Quotient inductiveinductive typesTools Altenkirch, Thorsten and Capriotti, Paolo and Dijkstra, Gabe and Nordvall Forsberg, Fredrik Nordvall Forsberg (2017) Quotient inductiveinductive types. In: FoSSaCS 2018: 21st International Conference on Foundations of Software Science and Computation Structures, 1420 April 2018, Thessaloniki, Greece. Full text not available from this repository.AbstractHigher 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 inductiveinductive definitions. We call those HITs quotient inductiveinductive types (QIITs).
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