On weak zariski decompositions and termination of flips

We prove that assuming the existence ofweak Zariski decompositions, the termination of flips for n-dimensional pseudo-effective generalized pairs is implied by the termination of flips for n􀀀1-dimensional generalized pairs. Moreover, using the existence of minimal models for pseudo-effective 4-folds and classic techniques for the termination of 3-fold flips, we prove that any sequence of pseudo-effective 4-fold flips terminates.