Universe of Binding and Computation

摘要

We construct a logical framework supporting datatypes that mix binding and computation, implemented as a universe in the depen-dently typed programming language Agda 2. We represent binding pronominally, using well-scoped de Bruijn indices, so that types can be used to reason about the scoping of variables. We equip our universe with datatype-generic implementations of weaken-ing, substitution, exchange, contraction, and subordination-based strengthening, so that programmers need not reimplement these op-erations for each individual language they define. In our mixed, pronominal setting, weakening and substitution hold only under some conditions on types, but we show that these conditions can be discharged automatically in many cases. Finally, we program a variety of standard difficult test cases from the literature, such as normalization-by-evaluation for the untyped A-calculus, demon-strating that we can express detailed invariants about variable usage in a program's type while still writing clean and clear code.