Martin-Lof a la Coq

We present an extensive mechanization of the metatheory of Martin-Lof Type Theory (MLTT) in the COQ proof assistant. Our development builds on pre-existing work in AGDA to show not only the decidability of conversion, but also the decidability of type checking, using an approach guided by bidirectional type checking. From our proof of decidability, we obtain a certified and executable type checker for a full-fledged version of MLTT with support for Pi, Sigma, N, and Id types, and one universe. Our development does not rely on impredicativity, induction-recursion or any axiom beyond MLTT extended with indexed inductive types and a handful of predicative universes, thus narrowing the gap between the object theory and the metatheory to a mere difference in universes. Furthermore, our formalization choices are geared towards a modular development that relies on COQ's features, e.g. universe polymorphism and metaprogramming with tactics.