SUBSETS_full up to universe inconsistency

[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 ec7db6df4899b3b49de18c8ccb4ab711994e76e9..74a7c7d7839bb26fee44529654fc3cc5ff653bb2 100644 (file)
--- 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).
@@ -44,9 +48,7 @@ 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^-1‡#)‡(#‡e1^-1)); assumption
@@ -96,6 +98,15 @@ definition REL: category1.
           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.
@@ -103,15 +114,6 @@ 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});
  intros; simplify;
@@ -187,7 +189,7 @@ 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^-1)‡#); assumption
      | apply (. (#‡e)‡#); assumption]
@@ -201,7 +203,7 @@ 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); assumption
      | apply (. #‡e ^ -1); assumption]
@@ -212,7 +214,7 @@ 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 ‡#); assumption
      | apply (. e^ -1‡#); assumption]
@@ -224,7 +226,7 @@ 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 ^ -1‡#)‡#); assumption
      | apply (. (e‡#)‡#); assumption]