On structural proof theory of the modal logic K+ extended with infinitary derivations

We consider an extension of the modal logic of transitive closure K+ with certain infinitary derivations and present a sequent calculus for this extension, which allows non-well-founded proofs. We establish continuous cut-elimination for the given calculus using fixed-point theorems for contractive mappings. The infinitary derivations mentioned above are well founded and countably branching, while the non-well-founded proofs of the sequent calculus can only be finitely branching. The ordinary derivations in K+, as we show additionally, correspond to the non-well-founded proofs of the calculus that are regular and cut-free. Therefore, in this article, we explore the relationship between deductive systems for K+ with well-founded infinitely branching derivations and (regular) non-well-founded finitely branching proofs.