Refinement type

In type theory, a refinement type¹²³ is a type endowed with a predicate which is assumed to hold for any element of the refined type. Refinement types can express preconditions when used as function arguments or postconditions when used as return types: for instance, the type of a function which accepts natural numbers and returns natural numbers greater than 5 may be written as ${\displaystyle f:\mathbb {N} \rightarrow \{n\in \mathbb {N} \,|\,n>5\}}$. Refinement types are thus related to behavioral subtyping.

The concept of refinement types was first introduced in Freeman and Pfenning's 1991 Refinement types for ML,¹ which presents a type system for a subset of Standard ML. The type system "preserves the decidability of ML's type inference" whilst still "allowing more errors to be detected at compile-time". In more recent times, refinement type systems have been developed (primary in academia) for languages such as Haskell,⁴⁵ TypeScript,⁶ Rust,⁷ and as libraries for real world usage in Scala.⁸⁹