  We show how up-to techniques for (bi-)similarity can be used in the setting
of weighted systems. The problems we consider are language equivalence,
language inclusion and the threshold problem (also known as universality
problem) for weighted automata. We build a bisimulation relation on the fly and
work up-to congruence and up-to similarity. This requires to determine whether
a pair of vectors (over a semiring) is in the congruence closure of a given
relation of vectors. This problem is considered for rings and l-monoids, for
the latter we provide a rewriting algorithm and show its confluence and
termination. We then explain how to apply these up-to techniques to weighted
automata and provide runtime results.
