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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "formal_topology/basic_pairs_to_o-basic_pairs.ma".
16 include "formal_topology/apply_functor.ma".
18 definition rOBP ≝ Apply (category2_of_category1 BP) OBP BP_to_OBP.
20 include "formal_topology/o-basic_pairs_to_o-basic_topologies.ma".
22 lemma category2_of_category1_respects_comp_r:
23 ∀C:category1.∀o1,o2,o3:C.
24 ∀f:arrows1 ? o1 o2.∀g:arrows1 ? o2 o3.
25 (comp1 ???? f g) =_\ID (comp2 (category2_of_category1 C) o1 o2 o3 f g).
26 intros; constructor 1;
29 lemma category2_of_category1_respects_comp:
30 ∀C:category1.∀o1,o2,o3:C.
31 ∀f:arrows1 ? o1 o2.∀g:arrows1 ? o2 o3.
32 (comp2 (category2_of_category1 C) o1 o2 o3 f g) =_\ID (comp1 ???? f g).
33 intros; constructor 1;
37 ∀S,T:REL.∀f:arrows2 SET1 (POW S) (POW T).
40 constructor 1; constructor 1;
41 [ intros (x y); apply (y ∈ c {(x)});
42 | apply hide; intros; unfold FunClass_1_OF_Ocontinuous_relation;
48 ∀S,T:REL.∀f:arrows2 SET1 (POW S) (POW T).
49 exT22 ? (λg:arrows1 REL S T.or_f ?? (map_arrows2 ?? POW S T g) = f).
50 intros; letin g ≝ (? : carr1 (arrows1 REL S T)); [
51 constructor 1; constructor 1;
52 [ intros (x y); apply (y ∈ f {(x)});
53 | apply hide; intros; unfold FunClass_1_OF_Ocontinuous_relation;
56 intro; split; intro; simplify; intro;
57 [ whd in f1; change in f1:(? ? (λ_:?.? ? ? ? ? % ?)) with (a1 ∈ f {(x)});
58 cases f1; cases x; clear f1 x; change with (a1 ∈ f a);
59 lapply (f_image_monotone ?? (map_arrows2 ?? POW S T g) (singleton ? w) a ? a1);
61 change in Hletin:(? ? (λ_:?.? ? ? ? ? % ?))
62 with (a1 ∈ f {(x)}); cases Hletin; cases x;
63 [ intros 2; change in f3 with (eq1 ? w a2); change with (a2 ∈ a);
64 apply (. f3^-1‡#); assumption;
69 lapply (. (or_prop3 ?? (map_arrows2 ?? POW S T g) (singleton ? a1) a)^-1);
70 [ whd in Hletin:(? ? ? ? ? ? %);
71 change in Hletin:(? ? ? ? ? ? (? ? (? ? ? (λ_:?.? ? (λ_:?.? ? ? ? ? % ?)) ?)))
73 cases Hletin; cases x1; cases x2;
75 [ cases Hletin; change in x with (eq1 ? a1 w1); apply (. x‡#); assumption;
76 | exists; [apply w] assumption ]
80 cases f1; cases x; simplify in f2; change with (a1 ∈ (f a));
81 lapply depth=0 (let x ≝ POW in or_prop3 (POW S) (POW T) (map_arrows2 ?? POW S T g));
82 lapply (Hletin {(w)} {(a1)}).
83 lapply (if ?? Hletin1); [2: clear Hletin Hletin1;
84 exists; [apply a1] [whd; exists[apply w] split; [assumption;|change with (w = w); apply rule #]]
85 change with (a1=a1); apply rule #;]
86 clear Hletin Hletin1; cases Hletin2; whd in x2;
89 lemma curry: ∀A,B,C.(A × B ⇒_1 C) → A → (B ⇒_1 C).
93 | intros; apply (#‡e); ]
96 notation < "F x" left associative with precedence 65 for @{'map_arrows $F $x}.
97 interpretation "map arrows 2" 'map_arrows F x = (fun12 ? ? (map_arrows2 ? ? F ? ?) x).
