X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fbasic_pairs_to_o-basic_pairs.ma;h=d8813fdc1b1154c37458065c9f548d4ddddb3fea;hb=6ca487df4f361fd6b0a3b0734396bc6a62b520c3;hp=7fc6a07f46f0fc2f3c0bb43786a039dfc52b6d3e;hpb=2857d1c432f073379552e1572235a86509b665a4;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
index 7fc6a07f4..d8813fdc1 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
@@ -17,24 +17,96 @@ include "o-basic_pairs.ma".
 include "relations_to_o-algebra.ma".
 
 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
- intro;
+definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
+ intro b;
  constructor 1;
-  [ apply (SUBSETS (concr b));
-  | apply (SUBSETS (form b));
-  | apply (orelation_of_relation ?? (rel b)); ]
+  [ apply (map_objs2 ?? POW (concr b));
+  | apply (map_objs2 ?? POW (form b));
+  | apply (map_arrows2 ?? POW (concr b) (form b) (rel b)); ]
 qed.
 
 definition o_relation_pair_of_relation_pair:
- ∀BP1,BP2.cic:/matita/formal_topology/basic_pairs/relation_pair.ind#xpointer(1/1) BP1 BP2 →
-  relation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
+ ∀BP1,BP2. relation_pair BP1 BP2 →
+  Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
  intros;
  constructor 1;
-  [ apply (orelation_of_relation ?? (r \sub \c));
-  | apply (orelation_of_relation ?? (r \sub \f));
-  | lapply (commute ?? r);
-    lapply (orelation_of_relation_preserves_equality ???? Hletin);
-    apply (.= (orelation_of_relation_preserves_composition (concr BP1) ??? (rel BP2)) ^ -1);
-    apply (.= (orelation_of_relation_preserves_equality ???? (commute ?? r)));
-    apply (orelation_of_relation_preserves_composition ?? (form BP2)  (rel BP1) ?); ]
+  [ apply (map_arrows2 ?? POW (concr BP1) (concr BP2) (r \sub \c));
+  | apply (map_arrows2 ?? POW (form BP1) (form BP2) (r \sub \f));
+  | apply (.= (respects_comp2 ?? POW (concr BP1) (concr BP2) (form BP2)  r\sub\c (⊩\sub BP2) )^-1);
+    cut ( ⊩ \sub BP2 ∘ r \sub \c =_12 r\sub\f ∘ ⊩ \sub BP1) as H;
+    [ apply (.= †H);
+      apply (respects_comp2 ?? POW (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+    | apply commute;]]
 qed.
+
+definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
+ constructor 1;
+  [ apply o_basic_pair_of_basic_pair;
+  | intros; constructor 1;
+     [ apply (o_relation_pair_of_relation_pair S T);
+     | intros (a b Eab); split; unfold o_relation_pair_of_relation_pair; simplify;
+       unfold o_basic_pair_of_basic_pair; simplify;
+       [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
+       | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+       | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+       | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+       simplify;
+       apply (prop12);
+       apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
+       apply sym2;
+       apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
+       apply sym2;
+       apply prop12;
+       apply Eab;
+     ]
+  | simplify; intros; whd; split; 
+       [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
+       | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+       | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+       | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+    simplify;
+    apply prop12;
+    apply prop22;[2,4,6,8: apply rule #;]
+    apply (respects_id2 ?? POW (concr o)); 
+  | simplify; intros; whd; split;
+       [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
+       | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+       | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+       | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+    simplify;
+    apply prop12;
+    apply prop22;[2,4,6,8: apply rule #;]
+    apply (respects_comp2 ?? POW (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);]     
+qed.
+
+theorem BP_to_OBP_faithful:
+ ∀S,T.∀f,g:arrows2 (category2_of_category1 BP) S T.
+  map_arrows2 ?? BP_to_OBP ?? f = map_arrows2 ?? BP_to_OBP ?? g → f=g.
+ intros; change with ( (⊩) ∘ f \sub \c = (⊩) ∘ g \sub \c);
+ apply (POW_faithful);
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
+ apply sym2;
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
+ apply sym2;
+ apply e;
+qed.
+
+theorem BP_to_OBP_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
+ intros; 
+ cases (POW_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
+ cases (POW_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
+ exists[
+   constructor 1; [apply gc|apply gf]
+   apply (POW_faithful);
+   apply (let xxxx ≝POW in .= respects_comp2 ?? POW (concr S) (concr T) (form T) gc (rel T));
+   apply rule (.= Hgc‡#);
+   apply (.= Ocommute ?? f);
+   apply (.= #‡Hgf^-1);
+   apply (let xxxx ≝POW in (respects_comp2 ?? POW (concr S) (form S) (form T) (rel S) gf)^-1)]
+ split;
+  [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
+  | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+  | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+  | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+ simplify; apply (†(Hgc‡#));
+qed.