Up-to techniques for behavioural metrics via fibrations

Abstract Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in compositional way. In order obtain general framework, need systematic way lift functors: show that the Wasserstein lifting functor, introduced previous work, corresponds change base fibrational sense. This observation enables us reuse existing about setting. focus on fibrations predicates relations valued quantale. To illustrate our approach, an example distances regular languages.