Cut-elimination and Redundancy-elimination by Resolution

Cut-elimination is one of the most important techniques in proof theory. Roughly speaking, eliminating cuts from a proof generates a new proof without lemmas, which essentially consists of the syntactic material of the proven theorem. Traditionally cutelimination served the purpose to show consistency of calculi and thus played a central role in metamathematics. In this traditional context the aim is to define just a constructive method for eliminating cuts, its actual use as an algorithm is of minor importance. But in more recent time J.Y. Girard demonstrated that cut-elimination on real mathematical proofs may produce valuable mathematical information (Girard, 1987). In particular he showed, how a proof of van der Waerden’s theorem using concepts of topology can be transformed into an elementary combinatorial proof by means of cut-elimination. Also in many other cases, cut-elimination can be used to construct elementary proofs from nonelementary ones, for example to convert number theoretic proofs using knowledge about the field of complex numbers to proofs by induction only. (C.f. G.Takeuti’s analysis of the potentials of this approach in (Takeuti, 1978)). Another important application is the construction of Herbrand disjunctions from proofs, making information explicit, which is hidden in the mathematical argumentation. H. Luckhardt (Luckhardt, 1989) used this information to obtain bounds for Roth’s theorem, which are of the same level as those obtained by the most advanced number theoretic methods. Finally, it should be