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We describe a recent formalization of several results from the metatheory of Pure Type Systems (PTSs) in Coq, including Subject Reduction, Uniqueness of Types in a functional PTS, and the difficult proof of Strengthening.  The terms of the PTS are represented by an inductive family of types: "term n" is the type of all terms with at most n free variables.  This representation of terms has often been used to study syntax and substitution, but not the metatheory of a formal system.  We show how it requires many metatheorems to be stated in a somewhat unfamiliar "big-step" form, but then allows for very elegant and direct proofs

Formalized Metatheory with Terms Represented by an Indexed Family of Types