A finite alternation result for reversible boolean circuits

We say that a reversible boolean function on n bits has alternation depth d if it can be written as the sequential composition of d reversible boolean functions, each of which acts only on the top n−1 bits or on the bottom n−1 bits. Moreover, if the functions on n−1 bits are even, we speak of even alternation depth. We show that every even reversible boolean function of n⩾4 bits has alternation depth at most 9 and even alternation depth at most 13.