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The logic of bunched implications (BI) is a substructural logic that formsthe backbone of separation logic, the much studied logic for reasoning aboutheap-manipulating programs. Although the proof theory and metatheory of BI aremathematically involved, the formalization of important metatheoretical resultsis still incipient. In this paper we present a self-contained formalized, inthe Coq proof assistant, proof of a central metatheoretical property of BI: cutelimination for its sequent calculus.  The presented proof is *semantic*, in the sense that is obtained byinterpreting sequents in a particular "universal" model. This results in a moremodular and elegant proof than a standard Gentzen-style cut eliminationargument, which can be subtle and error-prone in manual proofs for BI. Inparticular, our semantic approach avoids unnecessary inversions on proofderivations, or the uses of cut reductions and the multi-cut rule.  Besides modular, our approach is also robust: we demonstrate how our methodscales, with minor modifications, to (i) an extension of BI with an arbitraryset of \emph{simple structural rules}, and (ii) an extension with an S4-like$\Box$ modality.

Semantic Cut Elimination for the Logic of Bunched Implications, Formalized in Coq