  For sub-Riemannian manifolds with a chosen complement, we first establish the
derivative formula and integration by parts formula on path space with respect
to a natural gradient operator. By using these formulae, we then show that
upper and lower bounds of the horizontal Ricci curvature correspond to
functional inequalities on path space analogous to what has been established in
Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev
and Poincar\'e inequalities.
