  In the context of formal verification in general and model checking in
particular, parity games serve as a mighty vehicle: many problems are encoded
as parity games, which are then solved by the seminal algorithm by Jurdzinski.
In this paper we identify the essence of this workflow to be the notion of
progress measure, and formalize it in general, possibly infinitary,
lattice-theoretic terms. Our view on progress measures is that they are to
nested/alternating fixed points what invariants are to safety/greatest fixed
points, and what ranking functions are to liveness/least fixed points. That is,
progress measures are combination of the latter two notions (invariant and
ranking function) that have been extensively studied in the context of
(program) verification.
  We then apply our theory of progress measures to a general model-checking
framework, where systems are categorically presented as coalgebras. The
framework's theoretical robustness is witnessed by a smooth transfer from the
branching-time setting to the linear-time one. Although the framework can be
used to derive some decision procedures for finite settings, we also expect the
proposed framework to form a basis for sound proof methods for some
undecidable/infinitary problems.
