  We consider Horndeski cosmological models, with a minisuperspace Lagrangian
linear in the field derivative, that are able to screen any vacuum energy and
material content leading to a spatially flat de Sitter vacuum fixed by the
theory itself. Furthermore, we investigate particular models with a cosmic
evolution independent of the material content and use them to understand the
general characteristics of this framework. We also consider more realistic
models, which we denote the "term-by-term" and "tripod" models, focusing
attention on cases in which the critical point is indeed an attractor solution
and the cosmological history is of particular interest.
