Strong Cut-Elimination Systems for Hudelmaier's Depth-Bounded Sequent Calculus for Implicational Logic

Inspired by the Curry-Howard correspondence, we study normalisation procedures in the depth-bounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cut-admissibility. We decorate proofs with terms and introduce various term-reduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cut-admissibility suggesting weakly normalising procedures for Cut-elimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence.