Injective type families for Haskell

Abstract

Haskell, as implemented by the Glasgow Haskell Compiler (GHC), allows expressive type-level programming. The most popular typelevel programming extension is TypeFamilies, which allows users to write functions on types. Yet, using type functions can cripple type inference in certain situations. In particular, lack of injectivity in type functions means that GHC can never infer an instantiation of a type variable appearing only under type functions.In this paper, we describe a small modification to GHC that allows type functions to be annotated as injective. GHC naturally must check validity of the injectivity annotations. The algorithm to do so is surprisingly subtle. We prove soundness for a simplification of our algorithm, and state and prove a completeness property, though the algorithm is not fully complete.As much of our reasoning surrounds functions defined by a simple pattern-matching structure, we believe our results extend beyond just Haskell. We have implemented our solution on a branch of GHC and plan to make it available to regular users with the next stable release of the compiler.

Higher-order type-level programming in Haskell

“…For example, suppose that the database representation of Email in the example above is also DbText: Happily, the details of these variations are not important for this paper, and we stick only to associated type families, declared within a class. 3 Haskell aficionados will know that the user can declare a type family to be injective [Stolarek et al 2015]. But they cannot be generative, so from the perspective of this paper, declaring injectivity adds nothing.…”

“…Indeed, we could track injectivity and generativity separately by having dedicated arrows for both. This would enable abstraction over injective type families [Stolarek et al 2015], but the practical applicability of such a scheme seems limited considering the additional complexity and notational burden it would incur.…”

Constrained type families

“…We would expect that if the duals of two session types are equal, then the session types themselves are equal as well. Injective type families (Stolarek et al 2015) can express such cases; duality could be characterized by the following type family: type family Dual s = r | r → s where the s → r annotation denotes the injectivity of duality. e compiler validates that the injectivity condition is upheld by the type family's de ning equations.…”

“…Recent work has focused on extending the expressiveness of type families themselves. introduced closed type families, which allow overlapping equations in type family de nitions, and Stolarek et al (2015) introduced injective type families, recovering additional equalities from applications of injective type families. ese features, particularly closed type families, have seen signi cant practical application.…”

“…ey allow programmers to de ne and use open mappings from types to types. ese have given rise to further extensions of the language, such as closed type families ) and injective type families (Stolarek et al 2015), and they have many applications, including encoding units of measure in scienti c calculation (Muranushi and Eisenberg 2014) and extensible variants (Bahr 2014;Morris 2015).…”