OR ?? (ℳ_2 g) = f).
26 intro; cases s (s_2 s_1 s_eq); clear s;
27 whd in ⊢ (?→? (? ? (? ?? ? %) ?)→?);
28 whd in ⊢ (?→?→? ? (λ_:?.? ? ? (? ? ? (? ? ? (? ? ? ? % ?) ?)) ?));;
29 include "logic/equality.ma".
31 match s_eq in eq return
32 (λright_1:?.(λmatched:(eq (objs2 OBP) (map_objs2 (category2_of_category1 BP) OBP BP_to_OBP s_1) right_1).
34 (∀f:(carr2 (arrows2 OBTop (map_objs2 OBP OBTop OR right_1) (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))).
35 (exT22 (carr2 (arrows2 rOBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched) t))
36 (λg:(carr2 (arrows2 rOBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched) t)).
37 (eq_rel1 (carr1 (unary_morphism1_setoid1 (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched)))) (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))))
38 (eq1 (unary_morphism1_setoid1 (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched)))) (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))))
39 (carr1_OF_Ocontinuous_relation
40 (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP (*XXX*)right_1 s_1 matched)))
41 (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t))
43 (arrows2 OBP (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched)) (F2 (category2_of_category1 BP) OBP BP_to_OBP t))
44 (arrows2 OBTop (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched))) (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))
45 (map_arrows2 OBP OBTop OR right_1 (F2 (category2_of_category1 BP) OBP BP_to_OBP t))
46 (Fm2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP right_1 s_1 matched) t g)))
52 (eq_rel1 (carr1 (unary_morphism1_setoid1 (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched)))) (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))))
53 (eq1 (unary_morphism1_setoid1 (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched)))) (objs2_OF_Obasic_topology (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)))))
54 (carr1_OF_Ocontinuous_relation (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched))) (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)) (fun12 (arrows2 OBP (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched)) (F2 (category2_of_category1 BP) OBP BP_to_OBP t)) (arrows2 OBTop (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched))) (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t))) (map_arrows2 OBP OBTop OR ? (F2 (category2_of_category1 BP) OBP BP_to_OBP t)) (Fm2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched) t g)))
55 (carr1_OF_Ocontinuous_relation (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP (mk_Fo (category2_of_category1 BP) OBP BP_to_OBP ? s_1 matched))) (map_objs2 OBP OBTop OR (F2 (category2_of_category1 BP) OBP BP_to_OBP t)) f)))))))) with
58 cases s_eq; clear s_eq s_2;
59 intro; cases t (t_2 t_1 t_eq); clear t; cases t_eq; clear t_eq t_2;
60 whd in ⊢ (%→?); whd in ⊢ (? (? ? ? ? %) (? ? ? ? %)→?);
61 intro; whd in s_1 t_1;
62 letin R ≝ (? : (carr2 (arrows2 (category2_of_category1 BP) s_1 t_1)));
66 [2: simplify; apply R;
67 | simplify; apply (fun12 ?? (map_arrows2 ?? BP_to_OBP s_1 t_1)); apply R;
68 | simplify; apply rule #; ]]
71 [2: constructor 1; constructor 1;
72 [ intros (x y); apply (y ∈ f (singleton ? x));
73 | apply hide; intros; unfold FunClass_1_OF_Ocontinuous_relation;
