  We consider the almost-sure (a.s.) termination problem for probabilistic
programs, which are a stochastic extension of classical imperative programs.
Lexicographic ranking functions provide a sound and practical approach for
termination of non-probabilistic programs, and their extension to probabilistic
programs is achieved via lexicographic ranking supermartingales (LexRSMs).
However, LexRSMs introduced in the previous work have a limitation that impedes
their automation: all of their components have to be non-negative in all
reachable states. This might result in LexRSM not existing even for simple
terminating programs. Our contributions are twofold: First, we introduce a
generalization of LexRSMs which allows for some components to be negative. This
standard feature of non-probabilistic termination proofs was hitherto not known
to be sound in the probabilistic setting, as the soundness proof requires a
careful analysis of the underlying stochastic process. Second, we present
polynomial-time algorithms using our generalized LexRSMs for proving a.s.
termination in broad classes of linear-arithmetic programs.
