In the field of type theory in computer science, a quotient type is a data type which respects a user-defined equality relation. A quotient type defines an equivalence relation on elements of the type - for example, we might say that two values of the type Person are equivalent if they have the same name; formally p1 == p2 if p1.name == p2.name. In type theories which allow quotient types, an additional requirement is made that all operations must respect the equivalence between elements. For example, if f is a function on values of type Person, it must be the case that for two Persons p1 and p2, if p1 == p2 then f(p1) == f(p2).
Quotient types are part of a general class of types known as algebraic data types. In the early 1980s, quotient types were defined and implemented as part of the Nuprl proof assistant, in work led by Robert L. Constable and others.[1][2] Quotient types have been studied in the context of Martin-Löf type theory,[3] dependent type theory,[4] higher-order logic,[5] and homotopy type theory.[6]
^Constable, Robert L. (1986). Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall. ISBN 978-0-13-451832-9.
^Constable, R. L. (1984). "Mathematics as programming". In Clarke, Edmund; Kozen, Dexter (eds.). Logics of Programs. Lecture Notes in Computer Science. Vol. 164. Berlin, Heidelberg: Springer. pp. 116–128. doi:10.1007/3-540-12896-4_359. hdl:1813/6405. ISBN 978-3-540-38775-6.
^Li, Nuo (2015-07-15). "Quotient types in type theory". eprints.nottingham.ac.uk. Retrieved 2023-09-13.
^Hofmann, Martin (1995). "A simple model for quotient types". Typed Lambda Calculi and Applications. Lecture Notes in Computer Science. Vol. 902. Berlin, Heidelberg: Springer. pp. 216–234. doi:10.1007/BFb0014055. ISBN 978-3-540-49178-1.
^Homeier, Peter V. (2005). "A Design Structure for Higher Order Quotients". In Hurd, Joe; Melham, Tom (eds.). Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science. Vol. 3603. Berlin, Heidelberg: Springer. pp. 130–146. doi:10.1007/11541868_9. ISBN 978-3-540-31820-0.
^"The HoTT Book". Homotopy Type Theory. 2013-03-12. Retrieved 2023-09-13.
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