(* *)
(**************************************************************************)
+include "notation.ma".
include "o-basic_pairs.ma".
include "o-basic_topologies.ma".
-(* qui la notazione non va *)
-lemma leq_to_eq_join: ∀S:OA.∀p,q:S. p ≤ q → q = binary_join ? p q.
- intros;
- apply oa_leq_antisym;
- [ apply oa_density; intros;
- apply oa_overlap_sym;
- unfold binary_join; simplify;
- apply (. (oa_join_split : ?));
- exists; [ apply false ]
- apply oa_overlap_sym;
- assumption
- | unfold binary_join; simplify;
- apply (. (oa_join_sup : ?)); intro;
- cases i; whd in ⊢ (? ? ? ? ? % ?);
- [ assumption | apply oa_leq_refl ]]
-qed.
-
-lemma overlap_monotone_left: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
- intros;
- apply (. (leq_to_eq_join : ?)‡#);
- [ apply f;
- | skip
- | apply oa_overlap_sym;
- unfold binary_join; simplify;
- apply (. (oa_join_split : ?));
- exists [ apply true ]
- apply oa_overlap_sym;
- assumption; ]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_minus_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻ p ≤ R⎻ q.
- intros;
- apply (. (or_prop2 : ?));
- apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop2 : ?)^ -1); apply oa_leq_refl;]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_minus_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻* p ≤ R⎻* q.
- intros;
- apply (. (or_prop2 : ?)^ -1);
- apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop2 : ?)); apply oa_leq_refl;]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
- intros;
- apply (. (or_prop1 : ?));
- apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop1 : ?)^ -1); apply oa_leq_refl;]
-qed.
-
-(* Part of proposition 9.9 *)
-lemma f_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R* p ≤ R* q.
- intros;
- apply (. (or_prop1 : ?)^ -1);
- apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop1 : ?)); apply oa_leq_refl;]
-qed.
-
-lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
- intros;
- apply (. (or_prop2 : ?)^-1);
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
- intros;
- apply (. (or_prop2 : ?));
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_c: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R* (R p).
- intros;
- apply (. (or_prop1 : ?)^-1);
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_d: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* p) ≤ p.
- intros;
- apply (. (or_prop1 : ?));
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_3_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* (R⎻ p)) = R⎻ p.
- intros; apply oa_leq_antisym;
- [ apply lemma_10_2_b;
- | apply f_minus_image_monotone;
- apply lemma_10_2_a; ]
-qed.
-
-lemma lemma_10_3_b: ∀S,T.∀R:arrows2 OA S T.∀p. R* (R (R* p)) = R* p.
- intros; apply oa_leq_antisym;
- [ apply f_star_image_monotone;
- apply (lemma_10_2_d ?? R p);
- | apply lemma_10_2_c; ]
-qed.
-
-lemma lemma_10_3_c: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R p)) = R p.
- intros; apply oa_leq_antisym;
- [ apply lemma_10_2_d;
- | apply f_image_monotone;
- apply (lemma_10_2_c ?? R p); ]
-qed.
-
-lemma lemma_10_3_d: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
- intros; apply oa_leq_antisym;
- [ apply f_minus_star_image_monotone;
- apply (lemma_10_2_b ?? R p);
- | apply lemma_10_2_a; ]
-qed.
-
-lemma lemma_10_4_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* (R⎻ p))) = R⎻* (R⎻ p).
- intros;
- (* BAD *)
- lapply (†(lemma_10_3_a ?? R p)); [2: apply (R⎻* ); | skip | apply Hletin ]
-qed.
-
-(* VEERY BAD! *)
-axiom lemma_10_4_b: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R (R* p))) = R (R* p).
-(*
- intros;
- (* BAD *)
- lapply (†(lemma_10_3_b ?? R p)); [2: apply rule R; | skip | apply Hletin ]
-qed. *)
-
-lemma oa_overlap_sym': ∀o:OA.∀U,V:o. (U >< V) = (V >< U).
- intros; split; intro; apply oa_overlap_sym; assumption.
-qed.
+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.
- intro;
+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;
+ 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;
- alias symbol "refl" = "refl1".
- apply (.= †?); [1: apply (t \sub \f (((◊_BP1∘(⊩)* ) U))); |
- lapply (†e); [2: apply rule t \sub \f; | skip | apply Hletin]]
- change in ⊢ (? ? ? % ?) with ((◊_BP2 ∘(⊩)* ) ((t \sub \f ∘ (◊_BP1∘(⊩)* )) U));
- lapply (comp_assoc2 ????? (⊩)* (⊩) t \sub \f);
- apply (.= †(Hletin ?)); clear Hletin;
+ apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
- cut ?;
- [3: apply CProp1; |5: cases (commute ?? t); [2: apply (e3 ^ -1 ((⊩)* U));] | 2,4: skip]
- apply (.= †Hcut);
+ cut ((t \sub \f ∘ (⊩)) ((⊩)* U) = ((⊩) ∘ t \sub \c) ((⊩)* U)) as COM;[2:
+ 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))));
- apply (.= Hcut ^ -1);
+ apply (.= COM ^ -1);
change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
- apply (prop11 ?? t \sub \f);
- apply (e ^ -1);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
+ apply (†e^-1);
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
- apply (.= †?); [1: apply (t \sub \f⎻* ((((⊩)⎻* ∘ (⊩)⎻) U))); |
- lapply (†e); [2: apply rule (t \sub \f⎻* ); | skip | apply Hletin]]
- change in ⊢ (? ? ? % ?) with (((⊩)⎻* ∘(⊩)⎻ ) ((t \sub \f⎻* ∘ ((⊩)⎻*∘(⊩)⎻ )) U));
- lapply (comp_assoc2 ????? (⊩)⎻ (⊩)⎻* t \sub \f⎻* );
- apply (.= †(Hletin ?)); clear Hletin;
+ apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
- cut ?;
- [3: apply CProp1; |5: cases (commute ?? t); [2: apply (e1 ^ -1 ((⊩)⎻ U));] | 2,4: skip]
- apply (.= †Hcut);
+ cut ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U) = ((⊩)⎻* ∘ t \sub \c⎻* ) ((⊩)⎻ U)) as COM;[2:
+ 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))));
- apply (.= Hcut ^ -1);
+ apply (.= COM ^ -1);
change in ⊢ (? ? ? % ?) with (t \sub \f⎻* (((⊩)⎻* ∘ (⊩)⎻ ) U));
- apply (prop11 ?? t \sub \f⎻* );
- apply (e ^ -1); ]
-qed.
\ No newline at end of file
+ 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 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.
+
+definition Apply : ∀C1,C2: CAT2.arrows3 CAT2 C1 C2 → CAT2.
+intros (C1 C2 F);
+constructor 1;
+[ apply (exT22 ? (λx:C2.exT22 ? (λy:C1.map_objs2 ?? F y =_\ID x)));
+| intros (X Y); constructor 1;
+ [ apply (exT22 ? (λf:arrows2 C2 (\fst X) (\fst Y).
+ exT22 ? (λg:arrows2 C1 (\fst (\snd X)) (\fst (\snd Y)).
+ ? (map_arrows2 ?? F ?? g) = f)));
+ intro; apply hide; clear g f; cases X in c; cases Y; cases x; cases x1; clear X Y x x1;
+ simplify; cases H; cases H1; intros; assumption;
+ | apply sigma_equivalence_relation2;]
+| intro o; 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);]
+| 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
+
+*)
+
projects / helm.git / blobdiff