A Natural Proof System for Herbrand's Theorem

The reduction of undecidable first-order logic to decidable propositional logic via Herbrand's theorem has long been of interest to theoretical computer science, with the notion of a Herbrand proof motivating the definition of expansion proofs. The problem of building a natural proof system around expansion proofs, with composition of proofs and cut-free completeness, has been approached from a variety of different angles. In this paper we construct a simple deep inference system for first-order logic, KSh2, based around the notion of expansion proofs, as a starting point to developing a rich proof theory around this foundation. Translations between proofs in this system and expansion proofs are given, retaining much of the structure in each direction.