notation made half decent

[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 5678b6a892bbc75c2f5f97852dd3c820566349e5..eda2cfc6d7550bfb813ad667739d736d4eb77668 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -19,9 +19,9 @@ record binary_relation (A,B: SET) : Type1 ≝
 
 notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
 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).
+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)
@@ -36,9 +36,13 @@ 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).
+  (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
  intros;
  constructor 1;
   [ intros (R12 R23);
@@ -94,34 +98,31 @@ definition REL: category1.
           first [apply refl | assumption]]]
 qed.
 
-definition full_subset: ∀s:REL. Ω \sup s.
+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^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 rule 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});
+alias symbol "arrows1_SET" (instance 2) = "'arrows1_SET low".
+definition comprehension: ∀b:REL. (b ⇒_1 CPROP) → Ω^b.
+ apply (λb:REL. λP: b ⇒_1 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_morphism1 __ p _)).
+ (comprehension s (mk_unary_morphism1 ?? p ?)).
 
-definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
+definition ext: ∀X,S:REL. (arrows1 ? X S) × S ⇒_1 (Ω^X).
  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;
@@ -183,7 +184,7 @@ qed.
 *)
 
 (* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+definition image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^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 e1; exists [1,3: apply w]
@@ -197,7 +198,7 @@ definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \
 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).
+definition minus_star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^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 f;
@@ -208,7 +209,7 @@ definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \s
 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).
+definition star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
     intros; simplify; split; intros; apply f;
@@ -219,7 +220,7 @@ definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V)
 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).
+definition minus_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
  intros; constructor 1;
   [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
       exT ? (λy:V.x ♮r y ∧ y ∈ S) });