Bidirectional Elaboration of Dependently Typed Programs

Dependently typed programming languages allow programmers to express a rich set of invariants and verify them statically via type checking. To make programming with dependent types practical, dependently typed systems provide a compact language for programmers where one can omit some arguments, called implicit, which can be inferred. This source language is then usually elaborated into a core language where type checking and fundamental properties such as normalization are well understood. Unfortunately, this elaboration is rarely specified and in general is ill-understood. This makes it not only difficult for programmers to understand why a given program fails to type check, but also is one of the reasons that implementing dependently typed programming systems remains a black art known only to a few. In this paper, we specify the design of a source language for a dependently typed programming language where we separate the language of programs from the language of types and terms occurring in types. Total functions in our language correspond directly to first-order inductive proofs over a specific index domain. We then give a bi-directional elaboration algorithm to translate source terms where implicit arguments can be omitted to a fully explicit core language and prove soundness of our elaboration. Our framework provides post-hoc explanation for elaboration found in the programming and proof environment, Beluga.