Note that this is not refinement in the sense usually employed in the formal methods community as it is non-conservative. That is a property which holds when one assumes that the status-status mapping is always true, may cease to hold when the mapping is weakened. To understand the way status-status mappings are weakened, we make the temporal nature of the mapping explicit. Of course, this is precisely the sort of constraint that it is impossible to maintain in reality. Temporal Incoherence Unfortunately, even if sensible properties hold of simple status-status mappings it may not be true for more complex ones.
Cascaded mappings are no trouble.
Assume that x is always positive. Then, if the mappings were always true then z would also always be positive. In this case z may temporarily be negative. That is there is no value of x for which z is a consistent value. In previous work we have shown how the use of status to mediate event transfer leads to delays in interactive and cooperative systems.