# Abstract

In this paper, we present an explicit substitution calculus
which distinguishes between ordinary bound variables and
meta-variables. Its typing discipline is derived from contextual modal
type theory. We first present a dependently typed lambda calculus with
explicit substitutions for ordinary variables and explicit
meta-substitutions for meta-variables. We then present a weak head
normalization procedure which performs both substitutions lazily and
in a single pass thereby combining substitution walks for the two
different classes of variables. Finally, we describe a bidirectional
type checking algorithm which uses weak head normalization and prove
soundness.