(* *)
(**************************************************************************)
+include "notation.ma".
include "o-basic_pairs.ma".
include "o-basic_topologies.ma".
alias symbol "eq" = "setoid1 eq".
(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_o_basic_pair: BP → BTop.
+definition o_basic_topology_of_o_basic_pair: OBP → BTop.
intro t;
constructor 1;
- [ apply (form t);
+ [ apply (Oform t);
| apply (□_t ∘ Ext⎽t);
| apply (◊_t ∘ Rest⎽t);
- | intros 2; split; intro;
+ | apply hide; intros 2; split; intro;
[ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
apply (. (#‡(lemma_10_4_a ?? (⊩) V)^-1));
apply f_minus_star_image_monotone;
| change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
apply (. (or_prop2 : ?) ^ -1);
apply oa_leq_refl; ]]
- | intros 2; split; intro;
+ | apply hide; intros 2; split; intro;
[ change with (◊_t ((⊩) \sup * U) ≤ ◊_t ((⊩) \sup * V));
apply (. ((lemma_10_4_b ?? (⊩) U)^-1)‡#);
apply (f_image_monotone ?? (⊩) ? ((⊩)* V));
| change with ((⊩) ((⊩)* V) ≤ V);
apply (. (or_prop1 : ?));
apply oa_leq_refl; ]]
- | intros;
+ | apply hide; intros;
apply (.= (oa_overlap_sym' : ?));
change with ((◊_t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊_t ((⊩)* V))));
apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
qed.
definition o_continuous_relation_of_o_relation_pair:
- ∀BP1,BP2.arrows2 BP BP1 BP2 →
+ ∀BP1,BP2.arrows2 OBP BP1 BP2 →
arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
intros (BP1 BP2 t);
constructor 1;
[ apply (t \sub \f);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
cut ((t \sub \f ∘ (⊩)) ((⊩)* U) = ((⊩) ∘ t \sub \c) ((⊩)* U)) as COM;[2:
- cases (commute ?? t); apply (e3 ^ -1 ((⊩)* U));]
+ cases (Ocommute ?? t); apply (e3 ^ -1 ((⊩)* U));]
apply (.= †COM);
change in ⊢ (? ? ? % ?) with (((⊩) ∘ (⊩)* ) (((⊩) ∘ t \sub \c ∘ (⊩)* ) U));
apply (.= (lemma_10_3_c ?? (⊩) (t \sub \c ((⊩)* U))));
change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
apply (†e^-1);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩)⎻ U)) as COM;[2:
- cases (commute ?? t); apply (e1 ^ -1 ((⊩)⎻ U));]
+ cases (Ocommute ?? t); apply (e1 ^ -1 ((⊩)⎻ U));]
apply (.= †COM);
change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘ (⊩)⎻ ) (((⊩)⎻* ∘ t \sub \c⎻* ∘ (⊩)⎻ ) U));
apply (.= (lemma_10_3_d ?? (⊩) (t \sub \c⎻* ((⊩)⎻ U))));
change in e with (U=((⊩)⎻* ∘(⊩ \sub BP1)⎻ ) U);
apply (†e^-1);]
qed.
+
+
+definition OR : carr3 (arrows3 CAT2 OBP BTop).
+constructor 1;
+[ apply o_basic_topology_of_o_basic_pair;
+| intros; constructor 1;
+ [ apply o_continuous_relation_of_o_relation_pair;
+ | apply hide;
+ intros; whd; unfold o_continuous_relation_of_o_relation_pair; simplify;;
+ change with ((a \sub \f ⎻* ∘ A (o_basic_topology_of_o_basic_pair S)) =
+ (a' \sub \f ⎻*∘A (o_basic_topology_of_o_basic_pair S)));
+ whd in e; cases e; clear e e2 e3 e4;
+ change in ⊢ (? ? ? (? ? ? ? ? % ?) ?) with ((⊩\sub S)⎻* ∘ (⊩\sub S)⎻);
+ apply (.= (comp_assoc2 ? ???? ?? a\sub\f⎻* ));
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a\sub\f ∘ ⊩\sub S)⎻*;
+ apply (.= #‡†(Ocommute:?)^-1);
+ apply (.= #‡e1);
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (⊩\sub T ∘ a'\sub\c)⎻*;
+ apply (.= #‡†(Ocommute:?));
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a'\sub\f⎻* ∘ (⊩\sub S)⎻* );
+ apply (.= (comp_assoc2 ? ???? ?? a'\sub\f⎻* )^-1);
+ apply refl2;]
+| intros 2 (o a); apply rule #;
+| intros 6; apply refl1;]
+qed.
