  Probabilistic transition system specifications (PTSSs) in the ntmufnu/ntmuxnu
format provide structural operational semantics for Segala-type systems that
exhibit both probabilistic and nondeterministic behavior and guarantee that
isimilarity is a congruence.Similar to the nondeterministic case of rule format
tyft/tyxt, we show that the well-foundedness requirement is unnecessary in the
probabilistic setting. To achieve this, we first define an extended version of
the ntmufnu/ntmuxnu format in which quantitative premises and conclusions
include nested convex combinations of distributions. This format also
guarantees that bisimilarity is a congruence. Then, for a given (possibly
non-well-founded) PTSS in the new format, we construct an equivalent
well-founded transition system consisting of only rules of the simpler
(well-founded) probabilistic ntree format. Furthermore, we develop a
proof-theoretic notion for these PTSSs that coincides with the existing
stratification-based meaning in case the PTSS is stratifiable. This continues
the line of research lifting structural operational semantic results from the
nondeterministic setting to systems with both probabilistic and
nondeterministic behavior.
