  The null energy condition can be violated stably in generalized Galileon
theories, which gives rise to the possibilities of healthy non-singular
cosmologies. However, it has been reported that in many cases cosmological
solutions are plagued with instabilities or have some pathologies somewhere in
the whole history of the universe. Recently, this was shown to be generically
true in a certain subclass of the Horndeski theory. In this short paper, we
extend this no-go argument to the full Horndeski theory, and show that
non-singular models (with flat spatial sections) in general suffer either from
gradient instabilities or some kind of pathology in the tensor sector. This
implies that one must go beyond the Horndeski theory to implement healthy
non-singular cosmologies.
