X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Fbasic_pairs.ma;h=8e5421c1f0b39617a9896c045a7070b64c606061;hb=05cfeb82d2624860e66941421a937f308d66cf33;hp=b13317c67659c3963a9983b9ac3677b54463ad41;hpb=79ce67a7a7502462e827de098b1056516092c0a7;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 b13317c67..8e5421c1f 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -13,60 +13,28 @@
 (**************************************************************************)
 
 include "relations.ma".
+include "notation.ma".
 
-record basic_pair: Type1 ≝
- { concr: REL;
-   form: REL;
-   rel: arrows1 ? concr form
- }.
-
-notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
-notation "⊩" with precedence 60 for @{'Vdash}.
+record basic_pair: Type1 ≝ { 
+   concr: REL; form: REL; rel: concr ⇒_\r1 form
+}.
 
-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".
-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 ∘ ⊩
  }.
 
-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).
+interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r). 
+interpretation "formal relation" 'form_rel r = (form_rel ?? r).
 
-include "o-basic_pairs.ma".
-
-definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
- intro;
- constructor 1;
-  [ apply (SUBSETS (concr b));
-  | apply (SUBSETS (form b));
-  | constructor 1;
-  ]
-qed.
- 
-
-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.
@@ -77,16 +45,17 @@ 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 ∘ ⊩.
- intros 7 (o1 o2 r r' H c1 f2);
- split; intro H1;
-  [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
-    lapply (if ?? (H c1 f2) H2) as H3;
-    apply (if ?? (commute ?? r' c1 f2) H3);
-  | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
-    lapply (fi ?? (H c1 f2) H2) as H3;
-    apply (if ?? (commute ?? r c1 f2) H3);
-  ]
+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 =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+ intros 5 (o1 o2 r r' H);
+ apply (.= (commute ?? r)^-1);
+ change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+ apply rule (.= H);
+ apply (commute ?? r').
 qed.
 
 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
@@ -99,94 +68,156 @@ 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)
      | lapply (commute ?? r) as H;
        lapply (commute ?? r1) as H1;
-       apply (.= ASSOC1);
+       alias symbol "trans" = "trans1".
+       alias symbol "assoc" = "category1 assoc".
+       apply (.= ASSOC);
        apply (.= #‡H1);
-       apply (.= ASSOC1\sup -1);
+       alias symbol "invert" = "setoid1 symmetry".
+       apply (.= ASSOC ^ -1);
        apply (.= H‡#);
-       apply ASSOC1]
-  | intros;
+       apply ASSOC]
+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 H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
-    change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
-    apply (.= ASSOC1);
-    apply (.= #‡H1);
+    change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+    change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
+    apply (.= ASSOC);
+    apply (.= #‡e1);
     apply (.= #‡(commute ?? b'));
-    apply (.= ASSOC1 \sup -1);
-    apply (.= H‡#);
-    apply (.= ASSOC1);
+    apply (.= ASSOC ^ -1);
+    apply (.= e‡#);
+    apply (.= ASSOC);
     apply (.= #‡(commute ?? b')\sup -1);
-    apply (ASSOC1 \sup -1)]
+    apply (ASSOC ^ -1);
+qed.
+
+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 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‡#);
+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 ????)‡#);    
+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 ????)‡#);
+qed.
+
 definition BP: category1.
  constructor 1;
   [ apply basic_pair
   | apply relation_pair_setoid
   | apply id_relation_pair
-  | apply relation_pair_composition
-  | intros;
-    change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
-                 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
-    apply (ASSOC1‡#);
-  | intros;
-    change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_right1 ????)‡#);
-  | intros;
-    change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_left1 ????)‡#);]
+  | 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).
+(*
+definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
  intros; constructor 1;
   [ apply (ext ? ? (rel o));
   | intros;
-    apply (.= #‡H);
+    apply (.= #‡e);
     apply refl1]
 qed.
 
-definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
- λo.extS ?? (rel 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 (.= (†H)‡#); apply refl1
+    intros; simplify; apply (.= (†e)‡#); apply refl1
   | intros; split; simplify; intros;
-     [ apply (. #‡((†H)‡(†H1))); assumption
-     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+     [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+     | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
-interpretation "fintersects" 'fintersects U V = (fun1 ___ (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 });
-    intros; simplify; apply (.= (†H)‡#); apply refl1
+  [ 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 (. #‡((†H)‡(†H1))); assumption
-     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+     [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+     | apply (. #‡((†e)‡(†e1))); assumption]]
 qed.
 
-interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (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 ⊩ y);
-  | intros; split; intros; cases H2; exists [1,3: apply w]
-     [ apply (. (#‡H1)‡(H‡#)); assumption
-     | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
+  [ 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 = (fun1 (concr _) __ (relS _) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (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)).
+*)