Structural proof theory as rewriting

Abstract

The multiary version of the $\lambda$-calculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cut-elimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of non-terminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. well-behaved combinations of rules. We identify six of these ``combined'' normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of ``combined'' normal forms is the existence of three ways of expressing multiple application in the calculus.