9 Search Results for "van Doorn, Floris"

Document
DOI: 10.4230/LIPIcs.ITP.2025.4
A Formalization of Divided Powers in Lean

Authors: Antoine Chambert-Loir and María Inés de Frutos-Fernández

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Given an ideal I in a commutative ring A, a divided power structure on I is a collection of maps {γ_n : I → A}_{n ∈ ℕ}, subject to axioms that imply that it behaves like the family {x ↦ xⁿ/n!}_{n ∈ ℕ}, but which can be defined even when division by factorials is not possible in A. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.
Cite as

Antoine Chambert-Loir and María Inés de Frutos-Fernández. A Formalization of Divided Powers in Lean. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chambertloir_et_al:LIPIcs.ITP.2025.4,
  author =	{Chambert-Loir, Antoine and de Frutos-Fern\'{a}ndez, Mar{\'\i}a In\'{e}s},
  title =	{{A Formalization of Divided Powers in Lean}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{4:1--4:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.4},
  URN =		{urn:nbn:de:0030-drops-246038},
  doi =		{10.4230/LIPIcs.ITP.2025.4},
  annote =	{Keywords: Formal mathematics, algebraic number theory, commutative algebra, divided powers, Lean, Mathlib}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.30
Coherent Tietze Transformations of 1-Polygraphs in Homotopy Type Theory

Authors: Samuel Mimram and Émile Oleon

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

Abstract
Polygraphs play a fundamental role in algebra, geometry, and computer science, by generalizing group presentations to higher-dimensional structures and encoding coherence for those. They have recently been adapted by Kraus and von Raumer to the setting of homotopy type theory, where they are useful to define and study higher inductive types. Here, we develop the theory of 1-dimensional polygraphs, which correspond to presentations of sets in homotopy type theory. This requires us to introduce a dedicated notion of Tietze transformation, generalizing their well-known counterpart in group theory: the equivalence generated by those transformations characterizes situations where two 1-polygraphs present the same set. We also show a homotopy transfer theorem, which provides a way to transport coherence structures from one 1-polygraph to another. This work lays the foundations for a general theory of polygraphs in arbitrary dimensions, which should be useful for instance to define and study coherent group presentations, allowing for synthetic (co)homology computations. Most of the results in the article have been formalized with the Agda proof assistant using the cubical HoTT library.
Cite as

Samuel Mimram and Émile Oleon. Coherent Tietze Transformations of 1-Polygraphs in Homotopy Type Theory. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mimram_et_al:LIPIcs.FSCD.2025.30,
  author =	{Mimram, Samuel and Oleon, \'{E}mile},
  title =	{{Coherent Tietze Transformations of 1-Polygraphs in Homotopy Type Theory}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{30:1--30:17},
  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.2025.30},
  URN =		{urn:nbn:de:0030-drops-236456},
  doi =		{10.4230/LIPIcs.FSCD.2025.30},
  annote =	{Keywords: homotopy type theory, polygraph, Tietze transformation, coherence}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2024.3
A Foundation for Synthetic Stone Duality

Authors: Felix Cherubini, Thierry Coquand, Freek Geerligs, and Hugo Moeneclaey

Published in: LIPIcs, Volume 336, 30th International Conference on Types for Proofs and Programs (TYPES 2024)

Abstract
The language of homotopy type theory has proved to be an appropriate internal language for various higher toposes, for example for the Zariski topos in Synthetic Algebraic Geometry. This paper aims to do the same for the higher topos of light condensed anima of Dustin Clausen and Peter Scholze. This seems to be an appropriate setting for synthetic topology in the style of Martín Escardó. We use homotopy type theory extended with 4 axioms. We prove Markov’s principle, LLPO and the negation of WLPO. Then we define a type of open propositions, inducing a topology on any type such that any map is continuous. We give a synthetic definition of second countable Stone and compact Hausdorff spaces, and show that their induced topologies are as expected. This means that any map from e.g. the unit interval 𝕀 to itself is continuous in the usual epsilon-delta sense. With the usual definition of cohomology in homotopy type theory, we show that H¹(S,ℤ) = 0 for S Stone and that H¹(X,ℤ) for X compact Hausdorff can be computed using Čech cohomology. We use this to prove H¹(𝕀¹,ℤ) = 0 and H¹(𝕊¹,ℤ) = ℤ where 𝕊¹ is the set ℝ/ℤ. As an application, we give a synthetic proof of Brouwer’s fixed-point theorem.
Cite as

Felix Cherubini, Thierry Coquand, Freek Geerligs, and Hugo Moeneclaey. A Foundation for Synthetic Stone Duality. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cherubini_et_al:LIPIcs.TYPES.2024.3,
  author =	{Cherubini, Felix and Coquand, Thierry and Geerligs, Freek and Moeneclaey, Hugo},
  title =	{{A Foundation for Synthetic Stone Duality}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{3:1--3:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-376-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{336},
  editor =	{M{\o}gelberg, Rasmus Ejlers and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.3},
  URN =		{urn:nbn:de:0030-drops-233659},
  doi =		{10.4230/LIPIcs.TYPES.2024.3},
  annote =	{Keywords: Homotopy Type Theory, Synthetic Topology, Cohomology}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.46
Coslice Colimits in Homotopy Type Theory

Authors: Perry Hart and Kuen-Bang Hou (Favonia)

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between colimits in coslices of a universe, called coslice colimits, and colimits in the universe (i.e., ordinary colimits). To derive this characterization, we find an explicit construction of colimits in coslices that is tailored to reveal the connection. We use the construction to derive properties of colimits. Notably, we prove that the forgetful functor from a coslice creates colimits over trees. We also use the construction to examine how colimits interact with orthogonal factorization systems and with cohomology theories. As a consequence of their interaction with orthogonal factorization systems, all pointed colimits (special kinds of coslice colimits) preserve n-connectedness, which implies that higher groups are closed under colimits on directed graphs. We have formalized our main construction of the coslice colimit functor in Agda.
Cite as

