Elementary translation of cut proofs between epsilon calculus and first-order sequent calculi

Determine whether sequent-calculus derivations with cuts formulated in the epsilon calculus (the epsilon-format sequent calculus defined in Section 3, where strong quantifier inferences are replaced by substitutions using epsilon terms) can be translated into derivations with cuts in a standard first-order sequent calculus with only an elementary increase in proof size (i.e., bounded by an elementary function of the original derivation length).

Background

The paper establishes a non-elementary speed-up of cut-free proofs when using the epsilon calculus compared to usual first-order sequent calculi, relying on the extended first epsilon theorem and known lower bounds (Statman, Orevkov). It presents sequences of sequents whose cut-free epsilon proofs are polynomially bounded, while the corresponding cut-free first-order sequent proofs are not bounded by any elementary function.

After discussing translations and bounds for cut-free derivations, the authors raise the question of whether an elementary translation can be achieved for derivations that allow cuts, seeking to understand if the advantage of epsilon calculus persists or can be bridged when cuts are permitted.