X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Fbasic_pairs.ma;h=0d51724de7114ad3eb7c2d3936856a13933ed313;hb=430d6307ae5776ed000a78358a2881cb88936c37;hp=5cc67a250878233200b20a6eed63f55b0f4915e7;hpb=be9826d87207e8dcf6eb152bd54417b5a9e80ab9;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/basic_pairs.ma b/helm/software/matita/library/formal_topology/basic_pairs.ma
index 5cc67a250..0d51724de 100644
--- a/helm/software/matita/library/formal_topology/basic_pairs.ma
+++ b/helm/software/matita/library/formal_topology/basic_pairs.ma
@@ -12,16 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/subsets.ma".
-include "logic/cprop_connectives.ma".
-include "formal_topology/categories.ma".
+include "formal_topology/relations.ma".
+include "datatypes/categories.ma".
 
 record basic_pair: Type ≝
- { carr1: Type;
-   carr2: Type;
-   concr: Ω \sup carr1;
-   form: Ω \sup carr2;
-   rel: binary_relation ?? concr form
+ { concr: REL;
+   form: REL;
+   rel: arrows1 ? concr form
  }.
 
 notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
@@ -30,12 +27,10 @@ notation "⊩" with precedence 60 for @{'Vdash}.
 interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
 interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
 
-alias symbol "eq" = "equal relation".
-alias symbol "compose" = "binary relation composition".
 record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: binary_relation ?? (concr BP1) (concr BP2);
-   form_rel: binary_relation ?? (form BP1) (form BP2);
-   commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+ { concr_rel: arrows1 ? (concr BP1) (concr BP2);
+   form_rel: arrows1 ? (form BP1) (form BP2);
+   commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
  }.
 
 notation "hvbox (r \sub \c)"  with precedence 90 for @{'concr_rel $r}.
@@ -44,27 +39,24 @@ 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). 
 
-
 definition relation_pair_equality:
- ∀o1,o2. equivalence_relation (relation_pair o1 o2).
+ ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
  intros;
  constructor 1;
-  [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
   | simplify;
     intros;
-    apply refl_equal_relations;
+    apply refl1;
   | simplify;
-    intros;
-    apply sym_equal_relations;
-    assumption
+    intros 2;
+    apply sym1;
   | simplify;
-    intros;
-    apply (trans_equal_relations ??????? H);
-    assumption
+    intros 3;
+    apply trans1;
   ]      
 qed.
 
-definition relation_pair_setoid: basic_pair → basic_pair → setoid.
+definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
  intros;
  constructor 1;
   [ apply (relation_pair b b1)
@@ -72,13 +64,9 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid.
   ]
 qed.
 
-definition eq' ≝
- λo1,o2.λr,r':relation_pair o1 o2.⊩ ∘ r \sub\f = ⊩ ∘ r' \sub\f.
-
-alias symbol "eq" = "setoid eq".
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → eq' ?? r r'.
+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;
+ 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);
@@ -87,97 +75,105 @@ lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → eq' ?? r
     apply (if ?? (commute ?? r c1 f2) H3);
   ]
 qed.
-   
 
-definition id: ∀o:basic_pair. relation_pair o o.
+definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
  intro;
  constructor 1;
-  [1,2: constructor 1;
-    intros;
-    apply (s=s1)
-  | simplify; intros;
-    split;
-    intro;
-    cases H;
-    cases H1; clear H H1;
-     [ exists [ apply y ]
-       split
-        [ rewrite > H2; assumption
-        | reflexivity ]
-     | exists [ apply x ]
-       split
-        [2: rewrite < H3; assumption
-        | reflexivity ]]]
+  [1,2: apply id1;
+  | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+    lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
+    apply (.= H);
+    apply (H1 \sup -1);]
 qed.
 
 definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ ∀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);
     constructor 1;
-     [ apply (r \sub\c ∘ r1 \sub\c) 
-     | apply (r \sub\f ∘ r1 \sub\f)
+     [ 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 (equal_morphism ???? (r\sub\c ∘ (r1\sub\c ∘ ⊩)) ? ((⊩ ∘ r\sub\f) ∘ r1\sub\f));
-        [1,2: apply associative_composition]
-       apply (equal_morphism ???? (r\sub\c ∘ (⊩ ∘ r1\sub\f)) ? ((r\sub\c ∘ ⊩) ∘ r1\sub\f));
-        [1,2: apply composition_morphism; first [assumption | apply refl_equal_relations]
-        | apply sym_equal_relations;
-          apply associative_composition
-        ]]
+       apply (.= ASSOC1);
+       apply (.= #‡H1);
+       apply (.= ASSOC1\sup -1);
+       apply (.= H‡#);
+       apply ASSOC1]
   | intros;
-    alias symbol "eq" = "equal relation".
-    change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
-    apply (equal_morphism ???? (a\sub\c ∘ (b\sub\c ∘ ⊩)) ? (a'\sub\c ∘ (b'\sub\c ∘ ⊩)));
-     [ apply associative_composition
-     | apply sym_equal_relations; apply associative_composition]
-    apply (equal_morphism ???? (a\sub\c ∘ (b'\sub\c ∘ ⊩)) ? (a' \sub \c∘(b' \sub \c∘⊩)));
-     [2: apply refl_equal_relations;
-     |1: apply composition_morphism;
-          [ apply refl_equal_relations
-          | assumption]]
-    apply (equal_morphism ???? (a\sub\c ∘ (⊩ ∘ b'\sub\f)) ? (a'\sub\c ∘ (⊩ ∘ b'\sub\f)));
-     [1,2: apply composition_morphism;
-       [1,3: apply refl_equal_relations
-       | apply (commute ?? b');
-       | apply sym_equal_relations; apply (commute ?? b');]]
-    apply (equal_morphism ???? ((a\sub\c ∘ ⊩) ∘ b'\sub\f) ? ((a'\sub\c ∘ ⊩) ∘ b'\sub\f));
-     [2: apply associative_composition
-     |1: apply sym_equal_relations; apply associative_composition]
-    apply composition_morphism;
-     [ assumption
-     | apply refl_equal_relations]]
+    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);
+    apply (.= #‡(commute ?? b'));
+    apply (.= ASSOC1 \sup -1);
+    apply (.= H‡#);
+    apply (.= ASSOC1);
+    apply (.= #‡(commute ?? b')\sup -1);
+    apply (ASSOC1 \sup -1)]
 qed.
-
-definition BP: category.
+    
+definition BP: category1.
  constructor 1;
   [ apply basic_pair
   | apply relation_pair_setoid
-  | apply id
+  | apply id_relation_pair
   | apply relation_pair_composition
   | intros;
-    change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
-                 (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    apply associative_composition 
+    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 ((id o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    intros 2; unfold id; simplify;
-    split; intro;
-     [ cases H; cases H1; rewrite > H2; assumption
-     | exists; [assumption] split; [reflexivity| assumption]]
+    change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_right1 ????)‡#);
   | intros;
-    change with (a\sub\c ∘ (id o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    intros 2; unfold id; simplify;
-    split; intro;
-     [ cases H; cases H1; rewrite < H3; assumption
-     | exists; [assumption] split; [assumption|reflexivity]]
-  ]
+    change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_left1 ????)‡#);]
+qed.
+
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+  [ apply (ext ? ? (rel o));
+  | intros;
+    apply (.= #‡H);
+    apply refl1]
 qed.
+
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (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; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
+
+definition fintersectsS:
+ ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (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
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (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);
+  | intros; split; intros; cases H2; exists [1,3: apply w]
+     [ apply (. (#‡H1)‡(H‡#)); assumption
+     | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); 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 _)).