X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Frelations.ma;h=307797566dce8ea8280697246208f44e943a6fbf;hb=711b00a770c30056019a2b7204903e7fb5981940;hp=88af2926367242ba18372fd91060b8af9e81fe8b;hpb=32d2bb73b2ed863c988c61ce9d15404bb9d800ad;p=helm.git

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index 88af29263..307797566 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -21,7 +21,7 @@ notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{
 notation > "hvbox (x \natur term 90 r y)"  with precedence 45 for @{'satisfy $r $x $y}.
 interpretation "relation applied" 'satisfy r x y = (fun21 ___ (satisfy __ r) x y).
 
-definition binary_relation_setoid: SET → SET → SET1.
+definition binary_relation_setoid: SET → SET → setoid1.
  intros (A B);
  constructor 1;
   [ apply (binary_relation A B)
@@ -29,13 +29,17 @@ definition binary_relation_setoid: SET → SET → SET1.
      [ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
      | simplify; intros 3; split; intro; assumption
      | simplify; intros 5; split; intro;
-       [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption
+       [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
      | simplify;  intros 7; split; intro;
-        [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
-        [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+        [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
+        [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
        assumption]]
 qed.
 
+definition binary_relation_of_binary_relation_setoid : 
+  ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
+coercion binary_relation_of_binary_relation_setoid.
+
 definition composition:
  ∀A,B,C.
   binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
@@ -44,13 +48,11 @@ definition composition:
   [ intros (R12 R23);
     constructor 1;
     constructor 1;
-     [ alias symbol "and" = "and_morphism".
-       (* carr to avoid universe inconsistency *)  
-       apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+     [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
      | intros;
-       split; intro; cases H (w H3); clear H; exists; [1,3: apply w ]
-        [ apply (. (e‡#)‡(#‡e1)); assumption
-        | apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
+       split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
+        [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
+        | apply (. (e‡#)‡(#‡e1)); assumption]]
   | intros 8; split; intro H2; simplify in H2 ⊢ %;
     cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
     [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
@@ -87,15 +89,22 @@ definition REL: category1.
     split; assumption
   |6,7: intros 5; unfold composition; simplify; split; intro;
         unfold setoid1_of_setoid in x y; simplify in x y;
-        [1,3: cases H (w H1); clear H; cases H1; clear H1; unfold;
-          [ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
-          | apply (. #‡(e : eq1 ? w y)); assumption]
+        [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
+          [ apply (. (e : eq1 ? x w)‡#); assumption
+          | apply (. #‡(e : eq1 ? w y)^-1); assumption]
         |2,4: exists; try assumption; split;
           (* change required to avoid universe inconsistency *)
           change in x with (carr o1); change in y with (carr o2);
           first [apply refl | assumption]]]
 qed.
 
+definition setoid_of_REL : objs1 REL → setoid ≝ λx.x.
+coercion setoid_of_REL.
+
+definition binary_relation_setoid_of_arrow1_REL : 
+  ∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
+coercion binary_relation_setoid_of_arrow1_REL.
+
 definition full_subset: ∀s:REL. Ω \sup s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
@@ -103,26 +112,27 @@ qed.
 
 coercion full_subset.
 
-definition setoid1_of_REL: REL → setoid ≝ λS. S.
-
-coercion setoid1_of_REL.
-
-definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
- intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
+definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
+ intros; simplify;
+ alias symbol "trans" = "trans1".
+ alias symbol "prop1" = "prop11".
+ apply (.= †e); apply refl1.
 qed.
 
 interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
+ (comprehension s (mk_unary_morphism1 __ p _)).
 
 definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
- apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
-  [ intros; simplify; apply (.= (H‡#)); apply refl1
-  | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
-     [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
-     | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
+  [ intros; simplify; apply (.= (e‡#)); apply refl1
+  | intros; simplify; split; intros; simplify;
+     [ change with (∀x. x ♮a b → x ♮a' b'); intros;
+       apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
+     | change with (∀x. x ♮a' b' → x ♮a b); intros;
+       apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
 qed.
-
+(*
 definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
  (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
  intros (X S r); constructor 1;
@@ -172,32 +182,60 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (e
     cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
     assumption]
 qed.
+*)
 
 (* the same as ⋄ for a basic pair *)
 definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
-    intros; simplify; split; intro; cases H1; exists [1,3: apply w]
-     [ apply (. (#‡H)‡#); assumption
-     | apply (. (#‡H \sup -1)‡#); assumption]
-  | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
-     [ apply (. #‡(#‡H1)); cases x; split; try assumption;
-       apply (if ?? (H ??)); assumption
-     | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
-       apply (if ?? (H \sup -1 ??)); assumption]]
+  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+    intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+     [ apply (. (#‡e^-1)‡#); assumption
+     | apply (. (#‡e)‡#); assumption]
+  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+     [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
+       apply (if ?? (e ??)); assumption
+     | apply (. #‡(#‡e1)); cases x; split; try assumption;
+       apply (if ?? (e ^ -1 ??)); assumption]]
 qed.
 
 (* the same as □ for a basic pair *)
 definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
-    intros; simplify; split; intros; apply H1;
-     [ apply (. #‡H \sup -1); assumption
-     | apply (. #‡H); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
-    apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
+    intros; simplify; split; intros; apply f;
+     [ apply (. #‡e); assumption
+     | apply (. #‡e ^ -1); assumption]
+  | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
+    apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
 qed.
 
+(* the same as Rest for a basic pair *)
+definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+  [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+    intros; simplify; split; intros; apply f;
+     [ apply (. e ‡#); assumption
+     | apply (. e^ -1‡#); assumption]
+  | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
+    apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+  [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+      exT ? (λy:V.x ♮r y ∧ y ∈ S) });
+    intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+     [ apply (. (e ^ -1‡#)‡#); assumption
+     | apply (. (e‡#)‡#); assumption]
+  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+     [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+       apply (if ?? (e ??)); assumption
+     | apply (. #‡(#‡e1)); cases x; split; try assumption;
+       apply (if ?? (e ^ -1 ??)); assumption]]
+qed.
+
+(*
 (* minus_image is the same as ext *)
 
 theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
@@ -251,4 +289,5 @@ theorem extS_singleton:
   [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
     assumption
   | exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
\ No newline at end of file
+qed.
+*)