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=275b124e484510dc49141f86b5174f8bd0be7d97;hp=a7be5cc6c5db889bf10e791e477f149e0095182c;hpb=cb98bd7054893edee16aadd6741ec5210b04afbc;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 a7be5cc6c..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;
@@ -31,8 +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).
@@ -89,11 +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)
@@ -107,7 +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);
@@ -118,29 +127,64 @@ 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.
 
@@ -193,5 +237,5 @@ definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
      | 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)).