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

Abstract The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs.Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient.In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus, as well the extension of cut elimination to sequent calculus with arbitrary structural rules. The presented proof is semantic, in the sense that is obtained by interpreting sequents in a particular ``universal'' model.This results in a more modular and elegant proof than a standard Gentzen-style cut elimination argument, which can be subtle and error-prone in manual proofs for BI.In particular, our semantic approach avoids unnecessary inversions on proof derivations, or the uses of cut reductions and the multi-cut rule. Our prof is modular and also robust. We demonstrate how our method scales to (i) all extensions of BI with arbitrary structural rules, and (ii) an extension with an S4-like ▯ modality.