Nominal rewriting

Nominal rewriting is based on the observation that if we add support for alphaequivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order features such as lambda-calculus beta-reduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is just replacement in abstract syntax trees. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem for orthogonal systems. Nominal rewriting is designed to formalise higher-order “reduction schemes”, such as β-reduction, as rewrite rules. We give examples.