Multicomponent proof-theoretic method for proving interpolation properties

Proof-theoretic method has been successfully used almost from the inception of interpolation properties to provide efficient constructive proofs thereof. Until recently, the method was limited to sequent calculi (and their notational variants), despite the richness of generalizations of sequent structures developed in structural proof theory in the meantime. In this paper, we provide a systematic and uniform account of the recent extension of this proof-theoretic method to hypersequents, nested sequents, and labelled sequents for normal modal logic. The method is presented in terms and notation easily adaptable to other similar formalisms, and interpolant transformations are stated for typical rule types rather than for individual rules.