Normal forms in function fields

We consider function fields of functions of one variable augmented by the binary operation of composition of functions. It is shown that the straightforward axiomatization of this concept allows the introduction of a normal form for expressions denoting elements in such fields. While the description of this normal form seems relatively intuitive, it is surprisingly difficult to prove this fact. We present an algorithm for the normalization of expressions, formulated in the symbolic computer algebra language mathematica. This allows us to effectively decide compositional identities in such fields. Examples are given.