Cut-elimination: Experiments with CERES

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts. In this paper we present a systematic experiment with the implementation of CERES on a proof of reasonable size and complexity. It turns out that the proof with cuts can be transformed into two mathematically different proofs of the theorem. In particular, the application of positive and negative hyperresolution yield different mathematical arguments. As an unexpected side-effect the derived clauses of the resolution refutation proved particularly interesting as they can be considered as meaningful universal lemmas. Though the proof under investigation is intuitively simple, the experiment demonstrates that new (and relevant) mathematical information on proofs can be obtained by computational methods. It can be considered as a first step in the development of an experimental culture of computer-aided proof analysis in mathematics.