Perry Hart and Kuen-Bang Hou (Favonia). Coslice Colimits in Homotopy Type Theory. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hart_et_al:LIPIcs.CSL.2025.46,
  author =	{Hart, Perry and Hou (Favonia), Kuen-Bang},
  title =	{{Coslice Colimits in Homotopy Type Theory}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.46},
  URN =		{urn:nbn:de:0030-drops-228039},
  doi =		{10.4230/LIPIcs.CSL.2025.46},
  annote =	{Keywords: colimits, homotopy type theory, category theory, higher inductive types, synthetic homotopy theory}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.37
Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality

Authors: Floris van Doorn and Heather Macbeth

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
We introduce an abstraction which allows arguments involving iterated integrals to be formalized conveniently in type-theory-based proof assistants. We call this abstraction the marginal construction, since it is connected to the marginal distribution in probability theory. The marginal construction gracefully handles permutations to the order of integration (Tonelli’s theorem in several variables), as well as arguments involving an induction over dimension. We implement the marginal construction and several applications in the language Lean. The most difficult of these applications, the Gagliardo-Nirenberg-Sobolev inequality, is a foundational result in the theory of elliptic partial differential equations and has not previously been formalized.
Cite as

Floris van Doorn and Heather Macbeth. Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vandoorn_et_al:LIPIcs.ITP.2024.37,
  author =	{van Doorn, Floris and Macbeth, Heather},
  title =	{{Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.37},
  URN =		{urn:nbn:de:0030-drops-207657},
  doi =		{10.4230/LIPIcs.ITP.2024.37},
  annote =	{Keywords: Sobolev inequality, measure theory, Lean, formalized mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2023.15
Closure Properties of General Grammars – Formally Verified

Authors: Martin Dvorak and Jasmin Blanchette

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract
We formalized general (i.e., type-0) grammars using the Lean 3 proof assistant. We defined basic notions of rewrite rules and of words derived by a grammar, and used grammars to show closure of the class of type-0 languages under four operations: union, reversal, concatenation, and the Kleene star. The literature mostly focuses on Turing machine arguments, which are possibly more difficult to formalize. For the Kleene star, we could not follow the literature and came up with our own grammar-based construction.
Cite as

Martin Dvorak and Jasmin Blanchette. Closure Properties of General Grammars – Formally Verified. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dvorak_et_al:LIPIcs.ITP.2023.15,
  author =	{Dvorak, Martin and Blanchette, Jasmin},
  title =	{{Closure Properties of General Grammars – Formally Verified}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.15},
  URN =		{urn:nbn:de:0030-drops-183906},
  doi =		{10.4230/LIPIcs.ITP.2023.15},
  annote =	{Keywords: Lean, type-0 grammars, recursively enumerable languages, Kleene star}
}
Document
DOI: 10.4230/LIPIcs.ITP.2021.18
Formalized Haar Measure

Authors: Floris van Doorn

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
We describe the formalization of the existence and uniqueness of the Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean’s mathematical library mathlib, and discuss the construction of product measures and the proof of Fubini’s theorem for the Bochner integral.
Cite as

Floris van Doorn. Formalized Haar Measure. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vandoorn:LIPIcs.ITP.2021.18,
  author =	{van Doorn, Floris},
  title =	{{Formalized Haar Measure}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.18},
  URN =		{urn:nbn:de:0030-drops-139139},
  doi =		{10.4230/LIPIcs.ITP.2021.18},
  annote =	{Keywords: Haar measure, measure theory, Bochner integral, Lean, interactive theorem proving, formalized mathematics}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2019.8
Coherence for Monoidal Groupoids in HoTT

Authors: Stefano Piceghello

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncated higher inductive types, with constructors freely generating their defining objects, natural isomorphisms and commutative diagrams. All results included in this paper have been formalised in the proof assistant Coq.
Cite as

Stefano Piceghello. Coherence for Monoidal Groupoids in HoTT. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{piceghello:LIPIcs.TYPES.2019.8,
  author =	{Piceghello, Stefano},
  title =	{{Coherence for Monoidal Groupoids in HoTT}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.8},
  URN =		{urn:nbn:de:0030-drops-130722},
  doi =		{10.4230/LIPIcs.TYPES.2019.8},
  annote =	{Keywords: homotopy type theory, coherence, monoidal categories, groupoids, higher inductive types, formalisation, Coq}
}
Document
DOI: 10.4230/LIPIcs.ITP.2019.19
A Formalization of Forcing and the Unprovability of the Continuum Hypothesis

Authors: Jesse Michael Han and Floris van Doorn

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract
We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space 2^{omega_2 x omega} and formally verify the failure of the continuum hypothesis in the resulting model.
Cite as

Jesse Michael Han and Floris van Doorn. A Formalization of Forcing and the Unprovability of the Continuum Hypothesis. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{han_et_al:LIPIcs.ITP.2019.19,
  author =	{Han, Jesse Michael and van Doorn, Floris},
  title =	{{A Formalization of Forcing and the Unprovability of the Continuum Hypothesis}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.19},
  URN =		{urn:nbn:de:0030-drops-110742},
  doi =		{10.4230/LIPIcs.ITP.2019.19},
  annote =	{Keywords: Interactive theorem proving, formal verification, set theory, forcing, independence proofs, continuum hypothesis, Boolean-valued models, Lean}
}
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