set "baseuri" "cic:/matita/higher_order_defs/functions/".
include "logic/equality.ma".
-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. \exists 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 symmetric2: \forall A,B:Type.\forall f:A \to A\to B.Prop
+\def \lambda A,B.\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.
(* 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).
+definition distributive2: \forall A,B:Type.\forall f:A \to B \to B.
+\forall g: B\to B\to B. Prop
+\def \lambda A,B.\lambda f,g.\forall x:A.\forall y,z:B. f x (g y z) = g (f x y) (f x z).