Approximate Normalization and Eager Equality Checking for Gradual Inductive Families

Abstract

Harnessing the power of dependently typed languages can be difficult. Programmers must manually construct proofs to produce well-typed programs, which is not an easy task. In particular, migrating code to these languages is challenging. Gradual typing can make dependently-typed languages easier to use by mixing static and dynamic checking in a principled way. With gradual types, programmers can incrementally migrate code to a dependently typed language. However, adding gradual types to dependent types creates a new challenge: mixing decidable type-checking and incremental migration in a full-featured language is a precarious balance. Programmers expect type-checking to terminate, but dependent type-checkers evaluate terms at compile time, which is problematic because gradual types can introduce non-termination into an otherwise terminating language. Steps taken to mitigate this non-termination must not jeopardize the smooth transitions between dynamic and static. We present a gradual dependently-typed language that supports inductive type families, has decidable type-checking, and provably supports smooth migration between static and dynamic, as codified by the refined criteria for gradual typing proposed by Siek et al. (2015). Like Eremondi et al. (2019), we use approximate normalization for terminating compile-time evaluation. Unlike Eremondi et al., our normalization does not require comparison of variables, allowing us to show termination with a syntactic model that accommodates inductive types. Moreover, we design a novel a technique for tracking constraints on type indices, so that dynamic constraint violations signal run-time errors eagerly. To facilitate these checks, we define an algebraic notion of gradual precision, axiomatizing certain semantic properties of gradual terms.