X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Frelations.ma;h=b1589a827fbc50717a08be48ec3fc2f2adf3eae2;hb=82b1a205fdf9bc2c8029296ebe94c5667798845b;hp=c4502f3d080b61874360cfb4e650bde1f3b99694;hpb=49045bfd9b3038ce30a1911e2345f949ed38ec8a;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 c4502f3d0..b1589a827 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -19,9 +19,9 @@ record binary_relation (A,B: SET) : Type1 ≝
 
 notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
 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).
+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)
@@ -36,21 +36,23 @@ definition binary_relation_setoid: SET → SET → SET1.
        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).
+  (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
  intros;
  constructor 1;
   [ 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 e2 (w H3); clear e2; exists; [1,3: apply w ]
-        [ apply (. (e‡#)‡(#‡e1)); assumption
-        | apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
+        [ 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) ]
@@ -88,50 +90,53 @@ definition REL: category1.
   |6,7: intros 5; unfold composition; simplify; split; intro;
         unfold setoid1_of_setoid in x y; simplify in x y;
         [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
-          [ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
-          | apply (. #‡(e : eq1 ? w y)); assumption]
+          [ 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 full_subset: ∀s:REL. Ω \sup s.
+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.
+
+
+notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
+
+
+definition full_subset: ∀s:REL. Ω^s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
 qed.
 
 coercion full_subset.
 
-definition setoid1_of_REL: REL → setoid ≝ λS. S.
-coercion setoid1_of_REL.
-
-lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
- [ apply (setoid1_of_SET o1);
- | intros; apply t;]
-qed.
-coercion Type_OF_setoid1_of_REL.
-
-definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
+definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
+ apply (λb:REL. λP: b ⇒_1 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_morphism1 __ 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 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
-  [ intros; simplify; apply (.= (e‡#)); apply refl1
+definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
+ intros (X S); constructor 1; 
+  [ apply (λr:X ⇒_\r1 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)); whd in e; apply (if ?? (e ??)); assumption
+       apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
      | change with (∀x. x ♮a' b' → x ♮a b); intros;
-       apply (. (#‡e1\sup -1)); whd in e; apply (fi ?? (e ??));assumption]]
+       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});*)
@@ -185,75 +190,75 @@ qed.
 *)
 
 (* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:carr U. x ♮r y ∧ x ∈ S });
+  [ apply (λr:U ⇒_\r1 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)‡#); assumption
-     | apply (. (#‡e ^ -1)‡#); assumption]
+     [ apply (. (#‡e^-1)‡#); assumption
+     | apply (. (#‡e)‡#); assumption]
   | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
-     [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+     [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
        apply (if ?? (e ??)); assumption
-     | apply (. #‡(#‡e1 ^ -1)); cases x; split; try 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).
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
+  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
     intros; simplify; split; intros; apply f;
-     [ apply (. #‡e ^ -1); assumption
-     | apply (. #‡e); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+     [ 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).
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr V. x ♮r y → y ∈ S});
+  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
     intros; simplify; split; intros; apply f;
-     [ apply (. e ^ -1‡#); assumption
-     | apply (. e‡#); assumption]
-  | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+     [ 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).
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
-      exT ? (λy:carr V.x ♮r y ∧ y ∈ S) });
+  [ apply (λr:U ⇒_\r1 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‡#)‡#); assumption
-     | apply (. (e ^ -1‡#)‡#); assumption]
+     [ apply (. (e ^ -1‡#)‡#); assumption
+     | apply (. (e‡#)‡#); assumption]
   | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
-     [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+     [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
        apply (if ?? (e ??)); assumption
-     | apply (. #‡(#‡e1 ^ -1)); cases x; split; try 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.
  intros; unfold image; simplify; split; simplify; intros;
   [ change with (a ∈ U);
-    cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
+    cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
   | change in f with (a ∈ U);
-    exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
+    exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
 qed.
 
 theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
  intros; unfold minus_star_image; simplify; split; simplify; intros;
-  [ change with (a ∈ U); apply H; change with (a=a); apply refl
-  | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
+  [ change with (a ∈ U); apply f; change with (a=a); apply refl1
+  | change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
 qed.
 
+alias symbol "compose" (instance 2) = "category1 composition".
 theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
+ intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
  clear x; [ cases f; clear f; | cases f1; clear f1 ]
  exists; try assumption; cases x; clear x; split; try assumption;
  exists; try assumption; split; assumption.
@@ -263,10 +268,11 @@ theorem minus_star_image_comp:
  ∀A,B,C,r,s,X.
   minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
  intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
-  [ apply H; exists; try assumption; split; assumption
-  | change with (x ∈ X); cases f; cases x1; apply H; assumption]
+  [ apply f; exists; try assumption; split; assumption
+  | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
 qed.
 
+(*
 (*CSC: unused! *)
 theorem ext_comp:
  ∀o1,o2,o3: REL.