Handling Delimited Continuations with Dependent Types

Dependent types are a powerful tool for maintaining program invariants. To take advantage of this aspect in real-world programming, efforts have been put into enriching dependently typed languages with missing constructs, most notably, effects. This paper presents a language that has two practically interesting ingredients: dependent inductive types, and the delimited control constructs shift and reset. When integrating delimited control into a dependently typed language, however, two challenges arise. First, the dynamic nature of control operators, which is the source of their expressiveness, can break fundamental language properties such as logical consistency and subject reduction. Second, CPS translations, which we often use to define the semantics of control operators, do not scale straightforwardly to dependently typed languages. We solve the former issue by restricting dependency of types, and the latter using answer-type polymorphism of pure terms. The main contribution of this paper is to give a sound type system of our language, as well as a type-preserving CPS translation. We also discuss various extensions, which would make our language more like a full-spectrum proof assistant but pose non-trivial issues.