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=3e2fe8f354ee116ad0e851d568893eaa14182b2a;hb=e78cf74f8976cf0ca554f64baa9979d0423ee927;hp=7dbd3a80ae6bd0c9bbe9b6c687e9404dda6420d8;hpb=13114a0147a28f8c7359c9c19ee254716eb5f55a;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 7dbd3a80a..3e2fe8f35 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
@@ -20,9 +20,9 @@ include "relations_to_o-algebra.ma".
 definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
  intro b;
  constructor 1;
-  [ apply (map_objs2 ?? SUBSETS' (concr b));
-  | apply (map_objs2 ?? SUBSETS' (form b));
-  | apply (map_arrows2 ?? SUBSETS' (concr b) (form b) (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:
@@ -30,21 +30,22 @@ definition o_relation_pair_of_relation_pair:
   Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
  intros;
  constructor 1;
-  [ apply (map_arrows2 ?? SUBSETS' (concr BP1) (concr BP2) (r \sub \c));
-  | apply (map_arrows2 ?? SUBSETS' (form BP1) (form BP2) (r \sub \f));
-  | apply (.= (respects_comp2 ?? SUBSETS' (concr BP1) (concr BP2) (form BP2)  r\sub\c (⊩\sub BP2) )^-1);
-    cut (⊩ \sub BP2∘r \sub \c = r\sub\f ∘ ⊩ \sub BP1) as H;
+  [ 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 ?? SUBSETS' (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+      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;
+lemma o_relation_pair_of_relation_pair_is_morphism : 
+  ∀S,T:category2_of_category1 BP.    
+  ∀a,b:arrows2 (category2_of_category1 BP) S T.a=b → 
+   (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T))) 
+    (o_relation_pair_of_relation_pair S T a) (o_relation_pair_of_relation_pair S T b).
+intros 2 (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);
@@ -52,14 +53,29 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
        | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
        simplify;
        apply (prop12);
-       apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
+       apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
        apply sym2;
-       apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
+       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; 
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism : 
+  ∀S,T:category2_of_category1 BP.
+  unary_morphism2 (arrows2 (category2_of_category1 BP) S T)
+   (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)).
+intros (S T);
+   constructor 1;
+     [ apply (o_relation_pair_of_relation_pair S T);
+     | apply (o_relation_pair_of_relation_pair_is_morphism S T)]
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism_respects_id:
+ ∀o:category2_of_category1 BP.
+  o_relation_pair_of_relation_pair_morphism o o (id2 (category2_of_category1 BP) o)
+  = id2 OBP (o_basic_pair_of_basic_pair 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);
@@ -67,8 +83,19 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
     simplify;
     apply prop12;
     apply prop22;[2,4,6,8: apply rule #;]
-    apply (respects_id2 ?? SUBSETS' (concr o)); 
-  | simplify; intros; whd; split;
+    apply (respects_id2 ?? POW (concr o));
+qed. 
+
+lemma o_relation_pair_of_relation_pair_morphism_respects_comp:
+  ∀o1,o2,o3:category2_of_category1 BP.
+  ∀f1:arrows2 (category2_of_category1 BP) o1 o2.
+  ∀f2:arrows2 (category2_of_category1 BP) o2 o3.
+  (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair o1) (o_basic_pair_of_basic_pair o3)))
+    (o_relation_pair_of_relation_pair_morphism o1 o3 (f2 ∘ f1))
+    (comp2 OBP ???
+      (o_relation_pair_of_relation_pair_morphism o1 o2 f1)
+      (o_relation_pair_of_relation_pair_morphism o2 o3 f2)).
+   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);
@@ -76,26 +103,46 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
     simplify;
     apply prop12;
     apply prop22;[2,4,6,8: apply rule #;]
-    apply (respects_comp2 ?? SUBSETS' (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);]     
+    apply (respects_comp2 ?? POW (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);
 qed.
 
+definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
+ constructor 1;
+  [ apply o_basic_pair_of_basic_pair;
+  | intros; apply o_relation_pair_of_relation_pair_morphism;
+  | apply o_relation_pair_of_relation_pair_morphism_respects_id;
+  | apply o_relation_pair_of_relation_pair_morphism_respects_comp;]
+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; unfold BP_to_OBP in e; simplify in e; cases e;
- unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
- intros 2; change in match or_f_ in e3 with (λq,w,x.fun12 ?? (or_f q w) x);
- simplify in e3; STOP lapply (e3 (singleton ? x)); cases Hletin;
- split; intro; [ lapply (s y); | lapply (s1 y); ]
-  [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
-  |*: cases Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+ 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 SUBSETS_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
- intros; exists;
- 
-*)
\ No newline at end of file
+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.