Notation for equality used everywhere.

[helm.git] / helm / matita / library / higher_order_defs / functions.ma

diff --git a/helm/matita/library/higher_order_defs/functions.ma b/helm/matita/library/higher_order_defs/functions.ma
index bdea562cedeff817a299edbe619a69b07084a975..e8fbb6dd9b4b63beee7d9cba581f325e368d6cbf 100644 (file)
--- a/helm/matita/library/higher_order_defs/functions.ma
+++ b/helm/matita/library/higher_order_defs/functions.ma
@@ -19,18 +19,18 @@ include "logic/connectives.ma".
 
 definition injective: \forall A,B:Type.\forall f:A \to B.Prop
 \def \lambda A,B.\lambda f.
-  \forall x,y:A.eq B (f x) (f y) \to (eq A x y).
+  \forall x,y:A.f x = f y \to x=y.
 
 (* we have still to attach exists *)
 definition surjective: \forall A,B:Type.\forall f:A \to B.Prop
 \def \lambda A,B.\lambda f.
-  \forall z:B.ex A (\lambda x:A.(eq B z (f x))).
+  \forall z:B.ex A (\lambda x:A.z=f x).
 
 definition symmetric: \forall A:Type.\forall f:A \to A\to A.Prop
-\def \lambda A.\lambda f.\forall x,y.eq A (f x y) (f y x).
+\def \lambda A.\lambda f.\forall x,y.f x y = f y x.
 
 definition associative: \forall A:Type.\forall f:A \to A\to A.Prop
-\def \lambda A.\lambda f.\forall x,y,z.eq A (f (f x y) z) (f x (f y z)).
+\def \lambda A.\lambda f.\forall x,y,z.f (f x y) z = f x (f y z).
 
 (* functions and relations *)
 definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
@@ -39,6 +39,6 @@ definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
 
 (* functions and functions *)
 definition distributive: \forall A:Type.\forall f,g:A \to A \to A.Prop
-\def \lambda A.\lambda f,g.\forall x,y,z:A.eq A (f x (g y z)) (g (f x y) (f x z)).
+\def \lambda A.\lambda f,g.\forall x,y,z:A. f x (g y z) = g (f x y) (f x z).