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Type Theory with Explicit Universe Polymorphism

Authors Marc Bezem , Thierry Coquand , Peter Dybjer , Martín Escardó

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Author Details

Marc Bezem
  • University of Bergen, Norway
Thierry Coquand
  • University of Gothenburg, Sweden
Peter Dybjer
  • Chalmers University of Technology, Gothenburg, Sweden
Martín Escardó
  • University of Birmingham, UK


The authors are grateful to the anonymous referees for useful feedback, and to Matthieu Sozeau for an update on the current state of universe polymorphism in Coq. We acknowledge the support of the Centre for Advanced Study (CAS) at the Norwegian Academy of Science and Letters in Oslo, Norway, which funded and hosted the research project Homotopy Type Theory and Univalent Foundations during the academic year 2018/19.

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Marc Bezem, Thierry Coquand, Peter Dybjer, and Martín Escardó. Type Theory with Explicit Universe Polymorphism. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The aim of this paper is to refine and extend proposals by Sozeau and Tabareau and by Voevodsky for universe polymorphism in type theory. In those systems judgments can depend on explicit constraints between universe levels. We here present a system where we also have products indexed by universe levels and by constraints. Our theory has judgments for internal universe levels, built up from level variables by a successor operation and a binary supremum operation, and also judgments for equality of universe levels.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • type theory
  • universes in type theory
  • universe polymorphism
  • level-indexed products
  • constraint-indexed products


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