Up-to techniques are a well-known method for accelerating proofs of

behavioural equivalences between systems. In this talk, I 
introduce up-to techniques for behavioural metrics in a coalgebraic
setting and provide general results that show under which conditions
such up-to techniques are sound.

For a system modelled as a coalgebra for a certain functor, the
behavioural distance can be seen as a coinductive predicate using a
suitable lifting of the functor. I will focus on the so called
Wasserstein lifting of a functor for which we provide a new
characterization in a fibrational setting. This is useful for
automatically proving the soundness of up-to techniques via a
generic framework developed in a previous CSL-LICS'14 paper.

I will use  fibrations of predicates and relations valued in a
quantale, for which pseudo-metric spaces are an example. To
illustrate our framework I provide an example on distances between
regular languages.