X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-basic_pairs_to_o-basic_topologies.ma;h=bfd76f99e31ed73bd6869051cceb62fd124d0741;hb=ccac4e720ff2a9bee8e1c9d5ba1ea6474db72572;hp=f2f6af0208c5c914fa25ab41db74f806e7da9804;hpb=8c0ccf03dbefd83818bc3b6849634f422f8ec727;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
index f2f6af020..bfd76f99e 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
@@ -12,145 +12,20 @@
 (*                                                                        *)
 (**************************************************************************)
 
+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;
@@ -162,7 +37,7 @@ definition o_basic_topology_of_o_basic_pair: BP → BTop.
         | 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));
@@ -174,7 +49,7 @@ definition o_basic_topology_of_o_basic_pair: BP → BTop.
         | 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) ?));
@@ -184,44 +59,172 @@ definition o_basic_topology_of_o_basic_pair: BP → BTop.
 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
+
+*)
+