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Quotient inductive-inductive types

Altenkirch, Thorsten; Capriotti, Paolo; Dijkstra, Gabe; Kraus, Nicolai; Nordvall Forsberg, Fredrik

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Paolo Capriotti

Gabe Dijkstra

Fredrik Nordvall Forsberg


Christel Baier

Ugo Dal Lago


Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.


Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., & Nordvall Forsberg, F. (2018). Quotient inductive-inductive types. In C. Baier, & U. Dal Lago (Eds.), FoSSaCS 2018: Foundations of Software Science and Computation Structures (293-310). Cham: Springer Publishing Company.

Conference Name FoSSaCS 2018: 21st International Conference on Foundations of Software Science and Computation Structures
Conference Location Thessaloniki, Greece
Start Date Apr 14, 2018
End Date Apr 20, 2018
Acceptance Date Dec 22, 2017
Online Publication Date Apr 14, 2018
Publication Date 2018
Deposit Date Apr 18, 2018
Publicly Available Date Apr 18, 2018
Publisher Springer Publishing Company
Peer Reviewed Peer Reviewed
Pages 293-310
Series Title Lecture notes in computer science
Series Number 10803
Book Title FoSSaCS 2018: Foundations of Software Science and Computation Structures
ISBN 9783319893655
Keywords Logic in Computer Science;
Public URL
Publisher URL


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