minor fixes

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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
index d0e4ffd4a67607650efe5cbc7127fc1dbc333b2d..84f48c894282c7a1971ed4f52c63d9ab6f0a0a20 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -13,6 +13,7 @@
 (**************************************************************************)
 
 include "relations.ma".
+include "notation.ma".
 
 record basic_pair: Type1 ≝
  { concr: REL;
@@ -20,11 +21,8 @@ record basic_pair: Type1 ≝
    rel: arrows1 ? concr form
  }.
 
-notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
-notation "⊩" with precedence 60 for @{'Vdash}.
-
-interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
-interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
+interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ___ (rel c) x y).
+interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
 
 alias symbol "eq" = "setoid1 eq".
 alias symbol "compose" = "category1 composition".
@@ -34,8 +32,6 @@ record relation_pair (BP1,BP2: basic_pair): Type1 ≝
    commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
  }.
 
-notation "hvbox (r \sub \c)"  with precedence 90 for @{'concr_rel $r}.
-notation "hvbox (r \sub \f)"  with precedence 90 for @{'form_rel $r}.
 
 interpretation "concrete relation" 'concr_rel r = (concr_rel __ r). 
 interpretation "formal relation" 'form_rel r = (form_rel __ r).
@@ -65,7 +61,12 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
   ]
 qed.
 
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+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 7 (o1 o2 r r' H c1 f2);
  split; intro H1;
   [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
@@ -139,6 +140,13 @@ definition BP: category1.
     apply ((id_neutral_left1 ????)‡#);]
 qed.
 
+definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
+coercion basic_pair_of_BP.
+
+definition relation_pair_setoid_of_arrows1_BP :
+  ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
+coercion relation_pair_setoid_of_arrows1_BP.
+
 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
  intros; constructor 1;
   [ apply (ext ? ? (rel o));
@@ -158,8 +166,8 @@ definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (f
   [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
     intros; simplify; apply (.= (†e)‡#); apply refl1
   | intros; split; simplify; intros;
-     [ apply (. #‡((†e)‡(†e1))); assumption
-     | apply (. #‡((†e\sup -1)‡(†e1\sup -1))); assumption]]
+     [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+     | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
 interpretation "fintersects" 'fintersects U V = (fun21 ___ (fintersects _) U V).
@@ -170,19 +178,19 @@ definition fintersectsS:
   [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
     intros; simplify; apply (.= (†e)‡#); apply refl1
   | intros; split; simplify; intros;
-     [ apply (. #‡((†e)‡(†e1))); assumption
-     | apply (. #‡((†e\sup -1)‡(†e1\sup -1))); assumption]]
+     [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+     | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
 interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (fintersectsS _) U V).
 
 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
  intros (o); constructor 1;
-  [ apply (λx:concr o.λS: Ω \sup (form o).∃y:carr (form o).y ∈ S ∧ x ⊩ y);
+  [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩_o y);
   | intros; split; intros; cases e2; exists [1,3: apply w]
-     [ apply (. (#‡e1)‡(e‡#)); assumption
-     | apply (. (#‡e1 \sup -1)‡(e \sup -1‡#)); assumption]]
+     [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
+     | apply (. (#‡e1)‡(e‡#)); assumption]]
 qed.
 
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun21 (concr _) __ (relS _) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun21 ___ (relS _)).
+interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr _) __ (relS c) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ___ (relS c)).