A Syntax for Higher Inductive-Inductive Types

Authors Ambrus Kaposi , András Kovács

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Ambrus Kaposi
  • Faculty of Informatics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary
András Kovács
  • Faculty of Informatics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary

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Ambrus Kaposi and András Kovács. A Syntax for Higher Inductive-Inductive Types. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Higher inductive-inductive types (HIITs) generalise inductive types of dependent type theories in two directions. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalising higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy reals and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a domain-specific type theory. A context in this small type theory encodes a HIIT by listing the type formation rules and constructors. The type of the elimination principle and its beta-rules are computed from the context using a variant of the syntactic logical relation translation. We show that for indexed W-types and various examples of HIITs the computed elimination principles are the expected ones. Showing that the thus specified HIITs exist is left as future work. The type theory specifying HIITs was formalised in Agda together with the syntactic translations. A Haskell implementation converts the types of sorts and constructors into valid Agda code which postulates the elimination principles and computation rules.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • homotopy type theory
  • inductive-inductive types
  • higher inductive types
  • quotient inductive types
  • logical relations


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