Aspects of the Cut-Elimination Theorem

I give a proof of the cut-elimination theorem (Gentzen's Hauptsatz ) for an intuitionistic multi-succedent calculus. The proof follows the strategy of eliminating topmost maximal-rank cuts that allows for a straightforward way to measure the upper bound of the increase of derivations during the procedure. The elimination of all cut inferences generates a superexponential increase. I follow the structure of the proof for classical logic given in Švejdar's [18], modifying only the critical cases related to two restricted rules. Motivated by the diversity found in the early literature on this topic, I survey selected aspects of various formulations of sequent calculi. These are reflected in the proof of the Hauptsatz and its preliminaries. In the end I give one corollary of cut elimination, the Midsequent theorem, which is one of the three applications to be found already in Gentzen's [10].