Higher-order unification is equational unification for βη-conversion. But it is not first-order equational unification, as substitution has to avoid capture. In this paper higher-order unification is reduced to first-order equational unification in a suitable theory: the λσ-calculus of explicit substitutions.

Fraenkel-Mostowski (FM) set theory delivers a model of names and alpha-equivalence. This model, now generally called the ‘nominal ’ model, delivers inductive datatypes of syntax with alpha-equivalence — rather than inductive datatypes of syntax, quotiented by alpha-equivalence. The treatment of names and alpha-equivalence extends to the entire sets universe. This has proven useful for developing ‘nominal ’ theories of reasoning and programming on syntax with alpha-equivalence, because a sets universe includes elements representing functions, predicates, and behaviour. Often, we want names and alpha-equivalence to model capture-avoiding substitution. In this paper we show that FM set theory models capture-avoiding subsitution for names in much the same way as it models alpha-equivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes. In fact, more than one substitution action is possible (they all agree on sets representing