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Modular Focused Proof Systems for Intuitionistic Modal Logics

Focusing is a general technique for syntactically compartmentalizing 
the non-deterministic choices in a proof system, which not only 
improves proof search but also has the representational benefit of 
distilling sequent proofs into synthetic normal forms. However, since 
focusing is usually specified as a restriction of the sequent 
calculus, the technique has not been transferred to logics that lack a 
(shallow) sequent presentation, as is the case for some of the logics 
of the modal cube. We have recently extended the focusing technique 
to classical nested sequents, a generalization of ordinary sequents. 
In this work we further extend focusing to intuitionistic nested 
sequents, which can capture all the logics of the intuitionistic S5 
cube in a modular fashion. We present an internal cut-elimination 
procedure for the focused system which in turn is used to show its 
completeness.