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 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
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).
+