X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Frelations.ma;h=74a7c7d7839bb26fee44529654fc3cc5ff653bb2;hb=b1dbb34e1e2388a3987710e128e6f19b7d8fe5fc;hp=c9685f2207f4cf0e93db0d23f2d89fde0fbd2bf7;hpb=81432e2003b9c1514975e006775fe59056e125a4;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 c9685f220..74a7c7d78 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)
@@ -36,6 +36,10 @@ 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).
@@ -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 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) ]
@@ -88,49 +90,51 @@ 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 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.
 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. (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;
@@ -185,36 +189,36 @@ 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:carr U. x ♮r y ∧ x ∈ S });
+  [ 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)‡#); 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).
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
+  [ apply (λr: arrows1 ? U 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).
  intros; constructor 1;
-  [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr V. x ♮r y → y ∈ S});
+  [ 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 ^ -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.
 
@@ -222,14 +226,14 @@ qed.
 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:carr V.x ♮r y ∧ y ∈ 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.
 
@@ -289,121 +293,3 @@ theorem extS_singleton:
   | exists; try assumption; split; try assumption; change with (x = x); apply refl]
 qed.
 *)
-
-include "o-algebra.ma".
-
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
- intros;
- constructor 1;
-  [ constructor 1; 
-     [ apply (λU.image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.minus_star_image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.star_image ?? t U);
-     | intros; apply (#‡e); ]
-  | constructor 1;
-     [ apply (λU.minus_image ?? t U);
-     | intros; apply (#‡e); ]
-  | intros; split; intro;
-     [ change in f with (∀a. a ∈ image ?? t p → a ∈ q);
-       change with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
-       intros 4; apply f; exists; [apply a] split; assumption;
-     | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
-       change with (∀a. a ∈ image ?? t p → a ∈ q);
-       intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
-  | intros; split; intro;
-     [ change in f with (∀a. a ∈ minus_image ?? t p → a ∈ q);
-       change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
-       intros 4; apply f; exists; [apply a] split; assumption;
-     | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
-       change with (∀a. a ∈ minus_image ?? t p → a ∈ q);
-       intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
-  | intros; split; intro; cases f; clear f;
-     [ cases x; cases x2; clear x x2; exists; [apply w1]
-        [ assumption;
-        | exists; [apply w] split; assumption]
-     | cases x1; cases x2; clear x1 x2; exists; [apply w1]
-        [ exists; [apply w] split; assumption;
-        | assumption; ]]]
-qed.
-
-lemma orelation_of_relation_preserves_equality:
- ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. eq1 ? t t' → orelation_of_relation ?? t = orelation_of_relation ?? t'.
- intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
- simplify; whd in o1 o2;
-  [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
-    apply (. #‡(e‡#));
-  | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
-    apply (. #‡(e ^ -1‡#));
-  | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
-    apply (. #‡(e‡#));
-  | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
-    apply (. #‡(e ^ -1‡#));
-  | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
-    apply (. #‡(e‡#));
-  | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
-    apply (. #‡(e ^ -1‡#));
-  | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
-    apply (. #‡(e‡#));
-  | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
-    apply (. #‡(e ^ -1‡#)); ]
-qed.
-
-lemma hint: ∀o1,o2:OA. Type_OF_setoid2 (arrows2 ? o1 o2) → carr2 (arrows2 OA o1 o2).
- intros; apply t;
-qed.
-coercion hint.
-
-lemma orelation_of_relation_preserves_identity:
- ∀o1:REL. orelation_of_relation ?? (id1 ? o1) = id2 OA (SUBSETS o1).
- intros; split; intro; split; whd; intro; 
-  [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
-    apply (f a1); change with (a1 = a1); apply refl1;
-  | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
-    change in f1 with (x = a1); apply (. f1 ^ -1‡#); apply f;
-  | alias symbol "and" = "and_morphism".
-    change with ((∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
-    intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
-    apply (. f^-1‡#); apply f1;
-  | change with (a1 ∈ a → ∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a);
-    intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
-  | change with ((∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
-    intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
-    apply (. f‡#); apply f1;
-  | change with (a1 ∈ a → ∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a);
-    intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
-  | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
-    apply (f a1); change with (a1 = a1); apply refl1;
-  | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
-    change in f1 with (a1 = y); apply (. f1‡#); apply f;]
-qed.
-
-lemma hint2: ∀S,T. carr2 (arrows2 OA S T) → Type_OF_setoid2 (arrows2 OA S T).
- intros; apply c;
-qed.
-coercion hint2.
-
-(* CSC: ???? forse un uncertain mancato *)
-lemma orelation_of_relation_preserves_composition:
- ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
-  orelation_of_relation ?? (G ∘ F) =
-   comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
-    ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
- [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
- intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
-  [ whd; intros; apply f; exists; [ apply x] split; assumption; 
-  | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
-  | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
-    split; [ assumption | exists; [apply w] split; assumption ]
-  | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
-    split; [ exists; [apply w] split; assumption | assumption ]
-  | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
-    split; [ assumption | exists; [apply w] split; assumption ]
-  | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
-    split; [ exists; [apply w] split; assumption | assumption ]
-  | whd; intros; apply f; exists; [ apply y] split; assumption;
-  | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
-qed.
\ No newline at end of file