+
+axiom DDD : False.
+
+definition Fo :=
+ λC1,C2: CAT2.λF:arrows3 CAT2 C1 C2.
+ (exT22 ? (λx:C2.exT22 ? (λy:C1.map_objs2 ?? F y =_\ID x))).
+
+definition sigma_equivalence_relation2:
+ ∀C2:CAT2.∀Q.∀X,Y:exT22 ? (λy:C2.Q y).∀P.
+ equivalence_relation2 (exT22 ? (λf:arrows2 C2 (\fst X) (\fst Y).P f)).
+intros; constructor 1;
+ [ intros(F G); apply (\fst F =_2 \fst G);
+ | intro; apply refl2;
+ | intros 3; apply sym2; assumption;
+ | intros 5; apply (trans2 ?? ??? x1 x2);]
+qed.
+
+lemma REW : ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.∀X,Y:Fo ?? F.
+ arrows2 C2 (F (\fst (\snd X))) (F (\fst (\snd Y))) →
+ arrows2 C2 (\fst X) (\fst Y).
+intros 5; cases X; cases Y; cases x; cases x1; clear X Y x x1;
+cases H; cases H1; intros; assumption;
+qed.
+
+definition Fm :
+ ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.
+ Fo ?? F → Fo ?? F → setoid2.
+intros (C1 C2 F X Y); constructor 1;
+ [ apply (exT22 ? (λf:arrows2 C2 (\fst X) (\fst Y).
+ exT22 ? (λg:arrows2 C1 (\fst (\snd X)) (\fst (\snd Y)).
+ REW ?? F X Y (map_arrows2 ?? F ?? g) = f)));
+ | apply sigma_equivalence_relation2;]
+qed.
+
+definition F_id :
+ ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.∀o.Fm ?? F o o.
+intros; constructor 1;
+ [ apply (id2 C2 (\fst o));
+ | exists[apply (id2 C1 (\fst (\snd o)))]
+ cases o; cases x; cases H; unfold hide; simplify;
+ apply (respects_id2 ?? F);]
+qed.
+
+definition F_comp :
+ ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.∀o1,o2,o3.
+ binary_morphism2 (Fm ?? F o1 o2) (Fm ?? F o2 o3) (Fm ?? F o1 o3).
+intros; constructor 1;
+[ intros (f g); constructor 1;
+ [ apply (comp2 C2 (\fst o1) (\fst o2) (\fst o3) (\fst f) (\fst g));
+ | exists[apply (comp2 C1 (\fst (\snd o1)) (\fst (\snd o2)) (\fst (\snd o3)) (\fst (\snd f)) (\fst (\snd g)))]
+ cases o1 in f; cases o2 in g; cases o3; clear o1 o2 o3;
+ cases x; cases x1; cases x2; clear x x1 x2;
+ cases H; cases H1; cases H2; simplify; intros 2;
+ cases c; cases c1; cases x; cases x1; clear x x1 c c1; simplify;
+ apply (.= (respects_comp2:?));
+ apply (x3‡x2);]
+| (* DISABILITARE INNERTYPES *)
+ STOP
+ cases x3; cases x2;
+ apply refl2;
+ simplify;
+
+definition Apply : ∀C1,C2: CAT2.arrows3 CAT2 C1 C2 → CAT2.
