Cut-Elimination in Schematic Proofs and Herbrand Sequents

In a recent paper [8], a procedure was developed extending the first-order CERES method [4] so that it can handle cut-elimination in a schematic first-order calculus. The goal of this work was to circumvent the problems reductive cut elimination methods face when the LK calculus is extended by an induction rule. The schematic calculus can be considered a replacement for certain types of induction. In this work, we used the schematic CERES method to analyse a proof formalized in a schematic sequent calculus. The statement being proved is a simple mathematical statement about total functions with a finite range. The goal of proof analysis using the first-order CERES method [4] has been to produce an ACNF (Atomic Cut Normal Form) as the final output of cut-elimination. However, due to the complexity of the schematic method, the value and usefulness of an ACNF quickly vanishes; it is not easily parsable by humans. The Herbrand sequent corresponding to an ACNF turned out to be a valuable, compact and informative structure, which may be considered the essence of a cut-free proof in first-order logic [10]. We provide a method for extracting a schematic Herbrand sequent from the formalized proof and hint at how ,in future work we can generalize the procedure to handle a class of proofs by a suitable schematic language and calculus, and not just for a particular instance.