The Soundness of Explicit Substitution with Nameless Variables

We show the soundness of a-calculus B where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the rst at semantics of explicit substitution and step-wise reduction and the rst clear account of exactly when-reduction is needed.