# Refinement type

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In type theory, a refinement type [1] [2] [3] 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.

## History

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 for languages such as Haskell, [4] [5] TypeScript [6] and Scala.

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## References

1. Freeman, T.; Pfenning, F. (1991). "Refinement types for ML" (PDF). Proceedings of the ACM Conference on Programming Language Design and Implementation. pp. 268–277. doi:10.1145/113445.113468.
2. Hayashi, S. (1993). "Logic of refinement types". Proceedings of the Workshop on Types for Proofs and Programs. pp. 157–172. CiteSeerX  . doi:10.1007/3-540-58085-9_74.
3. Denney, E. (1998). "Refinement types for specification". Proceedings of the IFIP International Conference on Programming Concepts and Methods. 125. Chapman & Hall. pp. 148–166. CiteSeerX  .
4. Vazou, Niki. Liquid Haskell: Refinement Types for Haskell. The 45th ACM SIGPLAN Symposium on Principles of Programming Languages (POPL 2018).
5. Volkov, Nikita (2015). "Refinement types as a Haskell library".
6. Panagiotis, Vekris; Cosman, Benjamin; Jhala, Ranjit (2016). "Refinement types for TypeScript". Proceedings of the 37th ACM SIGPLAN Conference on Programming Language Design and Implementation. pp. 310–325. arXiv:. doi:10.1145/2908080.2908110.