Proof Fragments, Cut-Elimination and Cut-Introduction

Cut-elimination is usually presented as a set of local proof reduction steps together with a terminating strategy thus showing the existence of cut-free proofs for all provable sequents. Viewing cut-elimination as a transformation of mathematical proofs, not only the existence but also the structure and content of the cut-free proofs deserves investigation. In this paper we use proof skeletons to describe the abstract structure of a proof and the changes it undergoes during cut-elimination. We show that a proof can be split up into several minimal fragments which themselves are not modified but merely rearranged and instantiated. This result allows to characterize a certain kind of redundancy whose presence is necessary for a cut-free proof to allow compression by introduction of cuts. We formulate the cut-introduction problem in terms of a variant of Kolmogorov complexity and prove a lower bound based on this characterization.