99 definition preserve_sup : ∀S,T.∀ f:Ω^S ⇒_1 Ω^T. CProp[1].
100 intros (S T f); apply (∀X:Ω \sup S. (f X) =_1 ?);
101 constructor 1; constructor 1;
102 [ intro y; alias symbol "singl" = "singleton". alias symbol "and" = "and_morphism".
103 apply (∃x:S. x ∈ X ∧ y ∈ f {(x)});
104 | intros (a b H); split; intro E; cases E; clear E; exists; [1,3:apply w]
105 [ apply (. #‡(H^-1‡#)); | apply (. #‡(H‡#));] assumption]
108 alias symbol "singl" = "singleton".
109 lemma eq_cones_to_eq_rel:
110 ∀S,T. ∀f,g: arrows1 REL S T.
111 (∀x. curry ??? (image ??) f {(x)} = curry ??? (image ??) g {(x)}) → f = g.
112 intros; intros 2 (a b); split; intro;
113 [ cases (f1 a); lapply depth=0 (s b); clear s s1;
114 lapply (Hletin); clear Hletin;
115 [ cases Hletin1; cases x; change in f4 with (a = w);
116 change with (a ♮g b); apply (. f4‡#); assumption;
117 | exists; [apply a] split; [ assumption | change with (a=a); apply rule #;]]
118 | cases (f1 a); lapply depth=0 (s1 b); clear s s1;
119 lapply (Hletin); clear Hletin;
120 [ cases Hletin1; cases x; change in f4 with (a = w);
121 change with (a ♮f b); apply (. f4‡#); assumption;
122 | exists; [apply a] split; [ assumption | change with (a=a); apply rule #;]]]
125 variant eq_cones_to_eq_rel':
126 ∀S,T. ∀f,g: arrows1 REL S T.
127 (∀x:S. or_f ?? (map_arrows2 ?? POW S T f) {(x)} = or_f ?? (map_arrows2 ?? POW S T g) {(x)}) →
129 ≝ eq_cones_to_eq_rel.
133 ∀s,t:rOBP.∀f:arrows2 OBTop (OR (ℱ_2 s)) (OR (ℱ_2 t)).
134 exT22 ? (λg:arrows2 rOBP s t.
135 map_arrows2 ?? OR ?? (ℳ_2 g) = f).
136 intros 2 (s t); cases s (s_2 s_1 s_eq); clear s;
137 change in match (F2 ??? (mk_Fo ??????)) with s_2;
138 cases s_eq; clear s_eq s_2;
139 letin s1 ≝ (BP_to_OBP s_1); change in match (BP_to_OBP s_1) with s1;
140 cases t (t_2 t_1 t_eq); clear t;
141 change in match (F2 ??? (mk_Fo ??????)) with t_2;
142 cases t_eq; clear t_eq t_2;
143 letin t1 ≝ (BP_to_OBP t_1); change in match (BP_to_OBP t_1) with t1;
144 whd in ⊢ (%→?); whd in ⊢ (? (? ? ? ? %) (? ? ? ? %)→?);
145 intro; whd in s_1 t_1;
146 letin R ≝ (? : (carr2 (arrows2 (category2_of_category1 BP) s_1 t_1)));
150 [2: simplify; apply R;
151 | simplify; apply (fun12 ?? (map_arrows2 ?? BP_to_OBP s_1 t_1)); apply R;
152 | simplify; apply rule #; ]]
155 [2: apply (pi1exT22 ?? (POW_full (form s_1) (form t_1) f));
156 |1: letin u ≝ (or_f_star ?? (map_arrows2 ?? POW (concr t_1) (form t_1) (⊩ \sub t_1)));
157 letin r ≝ (u ∘ (or_f ?? f));
158 letin xxx ≝ (or_f ?? (map_arrows2 ?? POW (concr s_1) (form s_1) (⊩ \sub s_1)));
159 letin r' ≝ (r ∘ xxx); clearbody r';
160 apply (POW_full' (concr s_1) (concr t_1) r');
161 | simplify in ⊢ (? ? ? (? ? ? ? ? % ?) ?);
162 apply eq_cones_to_eq_rel'; intro;
164 (cic:/matita/logic/equality/eq_elim_r''.con ?????
