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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.

Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu. Connecting Constructive Notions of Ordinals in Homotopy Type Theory. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kraus_et_al:LIPIcs.MFCS.2021.70, author = {Kraus, Nicolai and Nordvall Forsberg, Fredrik and Xu, Chuangjie}, title = {{Connecting Constructive Notions of Ordinals in Homotopy Type Theory}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {70:1--70:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.70}, URN = {urn:nbn:de:0030-drops-145100}, doi = {10.4230/LIPIcs.MFCS.2021.70}, annote = {Keywords: Constructive ordinals, Cantor normal forms, Brouwer trees} }

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**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

We introduce a syntactic translation of Gödel’s System 𝖳 parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzen’s negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of 𝖳-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.

Chuangjie Xu. A Gentzen-Style Monadic Translation of Gödel’s System T. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{xu:LIPIcs.FSCD.2020.25, author = {Xu, Chuangjie}, title = {{A Gentzen-Style Monadic Translation of G\"{o}del’s System T}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.25}, URN = {urn:nbn:de:0030-drops-123472}, doi = {10.4230/LIPIcs.FSCD.2020.25}, annote = {Keywords: monadic translation, G\"{o}del’s System T, logical relation, negative translation, majorizability, continuity, bar recursion, Agda} }

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**Published in:** LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Sigma type via the Curry-Howard interpretation. But with an intuitionistic notion of anonymous existence, defined as the propositional truncation of Sigma, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N -> 2) -> N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn't show Agda code.

Martín Hötzel Escardó and Chuangjie Xu. The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 153-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hotzelescardo_et_al:LIPIcs.TLCA.2015.153, author = {H\"{o}tzel Escard\'{o}, Mart{\'\i}n and Xu, Chuangjie}, title = {{The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {153--164}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-87-3}, ISSN = {1868-8969}, year = {2015}, volume = {38}, editor = {Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.153}, URN = {urn:nbn:de:0030-drops-51618}, doi = {10.4230/LIPIcs.TLCA.2015.153}, annote = {Keywords: Dependent type, intensional Martin-L\"{o}f type theory, Curry-Howard interpretation, constructive mathematics, Brouwerian continuity axioms, anonymous exi} }