A fully labelled proof system for intuitionistic modal logics

International audienceLabelled proof theory has been famously successful for modal logics by mimicking their relational seman-tics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionisticmodal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequentsystem for intuitionistic modal logics such that there is not only one, but two relation symbols appearingin sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logicsand one for the preorder relation associated with the Kripke semantics for intuitionistic logic. This putsour system in close correspondence with the standard birelational Kripke semantics for intuitionistic modallogics. As a consequence it can be extended with arbitrary intuitionistic Scott-Lemmon axioms. We showsoundness and completeness, together with an internal cut elimination proof, encompassing a wider array ofintuitionistic modal logics than any existing labelled system