  Vector addition systems (VAS), also known as Petri nets, are a popular model
of concurrent systems. Many problems from many areas reduce to the reachability
problem for VAS, which consists of deciding whether a target configuration of a
VAS is reachable from a given initial configuration. One of the main approaches
to solve the problem on practical instances is called flattening, intuitively
removing nested loops. This technique is known to terminate for semilinear VAS.
In this paper, we prove that also for VAS with nested zero tests, called
Priority VAS, flattening does in fact terminate for all semilinear reachability
relations. Furthermore, we prove that Priority VAS admit semilinear inductive
invariants. Both of these results are obtained by defining a well-quasi-order
on runs of Priority VAS which has good pumping properties.
