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The Interval Domain in Homotopy Type Theory

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Logics and Type Systems in Theory and Practice


Even though the real numbers are the cornerstone of many fields in mathematics, it is challenging to formalize them in a constructive setting, and in particular, homotopy type theory. Several approaches have been established to define the real numbers, and the most prominent of them are based on Dedekind cuts and on Cauchy sequences. In this paper, we study a different approach towards defining the real numbers. Our approach is based on domain theory, and in particular, the interval domain, and we build forth on recent work on domain theory in univalent foundations. All the results in this paper have been formalized in Coq as part of the UniMath library.

This paper is dedicated to our supervisor and teacher, Herman Geuvers, on the occasion of his birthday. Herman, you teachings and research in domain theory, type theory, and mechanized proofs have greatly inspired us, and we hope that you enjoy this little expedition in the world of constructive real numbers, combining all the topics above.

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We would like to thank the anonymous reviewers for their comments and suggestions. The authors also thank Andrej Bauer and Tom de Jong for useful pointers to the literature.

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Correspondence to Niels van der Weide .

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van der Weide, N., Frumin, D. (2024). The Interval Domain in Homotopy Type Theory. In: Capretta, V., Krebbers, R., Wiedijk, F. (eds) Logics and Type Systems in Theory and Practice. Lecture Notes in Computer Science, vol 14560. Springer, Cham.

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