Valentini's cut-elimination for provability logic resolved

In 1983, Valentini presented a syntactic proof of cut-elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for "Valentini". The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are 'hidden' in these set-based proofs of cut-elimination. There is often an underlying assumption that the move to a proof of cut-elimination for sequents built from multisets is easy. Recently, however, it has been claimed that Valentini's arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut-elimination for GL in a multiset setting. Here we refute this claim by placing Valentini's arguments in a formal setting and proving cut-elimination for sequents built from multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini's original proof relies on a novel induction parameter called "width" which is computed 'globally'. It is difficult to verify the correctness of his induction argument based on "width". In our formulation however, verification of the induction argument is straightforward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut-formula is boxed. We deal with this using a new transformation based on Valentini's original arguments. Finally, we show that the algorithm purporting to show the non-termination of Valentini's arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient.