From befe31089d1d45360b5b7681556c8a762800b3a2 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Tue, 6 Jan 2009 13:13:23 +0000
Subject: [PATCH] orelation_of_relation preserves equality and identities.

---
 .../formal_topology/overlap/relations.ma      | 62 ++++++++++++++++++-
 1 file changed, 60 insertions(+), 2 deletions(-)

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index ed8b83bcb..738035be0 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -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.
@@ -285,7 +292,7 @@ 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; 
@@ -322,3 +329,54 @@ definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (S
         [ 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
-- 
2.39.2