  Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone
function over a complete lattice as the largest post-fixpoint, naturally leads
to the so-called coinduction proof principle for showing that some element is
below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The
dual principle, used for showing that an element is above the least fixpoint,
is related to inductive invariants. In this paper we provide proof rules which
are similar in spirit but for showing that an element is above the greatest
fixpoint or, dually, below the least fixpoint. The theory is developed for
non-expansive monotone functions on suitable lattices of the form
$\mathbb{M}^Y$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and
it is based on the construction of (finitary) approximations of the original
functions. We show that our theory applies to a wide range of examples,
including termination probabilities, metric transition systems, behavioural
distances for probabilistic automata and bisimilarity. Moreover it allows us to
determine original algorithms for solving simple stochastic games.
