We introduce a method to associate calculi of proof terms and rewrite rules with cut
elimination procedures for logical deduction systems (i.e., Gentzen-style sequent calculi) in
the case of intuitionistic logic. We illustrate this method using two different versions of the
cut rule for a variant of the intuitionistic fragment of Kleene's logical deduction system G3.
Our systems are in fact calculi of explicit substitution, where the cut rule introduces an
explicit substitution and the left-→ rule introduces a binding of the result of a function
application. Cut propagation steps of cut elimination correspond to propagation of explicit
substitutions, and propagation of weakening (to eliminate it) corresponds to propagation of
index-updating operations. We prove various subject reduction, termination, and confluence
properties for our calculi.
Our calculi improve on some earlier calculi for logical deduction systems in a number of
ways. By using de Bruijn indices, our calculi qualify as first-order term rewriting systems
(TRS's), allowing us to use correctly certain results for TRS's about termination. Unlike in
some other calculi, each of our calculi has only one cut rule and we do not need unusual
features of sequents.
We show that the substitution and index-updating mechanisms of our calculi work the same
way as the substitution and index-updating mechanisms of Kamareddine and Ríos' λs and
λt, two well-known systems of explicit substitution for the standard λ-calculus. By a change
in the format of sequents, we obtain similar results for a known λ-calculus with variables
and explicit substitutions, Rose's λbxgc.