  We present a canonical formalism of the $f(R)$-type gravity using the Lie
derivatives instead of the time derivatives by refining the formalism of our
group. The previous formalism is a direct generalization of the Ostrogradski's
formalism. However the use of the Lie derivatives was not sufficient in that
Lie derivatives and time derivatives are used in a mixed way, so that the
expressions are somewhat complicated. In this paper, we use the Lie derivatives
and foliation structure of the spacetime thoroughly, which makes the procedure
and the expressions far more concise.
