  In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of
a polynomial ring. In the present paper, we reconstruct their theory by using
the "generalized Shestakov-Umirbaev inequality", which was recently given by
the author. As a consequence, we obtain a more precise tameness criterion for
polynomial automorphisms. In particular, we show that no tame automorphism of a
polynomial ring admits a reduction of type IV.
