X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fo-basic_pairs.ma;h=3ac2f62eec9831d6b9402213309f5743e5826030;hb=071d7a246190074c97e78192839e4bb5d5a1eef4;hp=9bd76ebeb78a45fb6ff3977a6080c1ea013b04f6;hpb=62e9e8296d172d6497f9ad29bad402fbad2014c3;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
index 9bd76ebeb..3ac2f62ee 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
@@ -13,27 +13,25 @@
 (**************************************************************************)
 
 include "o-algebra.ma".
-include "datatypes/categories.ma".
 
-record basic_pair: Type ≝
+record basic_pair: Type2 ≝
  { concr: OA;
    form: OA;
-   rel: arrows1 ? concr form
+   rel: arrows2 ? concr form
  }.
 
-notation > "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y ?}.
-notation < "x (⊩  \below c) y" with precedence 45 for @{'Vdash2 $x $y $c}.
-notation < "⊩ \sub c" with precedence 60 for @{'Vdash $c}.
-notation > "⊩ " with precedence 60 for @{'Vdash ?}.
-
 interpretation "basic pair relation indexed" 'Vdash2 x y c = (rel c x y).
 interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
 
 alias symbol "eq" = "setoid1 eq".
 alias symbol "compose" = "category1 composition".
-record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: arrows1 ? (concr BP1) (concr BP2);
-   form_rel: arrows1 ? (form BP1) (form BP2);
+(*DIFFER*)
+
+alias symbol "eq" = "setoid2 eq".
+alias symbol "compose" = "category2 composition".
+record relation_pair (BP1,BP2: basic_pair): Type2 ≝
+ { concr_rel: arrows2 ? (concr BP1) (concr BP2);
+   form_rel: arrows2 ? (form BP1) (form BP2);
    commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
  }.
 
@@ -44,23 +42,24 @@ interpretation "concrete relation" 'concr_rel r = (concr_rel __ r).
 interpretation "formal relation" 'form_rel r = (form_rel __ r). 
 
 definition relation_pair_equality:
- ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ ∀o1,o2. equivalence_relation2 (relation_pair o1 o2).
  intros;
  constructor 1;
   [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
   | simplify;
     intros;
-    apply refl1;
+    apply refl2;
   | simplify;
     intros 2;
-    apply sym1;
+    apply sym2;
   | simplify;
     intros 3;
-    apply trans1;
+    apply trans2;
   ]      
 qed.
 
-definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
+(* qui setoid1 e' giusto: ma non lo e'!!! *)
+definition relation_pair_setoid: basic_pair → basic_pair → setoid2.
  intros;
  constructor 1;
   [ apply (relation_pair b b1)
@@ -68,27 +67,31 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
   ]
 qed.
 
+definition relation_pair_of_relation_pair_setoid: 
+  ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
+coercion relation_pair_of_relation_pair_setoid.
+
 lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
  intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
- apply (.= (commute ?? r \sup -1));
+ apply (.= ((commute ?? r) \sup -1));
  apply (.= H);
  apply (.= (commute ?? r'));
- apply refl1;
+ apply refl2;
 qed.
 
 
 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
  intro;
  constructor 1;
-  [1,2: apply id1;
-  | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
-    lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
+  [1,2: apply id2;
+  | lapply (id_neutral_right2 ? (concr o) ? (⊩)) as H;
+    lapply (id_neutral_left2 ?? (form o) (⊩)) as H1;
     apply (.= H);
     apply (H1 \sup -1);]
 qed.
 
 definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ ∀o1,o2,o3. binary_morphism2 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
  intros;
  constructor 1;
   [ intros (r r1);
@@ -97,26 +100,26 @@ definition relation_pair_composition:
      | apply (r1 \sub\f ∘ r \sub\f)
      | lapply (commute ?? r) as H;
        lapply (commute ?? r1) as H1;
-       apply (.= ASSOC1);
+       apply rule (.= ASSOC);
        apply (.= #‡H1);
-       apply (.= ASSOC1\sup -1);
+       apply rule (.= ASSOC ^ -1);
        apply (.= H‡#);
-       apply ASSOC1]
+       apply rule ASSOC]
   | intros;
     change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
-    change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
-    change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
-    apply (.= ASSOC1);
-    apply (.= #‡H1);
+    change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+    change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
+    apply rule (.= ASSOC);
+    apply (.= #‡e1);
     apply (.= #‡(commute ?? b'));
-    apply (.= ASSOC1 \sup -1);
-    apply (.= H‡#);
-    apply (.= ASSOC1);
+    apply rule (.= ASSOC \sup -1);
+    apply (.= e‡#);
+    apply rule (.= ASSOC);
     apply (.= #‡(commute ?? b')\sup -1);
-    apply (ASSOC1 \sup -1)]
+    apply rule (ASSOC \sup -1)]
 qed.
     
-definition BP: category1.
+definition BP: category2.
  constructor 1;
   [ apply basic_pair
   | apply relation_pair_setoid
@@ -125,15 +128,21 @@ definition BP: category1.
   | intros;
     change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
                  ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
-    apply (ASSOC1‡#);
+    apply rule (ASSOC‡#);
   | intros;
     change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_right1 ????)‡#);
+    apply ((id_neutral_right2 ????)‡#);
   | intros;
     change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_left1 ????)‡#);]
+    apply ((id_neutral_left2 ????)‡#);]
 qed.
 
+definition basic_pair_of_objs2_BP: objs2 BP → basic_pair ≝ λx.x.
+coercion basic_pair_of_objs2_BP.
+
+definition relation_pair_setoid_of_arrows2_BP: 
+  ∀P,Q.arrows2 BP P Q → relation_pair_setoid P Q ≝ λP,Q,c.c.
+coercion relation_pair_setoid_of_arrows2_BP.
 
 (*
 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
@@ -184,4 +193,20 @@ qed.
 
 interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
 interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
-*)
\ No newline at end of file
+*)
+
+notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
+notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
+interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (rel x)).
+ 
+notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
+notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
+interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (rel x)).
+
+notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
+notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
+interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (rel x)).
+
+notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
+notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (rel x)).
\ No newline at end of file