+intros (C1 C2 F);
+constructor 1;
+[ apply (Fo ?? F);
+| apply (Fm ?? F);
+| apply F_id;
+| apply F_comp; intros (o1 o2 o3); constructor 1;
+ [ intros (f g); whd in f g; constructor 1;
+ [ apply (comp2 C2 (\fst o1) (\fst o2) (\fst o3) (\fst f) (\fst g));
+ | exists[apply (comp2 C1 (\fst (\snd o1)) (\fst (\snd o2)) (\fst (\snd o3)) (\fst (\snd f)) (\fst (\snd g)))]
+ cases o1; cases x; cases H;
+
+(* scrivo gli statement qua cosi' verra' un conflitto :-)
+
+1. definire il funtore OR
+2. dimostrare che ORel e' faithful
+
+3. Definire la funzione
+ Apply:
+ \forall C1,C2: CAT2. F: arrows3 CAT2 C1 C2 -> CAT2
+ :=
+ constructor 1;
+ [ gli oggetti sono gli oggetti di C1 mappati da F
+ | i morfismi i morfismi di C1 mappati da F
+ | ....
+ ]
+
+ E : objs CATS === Σx.∃y. F y = x
+
+ Quindi (Apply C1 C2 F) (che usando da ora in avanti una coercion
+ scrivero' (F C1) ) e' l'immagine di C1 tramite F ed e'
+ una sottocategoria di C2 (qualcosa da dimostare qui??? vedi sotto
+ al punto 5)
+
+4. Definire rOBP (le OBP rappresentabili) come (BP_to_OBP BP)
+ [Si puo' fare lo stesso per le OA: rOA := Rel_to_OA REL ]
+
+5. Dimostrare che OR (il funtore faithful da OBP a OBTop) e' full
+ quando applicato a rOBP.
+ Nota: puo' darsi che faccia storie ad accettare lo statement.
+ Infatti rOBP e' (BP_to_OBP BP) ed e' "una sottocategoria di OBP"
+ e OR va da OBP a OBTop. Non so se tipa subito o se devi dare
+ una "proiezione" da rOBP a OBP.
+
+6. Definire rOBTop come (OBP_to_OBTop rOBP).
+
+7. Per composizione si ha un funtore full and faithful da BP a rOBTop:
+ basta prendere (OR \circ BP_to_OBP).
+
+8. Dimostrare (banale: quasi tutti i campi sono per conversione) che
+ esiste un funtore da rOBTop a BTop. Dimostrare che tale funtore e'
+ faithful e full (banale: tutta conversione).
+
+9. Per composizione si ha un funtore full and faithful da BP a BTop.
+
+10. Dimostrare che i seguenti funtori sono anche isomorphism-dense
+ (http://planetmath.org/encyclopedia/DenseFunctor.html):
+
+ BP_to_OBP
+ OBP_to_OBTop quando applicato alle rOBP
+ OBTop_to_BTop quando applicato alle rOBTop
+
+ Concludere per composizione che anche il funtore da BP a BTop e'
+ isomorphism-dense.
+
+====== Da qui in avanti non e' "necessario" nulla:
+
+== altre cose mancanti
+
+11. Dimostrare che le r* e le * orrizzontali
+ sono isomorfe dando il funtore da r* a * e dimostrando che componendo i
+ due funtori ottengo l'identita'
+
+12. La definizione di r* fa schifo: in pratica dici solo come ottieni
+ qualcosa, ma non come lo caratterizzeresti. Ora un teorema carino
+ e' che una a* (e.g. una aOBP) e' sempre una rOBP dove "a" sta per
+ atomic. Dimostrarlo per tutte le r*.
+
+== categorish/future works
+
+13. definire astrattamente la FG-completion e usare quella per
+ ottenere le BP da Rel e le OBP da OA.
+
+14. indebolire le OA, generalizzare le costruzioni, etc. come detto
+ con Giovanni
+
+*)
+