On the Syntax of Dependent Types and the Coherence Problem (working Draft)

We discuss diierent ways to represent the syntax of dependent types using Martin-LL of type theory as a metalanguage. In particular, we show how to give an intrinsic syntax in which meaningful contexts, types in a context, and terms of a certain type in a context, are generated directly without rst introducing raw terms, types, and contexts. In the rst representation we deene inductively the normal contexts, types, and terms of the pure theory of dependent types. Simultaneously substitution and lifting are deened recursively. Equality in the object language is here syntactic equality and is represented by the equality of the metalanguage. The second representation is a calculus of explicit substitutions. This is a pure inductive deenition and proofs of equalities are generated simultaneously. As for Curien's explicit syntax there are term constructors corresponding to applications of the type equality rules, and coherence conditions related to those appearing in category theory. We prove the equivalence of the two representations. This proof can be viewed as a solution of the coherence problem for the calculus of explicit substitutions.