- new notation.ma file with local and common notation

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / basic_pairs_to_o-basic_pairs.ma

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 2dd4147ae592f6568a644d06c713eebef6a63e7c..7dbd3a80ae6bd0c9bbe9b6c687e9404dda6420d8 100644 (file)
--- 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
@@ -17,42 +17,85 @@ 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 ?? SUBSETS' (concr b));
+  | apply (map_objs2 ?? SUBSETS' (form b));
+  | apply (map_arrows2 ?? SUBSETS' (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 ?? 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 (.= †H);
+      apply (respects_comp2 ?? SUBSETS' (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+    | apply commute;]]
 qed.
 
-(*
-definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 cic:/matita/formal_topology/basic_pairs/BP.con) BP).
+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; split; unfold o_relation_pair_of_relation_pair; simplify;
-       unfold o_basic_pair_of_basic_pair; simplify; ]
-  | simplify; intros; whd; split; unfold o_relation_pair_of_relation_pair; simplify;
-       unfold o_basic_pair_of_basic_pair;
-simplify in ⊢ (? ? ? (? % ? ?) ?);
-simplify in ⊢ (? ? ? (? ? % ?) ?);
-simplify in ⊢ (? ? ? ? (? % ? ?));
-simplify in ⊢ (? ? ? ? (? ? % ?));
-  | simplify; intros; whd; split;unfold o_relation_pair_of_relation_pair; simplify;
-       unfold o_basic_pair_of_basic_pair;simplify;
+     | 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 ?? SUBSETS' (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 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 ?? SUBSETS' (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 ?? SUBSETS' (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; 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;]
+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