74 unfold in ⊢ (? ? ? (? ? ? ? ? ? %) ?); apply (.= e1‡††e);
76 |1: constructor 1; constructor 1;
77 [ intros (x y); apply (y ∈ star_image ?? (⊩ \sub t_1) (f (image ?? (⊩ \sub s_1) (singleton ? x))));
78 | apply hide; intros; unfold FunClass_1_OF_Ocontinuous_relation;
79 unfold in ⊢ (? ? ? (? ? ? ? ? ? (? ? ? ? ? ? %)) ?);
80 apply (.= e1‡(#‡†(#‡†e))); apply rule #; ]
81 | whd; simplify; intros; simplify;
82 whd in ⊢ (? % %); simplify in ⊢ (? % %);
83 lapply (Oreduced ?? f (image (concr s_1) (form s_1) (⊩ \sub s_1) (singleton ? x)));
84 [ whd in Hletin; simplify in Hletin; cases Hletin; clear Hletin;
85 lapply (s y); clear s;
86 whd in Hletin:(? ? ? (? ? (? ? ? % ?)) ?); simplify in Hletin;
87 whd in Hletin; simplify in Hletin;
88 lapply (s1 y); clear s1;
89 split; intros; simplify; whd in f1 ⊢ %; simplify in f1 ⊢ %;
90 cases f1; clear f1; cases x1; clear x1;
99 (* Todo: rename BTop → OBTop *)
101 (* scrivo gli statement qua cosi' verra' un conflitto :-)
103 1. definire il funtore OR
104 2. dimostrare che ORel e' faithful
106 3. Definire la funzione
108 ∀C1,C2: CAT2. F: arrows3 CAT2 C1 C2 → CAT2
111 [ gli oggetti sono gli oggetti di C1 mappati da F
112 | i morfismi i morfismi di C1 mappati da F
116 E : objs CATS === Σx.∃y. F y = x
118 Quindi (Apply C1 C2 F) (che usando da ora in avanti una coercion
119 scrivero' (F C1) ) e' l'immagine di C1 tramite F ed e'
120 una sottocategoria di C2 (qualcosa da dimostare qui??? vedi sotto
123 4. Definire rOBP (le OBP rappresentabili) come (BP_to_OBP BP)
124 [Si puo' fare lo stesso per le OA: rOA ≝ Rel_to_OA REL ]
126 5. Dimostrare che OR (il funtore faithful da OBP a OBTop) e' full
127 quando applicato a rOBP.
128 Nota: puo' darsi che faccia storie ad accettare lo statement.
129 Infatti rOBP e' (BP_to_OBP BP) ed e' "una sottocategoria di OBP"
130 e OR va da OBP a OBTop. Non so se tipa subito o se devi dare
131 una "proiezione" da rOBP a OBP.
133 6. Definire rOBTop come (OBP_to_OBTop rOBP).
135 7. Per composizione si ha un funtore full and faithful da BP a rOBTop:
136 basta prendere (OR ∘ BP_to_OBP).
138 8. Dimostrare (banale: quasi tutti i campi sono per conversione) che
139 esiste un funtore da rOBTop a BTop. Dimostrare che tale funtore e'
140 faithful e full (banale: tutta conversione).
142 9. Per composizione si ha un funtore full and faithful da BP a BTop.
144 10. Dimostrare che i seguenti funtori sono anche isomorphism-dense
145 (http://planetmath.org/encyclopedia/DenseFunctor.html):
148 OBP_to_OBTop quando applicato alle rOBP
149 OBTop_to_BTop quando applicato alle rOBTop
151 Concludere per composizione che anche il funtore da BP a BTop e'
154 ====== Da qui in avanti non e' "necessario" nulla:
156 == altre cose mancanti
158 11. Dimostrare che le r* e le * orrizzontali
159 sono isomorfe dando il funtore da r* a * e dimostrando che componendo i
160 due funtori ottengo l'identita'
162 12. La definizione di r* fa schifo: in pratica dici solo come ottieni
163 qualcosa, ma non come lo caratterizzeresti. Ora un teorema carino
164 e' che una a* (e.g. una aOBP) e' sempre una rOBP dove "a" sta per
165 atomic. Dimostrarlo per tutte le r*.
167 == categorish/future works
169 13. definire astrattamente la FG-completion e usare quella per
170 ottenere le BP da Rel e le OBP da OA.
172 14. indebolire le OA, generalizzare le costruzioni, etc. come detto