From 5ede839a0cf3339568202750b4aae85ccc63fcb0 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Wed, 30 Jun 2010 09:44:22 +0000
Subject: [PATCH] ....

---
 .../formal_topology/overlap/basic_pairs.ma    | 40 ++++++-------------
 .../formal_topology/overlap/relations.ma      | 32 +++++++++------
 2 files changed, 31 insertions(+), 41 deletions(-)

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
index 9d4cbbed0..0734411f8 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -15,42 +15,26 @@
 include "relations.ma".
 include "notation.ma".
 
-record basic_pair: Type1 ≝
- { concr: REL;
-   form: REL;
-   rel: arrows1 ? concr form
- }.
+record basic_pair: Type1 ≝ { 
+   concr: REL; form: REL; rel: concr ⇒_\r1 form
+}.
 
 interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ??? (rel c) x y).
 interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
 
-alias symbol "eq" = "setoid1 eq".
-alias symbol "compose" = "category1 composition".
-record relation_pair (BP1,BP2: basic_pair): Type1 ≝
- { concr_rel: arrows1 ? (concr BP1) (concr BP2);
-   form_rel: arrows1 ? (form BP1) (form BP2);
-   commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
+record relation_pair (BP1,BP2: basic_pair): Type1 ≝ { 
+   concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
+   commute: ⊩ ∘ concr_rel =_1 form_rel ∘ ⊩
  }.
 
-
 interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r). 
 interpretation "formal relation" 'form_rel r = (form_rel ?? r).
 
-definition relation_pair_equality:
- ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
- intros;
- constructor 1;
-  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
-  | simplify;
-    intros;
-    apply refl1;
-  | simplify;
-    intros 2;
-    apply sym1;
-  | simplify;
-    intros 3;
-    apply trans1;
-  ]      
+definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+  | simplify; intros; apply refl1;
+  | simplify; intros 2; apply sym1;
+  | simplify; intros 3; apply trans1; ]      
 qed.
 
 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
@@ -66,7 +50,7 @@ definition relation_pair_of_relation_pair_setoid :
 coercion relation_pair_of_relation_pair_setoid.
 
 lemma eq_to_eq': 
-  ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+  ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
  intros 7 (o1 o2 r r' H c1 f2);
  split; intro H1;
   [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index eda2cfc6d..f7827939d 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -105,6 +105,12 @@ 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.
 
+
+notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
+
+
 definition full_subset: ∀s:REL. Ω^s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
@@ -112,8 +118,7 @@ qed.
 
 coercion full_subset.
 
-alias symbol "arrows1_SET" (instance 2) = "'arrows1_SET low".
-definition comprehension: ∀b:REL. (b ⇒_1 CPROP) → Ω^b.
+definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
  apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
  intros; simplify;
  apply (.= †e); apply refl1.
@@ -122,15 +127,16 @@ qed.
 interpretation "subset comprehension" 'comprehension s p =
  (comprehension s (mk_unary_morphism1 ?? p ?)).
 
-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
+definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
+ intros (X S); constructor 1; 
+  [ apply (λr:X ⇒_\r1 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});*)
@@ -184,9 +190,9 @@ qed.
 *)
 
 (* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
+definition image: ∀U,V:REL. (U ⇒_\r1 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 });
+  [ apply (λr:U ⇒_\r1 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]
@@ -198,9 +204,9 @@ definition image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
 qed.
 
 (* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 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});
+  [ apply (λr:U ⇒_\r1 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]
@@ -209,9 +215,9 @@ definition minus_star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
 qed.
 
 (* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
+definition star_image: ∀U,V:REL. (U ⇒_\r1 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});
+  [ apply (λr:U ⇒_\r1 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]
@@ -220,9 +226,9 @@ definition star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
 qed.
 
 (* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 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*)
+  [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
       exT ? (λy:V.x ♮r y ∧ y ∈ S) });
     intros; simplify; split; intro; cases e1; exists [1,3: apply w]
      [ apply (. (e ^ -1‡#)‡#); assumption
-- 
2.39.2