Refinement type

In type theory, a refinement type 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 $f:\mathbb {N} \rightarrow \{n:\mathbb {N} \,|\,n>5\}$. Refinement types are thus related to behavioral subtyping.

History

The concept of refinement types was first introduced in Freeman and Pfenning's 1991 Refinement types for ML , which describes a refinement type system for a subset of Standard ML. In more recent times, refinement type systems have been developed for languages such as Haskell, TypeScript and Scala.

See also

• Liquid Haskell