1) Some reorganization.

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / relations.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index c99239bebf18d663d0f9eb28c368a4d71a750203..dc3c091bbfe49d75ba005997b44ef38b386db4d6 100644 (file)
--- 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.
@@ -257,18 +289,3 @@ 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 → ORelation (SUBSETS o1) (SUBSETS o2).
- intros;
- constructor 1;
-  [ 
-  |
-  |
-  |
-  |
-  |
-  |
-  ]
-qed.
\ No newline at end of file