Embedding formalisms: hypersequents and two-level systems of rules

A system of rules consists of (possibly labelled) sequent rules connected to each other by some variables and subject to the condition of appearing in a certain order in the derivation. The formalism of systems of rules is quite powerful and allows, e.g., the definition of analytic labelled sequent calculi for intermediate and modal logics characterised by frame conditions beyond the geometric fragment. Using propositional intermediate logics as a case study, we show how to use hypersequent calculus derivations to construct derivations using two-level systems of sequent rules and vice versa. Our transformations (embeddings) show that the hypersequent calculus and this proper restriction of systems of rules have the same expressive power.