  Under the assumption of the minimal model theory for projective klt pairs of
dimension $n$, we establish the minimal model theory for lc pairs
$(X/Z,\Delta)$ such that the log canonical divisor is relatively log abundant
and its restriction to any lc center has relative numerical dimension at most
$n$. We also give another detailed proof of results by the second author, and
study termination of log MMP with scaling.
