  We present a new proof rule for verifying lower bounds on quantities of
probabilistic programs. Our proof rule is not confined to almost-surely
terminating programs -- as is the case for existing rules -- and can be used to
establish non-trivial lower bounds on, e.g., termination probabilities and
expected values, for possibly divergent probabilistic loops, e.g., the
well-known three-dimensional random walk on a lattice.
