basic topologies are trivially o-basic topologies

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

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 c5f125ac679dc3d66c704874d27736a87fa55a67..6517689a15696f000993fded2ecf2076aff73dde 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
@@ -20,16 +20,15 @@ record basic_pair: Type2 ≝
    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".
+(*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);
@@ -68,6 +67,10 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid2.
   ]
 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));
@@ -97,23 +100,23 @@ definition relation_pair_composition:
      | apply (r1 \sub\f ∘ r \sub\f)
      | lapply (commute ?? r) as H;
        lapply (commute ?? r1) as H1;
-       apply rule (.= ASSOC1);
+       apply rule (.= ASSOC);
        apply (.= #‡H1);
-       apply rule (.= ASSOC1\sup -1);
+       apply rule (.= ASSOC ^ -1);
        apply (.= H‡#);
-       apply rule ASSOC1]
+       apply rule ASSOC]
   | intros;
     change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
     change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
     change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
-    apply rule (.= ASSOC1);
+    apply rule (.= ASSOC);
     apply (.= #‡e1);
     apply (.= #‡(commute ?? b'));
-    apply rule (.= ASSOC1 \sup -1);
+    apply rule (.= ASSOC \sup -1);
     apply (.= e‡#);
-    apply rule (.= ASSOC1);
+    apply rule (.= ASSOC);
     apply (.= #‡(commute ?? b')\sup -1);
-    apply rule (ASSOC1 \sup -1)]
+    apply rule (ASSOC \sup -1)]
 qed.
     
 definition BP: category2.
@@ -125,7 +128,7 @@ definition BP: category2.
   | intros;
     change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
                  ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
-    apply rule (ASSOC1‡#);
+    apply rule (ASSOC‡#);
   | intros;
     change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
     apply ((id_neutral_right2 ????)‡#);
@@ -134,6 +137,12 @@ definition BP: category2.
     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)).