--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/higher_order_defs/functions/".
+
+include "logic/equality.ma".
+
+definition compose \def
+ \lambda A,B,C:Type.\lambda f:(B\to C).\lambda g:(A\to B).\lambda x:A.
+ f (g x).
+
+notation "hvbox(a break \circ b)"
+ left associative with precedence 70
+for @{ 'compose $a $b }.
+
+interpretation "function composition" 'compose f g =
+ (cic:/matita/higher_order_defs/functions/compose.con _ _ _ f g).
+
+definition injective: \forall A,B:Type.\forall f:A \to B.Prop
+\def \lambda A,B.\lambda f.
+ \forall x,y:A.f x = f y \to x=y.
+
+definition surjective: \forall A,B:Type.\forall f:A \to B.Prop
+\def \lambda A,B.\lambda f.
+ \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.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.f (f x y) z = f x (f y z).
+
+theorem eq_f_g_h:
+ \forall A,B,C,D:Type.
+ \forall f:C \to D.\forall g:B \to C.\forall h:A \to B.
+ f \circ (g \circ h) = (f \circ g) \circ h.
+ intros.
+ reflexivity.
+qed.
+
+(* functions and relations *)
+definition monotonic : \forall A:Type.\forall R:A \to A \to Prop.
+\forall f:A \to A.Prop \def
+\lambda A. \lambda R. \lambda f. \forall x,y:A.R x y \to R (f x) (f y).
+
+(* 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. 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).
+