X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Frelations.ma;h=ec7db6df4899b3b49de18c8ccb4ab711994e76e9;hb=3e51297756e2c2422db7e35ca03af7123ff2498d;hp=738035be0c0ac25b56dcb57c42687c11fa1face7;hpb=befe31089d1d45360b5b7681556c8a762800b3a2;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 738035be0..ec7db6df4 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -49,8 +49,8 @@ definition composition:
        apply (λs1:carr A.λs3:carr C.∃s2:carr 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) ]
@@ -88,22 +88,20 @@ 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.
  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.
@@ -114,23 +112,27 @@ lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 →
 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;
@@ -187,12 +189,12 @@ definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:carr 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.
 
@@ -201,9 +203,9 @@ definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \s
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr 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.
 
@@ -212,9 +214,9 @@ definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V)
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr 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.
 
@@ -224,12 +226,12 @@ definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V)
   [ 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) });
     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.
 
@@ -289,94 +291,3 @@ 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.
\ No newline at end of file