  We present a systematic approach to logical predicates based on universal
coalgebra and higher-order abstract GSOS, thus making a first step towards a
unifying theory of logical relations. We first observe that logical predicates
are special cases of coalgebraic invariants on mixed-variance functors. We then
introduce the notion of a locally maximal logical refinement of a given
predicate, with a view to enabling inductive reasoning, and identify sufficient
conditions on the overall setup in which locally maximal logical refinements
canonically exist. Finally, we develop induction-up-to techniques that simplify
inductive proofs via logical predicates on systems encoded as (certain classes
of) higher-order GSOS laws by identifying and abstracting away from their
boiler-plate part.
