  We extensively discuss the Rademacher and Sobolev-to-Lipschitz properties for
generalized intrinsic distances on strongly local Dirichlet spaces possibly
without square field operator. We present many non-smooth and
infinite-dimensional examples. As an application, we prove the integral
Varadhan short-time asymptotic with respect to a given distance function for a
large class of strongly local Dirichlet forms.
