Cut-Elimination for Full Intuitionistic Linear Logic

Abstract

We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logic
introduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignment
system to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending the
system to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut elimination
theorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure for
cut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut
elimination proofs, is a little involved and we have not seen it published anywhere.