some notation for map_arrows2

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / relations_to_o-algebra.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
index 9bcd51c7c0ece3c3acab8416844cf523e3f3df16..8bf57da98bfccfb58e524cd2e8219ae34d79965d 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
@@ -51,7 +51,7 @@ qed.
 definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω^A ≝ λA,x.x.
 coercion powerset_of_POW'.
 
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (POW' o1) (POW' o2).
+definition orelation_of_relation: ∀o1,o2:REL. o1 ⇒_\r1 o2 → (POW' o1) ⇒_\o2 (POW' o2).
  intros;
  constructor 1;
   [ constructor 1; 
@@ -90,7 +90,7 @@ definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (
 qed.
 
 lemma orelation_of_relation_preserves_equality:
- ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. 
+ ∀o1,o2:REL.∀t,t': o1 ⇒_\r1 o2. 
    t = t' → orelation_of_relation ?? t =_2 orelation_of_relation ?? t'.
  intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
  simplify; whd in o1 o2;
@@ -140,7 +140,7 @@ qed.
 alias symbol "eq" = "setoid2 eq".
 alias symbol "compose" = "category1 composition".
 lemma orelation_of_relation_preserves_composition:
- ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
+ ∀o1,o2,o3:REL.∀F: o1 ⇒_\r1 o2.∀G: o2 ⇒_\r1 o3.
   orelation_of_relation ?? (G ∘ F) = 
   comp2 OA ??? (orelation_of_relation ?? F) (orelation_of_relation ?? G).
  intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
@@ -172,7 +172,7 @@ qed.
 
 theorem POW_faithful:
  ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
-  map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f = g.
+   POW⎽⇒ f =_2 POW⎽⇒ g → f =_2 g.
  intros; unfold POW in e; simplify in e; cases e;
  unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
  intros 2; cases (e3 {(x)}); 
@@ -186,13 +186,7 @@ lemma currify: ∀A,B,C. (A × B ⇒_1 C) → A → (B ⇒_1 C).
 intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
 qed.
 
-(*
-interpretation "lifting singl" 'singl x = 
- (fun11 ? (objs2 (POW ?))  (singleton ?) x).
-*)
-
-
-theorem POW_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? POW S T g = f).
+theorem POW_full: ∀S,T.∀f: (POW S) ⇒_\o2 (POW T) . exT22 ? (λg. POW⎽⇒ g = f).
  intros; exists;
   [ constructor 1; constructor 1;
      [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});