X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fbasic_pairs.ma;h=c5546477b938831b5cd22363de05a5997b10686e;hb=1654d0ad0d8ba9e03ab403b59fd0893ed8ebd4f6;hp=2635d51cede62ae5eb25d995b516a019f0886f0e;hpb=7db606e36d5c17681a62cf5186bafde65cbfa3db;p=helm.git

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 2635d51ce..c5546477b 100644
--- 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;
@@ -87,11 +88,11 @@ definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
     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).
- intros;
- constructor 1;
-  [ intros (r r1);
+lemma relation_pair_composition: 
+  ∀o1,o2,o3: basic_pair.
+  relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
+intros 3 (o1 o2 o3);
+  intros (r r1);
     constructor 1;
      [ apply (r1 \sub\c ∘ r \sub\c) 
      | apply (r1 \sub\f ∘ r \sub\f)
@@ -105,7 +106,17 @@ definition relation_pair_composition:
        apply (.= ASSOC ^ -1);
        apply (.= H‡#);
        apply ASSOC]
-  | intros;
+qed.
+
+lemma relation_pair_composition_is_morphism:
+  ∀o1,o2,o3: basic_pair.
+  ∀a,a':relation_pair_setoid o1 o2.
+  ∀b,b':relation_pair_setoid o2 o3.
+   a=a' → b=b' →
+    relation_pair_composition o1 o2 o3 a b
+    = relation_pair_composition o1 o2 o3 a' b'.
+intros 3 (o1 o2 o3);
+    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);
@@ -116,29 +127,71 @@ definition relation_pair_composition:
     apply (.= e‡#);
     apply (.= ASSOC);
     apply (.= #‡(commute ?? b')\sup -1);
-    apply (ASSOC ^ -1)]
+    apply (ASSOC ^ -1);
 qed.
-    
-definition BP: category1.
+
+definition relation_pair_composition_morphism:
+ ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ intros;
  constructor 1;
-  [ apply basic_pair
-  | apply relation_pair_setoid
-  | apply id_relation_pair
-  | apply relation_pair_composition
-  | intros;
+  [ apply relation_pair_composition;
+  | apply relation_pair_composition_is_morphism;]
+qed.
+    
+lemma relation_pair_composition_morphism_assoc:
+Πo1:basic_pair
+.Πo2:basic_pair
+ .Πo3:basic_pair
+  .Πo4:basic_pair
+   .Πa12:relation_pair_setoid o1 o2
+    .Πa23:relation_pair_setoid o2 o3
+     .Πa34:relation_pair_setoid o3 o4
+      .relation_pair_composition_morphism o1 o3 o4
+       (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+       =relation_pair_composition_morphism o1 o2 o4 a12
+        (relation_pair_composition_morphism o2 o3 o4 a23 a34).
+   intros;
     change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
                  ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
     alias symbol "refl" = "refl1".
     alias symbol "prop2" = "prop21".
     apply (ASSOC‡#);
-  | intros;
+qed.    
+    
+lemma relation_pair_composition_morphism_respects_id:
+  ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+  relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
+   intros;
     change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_right1 ????)‡#);
-  | intros;
+    apply ((id_neutral_right1 ????)‡#);    
+qed.
+    
+lemma relation_pair_composition_morphism_respects_id_r:
+  ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+  relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.  
+  intros;
     change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_left1 ????)‡#);]
+    apply ((id_neutral_left1 ????)‡#);
 qed.
 
+definition BP: category1.
+ constructor 1;
+  [ apply basic_pair
+  | apply relation_pair_setoid
+  | apply id_relation_pair
+  | apply relation_pair_composition_morphism
+  | apply relation_pair_composition_morphism_assoc;
+  | apply relation_pair_composition_morphism_respects_id;
+  | apply relation_pair_composition_morphism_respects_id_r;]
+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));
@@ -178,11 +231,11 @@ 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: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^-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)).