DROPS

Document

DOI: 10.4230/LIPIcs.FSCD.2025.9

A Zoo of Continuity Properties in Constructive Type Theory

Authors: Martin Baillon, Yannick Forster, Assia Mahboubi, Pierre-Marie Pédrot, and Matthieu Piquerez

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract

Continuity principles stating that all functions are continuous play a central role in some schools of constructive mathematics. However, there are different ways to formalise the property of being continuous in constructive foundations. We analyse these continuity properties from the perspective of constructive reverse mathematics. We work in constructive type theory, which can be seen as a minimal foundation for constructive reverse mathematics. We treat continuity of functions F : (Q → A) → R, i.e. with question type Q, answer type A, and result type R. Concretely, we discuss continuity defined via moduli, making the relevant list L : LQ of questions explicit, dialogue trees, making the question-answer process explicit as inductive tree, and tree functions, making the question-answer process explicit as function. We prove equivalences where possible and isolate necessary and sufficient axioms for equivalence proofs. Many of the results we discuss are already present in the works of Hancock, Pattinson, Ghani, Kawai, Fujiwara, Brede, Herbelin, Escardó, and others. Our main contribution is their formulation over a uniform foundation, the observation that no choice axioms are necessary, the generalisation to arbitrary types from natural numbers where possible, and a mechanisation in the Coq/Rocq proof assistant.

Cite as

Martin Baillon, Yannick Forster, Assia Mahboubi, Pierre-Marie Pédrot, and Matthieu Piquerez. A Zoo of Continuity Properties in Constructive Type Theory. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{baillon_et_al:LIPIcs.FSCD.2025.9,
  author =	{Baillon, Martin and Forster, Yannick and Mahboubi, Assia and P\'{e}drot, Pierre-Marie and Piquerez, Matthieu},
  title =	{{A Zoo of Continuity Properties in Constructive Type Theory}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.9},
  URN =		{urn:nbn:de:0030-drops-236245},
  doi =		{10.4230/LIPIcs.FSCD.2025.9},
  annote =	{Keywords: type theory, constructive mathematics, continuity, Coq}
}

Document

DOI: 10.4230/LIPIcs.CSL.2022.5

Gardening with the Pythia A Model of Continuity in a Dependent Setting

Authors: Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

We generalize to a rich dependent type theory a proof originally developed by Escardó that all System 𝚃 functionals are continuous. It relies on the definition of a syntactic model of Baclofen Type Theory, a type theory where dependent elimination must be strict, into the Calculus of Inductive Constructions. The model is given by three translations: the axiom translation, that adds an oracle to the context; the branching translation, based on the dialogue monad, turning every type into a tree; and finally, a layer of algebraic binary parametricity, binding together the two translations. In the resulting type theory, every function f : (ℕ → ℕ) → ℕ is externally continuous.

Cite as

Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot. Gardening with the Pythia A Model of Continuity in a Dependent Setting. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{baillon_et_al:LIPIcs.CSL.2022.5,
  author =	{Baillon, Martin and Mahboubi, Assia and P\'{e}drot, Pierre-Marie},
  title =	{{Gardening with the Pythia A Model of Continuity in a Dependent Setting}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.5},
  URN =		{urn:nbn:de:0030-drops-157256},
  doi =		{10.4230/LIPIcs.CSL.2022.5},
  annote =	{Keywords: Type theory, continuity, syntactic model}
}

Search Results

Documents authored by Baillon, Martin

A Zoo of Continuity Properties in Constructive Type Theory

Abstract

Cite as

Gardening with the Pythia A Model of Continuity in a Dependent Setting

Abstract

Cite as

Thanks for your feedback!

Could not send message