X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fformal_topology%2Foverlap%2Frelations.ma;h=738035be0c0ac25b56dcb57c42687c11fa1face7;hb=befe31089d1d45360b5b7681556c8a762800b3a2;hp=c99239bebf18d663d0f9eb28c368a4d71a750203;hpb=79ce67a7a7502462e827de098b1056516092c0a7;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 c99239beb..738035be0 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -29,10 +29,10 @@ 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.
 
@@ -48,7 +48,7 @@ definition composition:
        (* carr to avoid universe inconsistency *)  
        apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
      | intros;
-       split; intro; cases H (w H3); clear H; exists; [1,3: apply w ]
+       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]]
   | intros 8; split; intro H2; simplify in H2 ⊢ %;
@@ -87,7 +87,7 @@ 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;
+        [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]
         |2,4: exists; try assumption; split;
@@ -103,11 +103,18 @@ definition full_subset: ∀s:REL. Ω \sup s.
 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.
@@ -174,34 +181,59 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (e
     assumption]
 qed.
 *)
-axiom daemon: False.
+
 (* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) ?(*(Ω \sup V)*).
-cases daemon; qed.
+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 H; exists [1,3: apply w]
+  [ 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]
-  | intros; split; simplify; intros; cases H; exists [1,3: apply w]
+  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
      [ apply (. #‡(#‡e1)); cases x; split; try assumption;
        apply (if ?? (e ??)); assumption
      | apply (. #‡(#‡e1 ^ -1)); 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]
+  [ 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 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});
+    intros; simplify; split; intros; apply f;
+     [ apply (. e ^ -1‡#); assumption
+     | apply (. e‡#); assumption]
+  | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+    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: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]
+  | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+     [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+       apply (if ?? (e ??)); assumption
+     | apply (. #‡(#‡e1 ^ -1)); 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.
@@ -260,15 +292,91 @@ qed.
 
 include "o-algebra.ma".
 
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
+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