165 (category2_of_category1_respects_comp_r : ?));
166 apply rule (.= (#‡#));
167 apply (.= (respects_comp2 ?? POW (concr s_1) (concr t_1) (form t_1) ? (⊩\sub t_1))‡#);
169 apply (.= (respects_comp2 ?? POW (concr s_1) (form s_1) (form t_1) (⊩\sub s_1) (pi1exT22 ?? (POW_full (form s_1) (form t_1) (Ocont_rel ?? f)))));
170 apply (let H ≝(\snd (POW_full (form s_1) (form t_1) (Ocont_rel ?? f))) in .= #‡H);
177 (* Todo: rename BTop → OBTop *)
179 (* scrivo gli statement qua cosi' verra' un conflitto :-)
181 1. definire il funtore OR
182 2. dimostrare che ORel e' faithful
184 3. Definire la funzione
186 ∀C1,C2: CAT2. F: arrows3 CAT2 C1 C2 → CAT2
189 [ gli oggetti sono gli oggetti di C1 mappati da F
190 | i morfismi i morfismi di C1 mappati da F
194 E : objs CATS === Σx.∃y. F y = x
196 Quindi (Apply C1 C2 F) (che usando da ora in avanti una coercion
197 scrivero' (F C1) ) e' l'immagine di C1 tramite F ed e'
198 una sottocategoria di C2 (qualcosa da dimostare qui??? vedi sotto
201 4. Definire rOBP (le OBP rappresentabili) come (BP_to_OBP BP)
202 [Si puo' fare lo stesso per le OA: rOA ≝ Rel_to_OA REL ]
204 5. Dimostrare che OR (il funtore faithful da OBP a OBTop) e' full
205 quando applicato a rOBP.
206 Nota: puo' darsi che faccia storie ad accettare lo statement.
207 Infatti rOBP e' (BP_to_OBP BP) ed e' "una sottocategoria di OBP"
208 e OR va da OBP a OBTop. Non so se tipa subito o se devi dare
209 una "proiezione" da rOBP a OBP.
211 6. Definire rOBTop come (OBP_to_OBTop rOBP).
213 7. Per composizione si ha un funtore full and faithful da BP a rOBTop:
214 basta prendere (OR ∘ BP_to_OBP).
216 8. Dimostrare (banale: quasi tutti i campi sono per conversione) che
217 esiste un funtore da rOBTop a BTop. Dimostrare che tale funtore e'
218 faithful e full (banale: tutta conversione).
220 9. Per composizione si ha un funtore full and faithful da BP a BTop.
222 10. Dimostrare che i seguenti funtori sono anche isomorphism-dense
223 (http://planetmath.org/encyclopedia/DenseFunctor.html):
226 OBP_to_OBTop quando applicato alle rOBP
227 OBTop_to_BTop quando applicato alle rOBTop
229 Concludere per composizione che anche il funtore da BP a BTop e'
232 ====== Da qui in avanti non e' "necessario" nulla:
234 == altre cose mancanti
236 11. Dimostrare che le r* e le * orrizzontali
237 sono isomorfe dando il funtore da r* a * e dimostrando che componendo i
238 due funtori ottengo l'identita'
240 12. La definizione di r* fa schifo: in pratica dici solo come ottieni
241 qualcosa, ma non come lo caratterizzeresti. Ora un teorema carino
242 e' che una a* (e.g. una aOBP) e' sempre una rOBP dove "a" sta per
243 atomic. Dimostrarlo per tutte le r*.
245 == categorish/future works
247 13. definire astrattamente la FG-completion e usare quella per
248 ottenere le BP da Rel e le OBP da OA.
250 14. indebolire le OA, generalizzare le costruzioni, etc. come detto