Internal Adequacy of Bookkeeping in Coq

We focus on a common problem encountered in encoding and formally reasoning about a wide range of formal systems, namely, the representation of a typing environment. In particular, we apply the bookkeeping technique to a well-known case study (i.e., System F<:'s type language), proving in Coq an internal correspondence with a more standard representation of the typing environment as a list of pairs. In order to keep the signature readable and concise, we make use of higher-order abstract syntax (HOAS), which allows us to deal smoothly with the representation of the universal binder of System F<: type language.