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[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 c5546477b938831b5cd22363de05a5997b10686e..0734411f8d07192881bcb29516b089959d80f2d8 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -15,42 +15,26 @@
 include "relations.ma".
 include "notation.ma".
 
-record basic_pair: Type1 ≝
- { concr: REL;
-   form: REL;
-   rel: arrows1 ? concr form
- }.
+record basic_pair: Type1 ≝ { 
+   concr: REL; form: REL; rel: concr ⇒_\r1 form
+}.
 
-interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ___ (rel c) x y).
+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".
-record relation_pair (BP1,BP2: basic_pair): Type1 ≝
- { concr_rel: arrows1 ? (concr BP1) (concr BP2);
-   form_rel: arrows1 ? (form BP1) (form BP2);
-   commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
+record relation_pair (BP1,BP2: basic_pair): Type1 ≝ { 
+   concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
+   commute: ⊩ ∘ concr_rel =_1 form_rel ∘ ⊩
  }.
 
+interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r). 
+interpretation "formal relation" 'form_rel r = (form_rel ?? r).
 
-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).
- intros;
- constructor 1;
-  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
-  | simplify;
-    intros;
-    apply refl1;
-  | simplify;
-    intros 2;
-    apply sym1;
-  | simplify;
-    intros 3;
-    apply trans1;
-  ]      
+definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+  | simplify; intros; apply refl1;
+  | simplify; intros 2; apply sym1;
+  | simplify; intros 3; apply trans1; ]      
 qed.
 
 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
@@ -66,7 +50,7 @@ definition relation_pair_of_relation_pair_setoid :
 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 ∘ ⊩.
+  ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 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;
@@ -192,7 +176,7 @@ 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).
+definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
  intros; constructor 1;
   [ apply (ext ? ? (rel o));
   | intros;
@@ -200,13 +184,13 @@ definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
     apply refl1]
 qed.
 
-definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
+definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(concr o).
  intros; constructor 1;
   [ apply (minus_image ?? (rel o));
   | intros; apply (#‡e); ]
 qed.
 
-definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
+definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
  intros (o); constructor 1;
   [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
     intros; simplify; apply (.= (†e)‡#); apply refl1
@@ -215,27 +199,27 @@ definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (f
      | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
-interpretation "fintersects" 'fintersects U V = (fun21 ___ (fintersects _) U V).
+interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
 
 definition fintersectsS:
- ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
+ ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(form o).
  intros (o); constructor 1;
-  [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
+  [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
     intros; simplify; apply (.= (†e)‡#); apply refl1
   | intros; split; simplify; intros;
      [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
      | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
-interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (fintersectsS _) U V).
+interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
 
-definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
+definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
  intros (o); constructor 1;
-  [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩_o y);
+  [ apply (λx:concr o.λS: Ω^(form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
   | intros; split; intros; cases e2; exists [1,3: apply w]
      [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
      | apply (. (#‡e1)‡(e‡#)); assumption]]
 qed.
 
-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)).
+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)).