8 Search Results for "van Doorn, Floris"

Document
DOI: 10.4230/LIPIcs.ITP.2024.8
The Directed Van Kampen Theorem in Lean

Authors: Henning Basold, Peter Bruin, and Dominique Lawson

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

Abstract
Directed topology augments the concept of a topological space with a notion of directed paths. This leads to a category of directed spaces, in which the morphisms are continuous maps respecting directed paths. Directed topology thereby enables an accurate representation of computation paths in concurrent systems that usually cannot be reversed. Even though ideas from algebraic topology have analogues in directed topology, the directedness drastically changes how spaces can be characterised. For instance, while an important homotopy invariant of a topological space is its fundamental groupoid, for directed spaces this has to be replaced by the fundamental category because directed paths are not necessarily reversible. In this paper, we present a Lean 4 formalisation of directed spaces and of a Van Kampen theorem for them, which allows the fundamental category of a directed space to be computed in terms of the fundamental categories of subspaces. Part of this formalisation is also a significant theory of directed spaces, directed homotopy theory and path coverings, which can serve as basis for future formalisations of directed topology. The formalisation in Lean can also be used in computer-assisted reasoning about the behaviour of concurrent systems that have been represented as directed spaces.
Cite as

Henning Basold, Peter Bruin, and Dominique Lawson. The Directed Van Kampen Theorem in Lean. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{basold_et_al:LIPIcs.ITP.2024.8,
  author =	{Basold, Henning and Bruin, Peter and Lawson, Dominique},
  title =	{{The Directed Van Kampen Theorem in Lean}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{8:1--8:18},
  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.2024.8},
  URN =		{urn:nbn:de:0030-drops-207368},
  doi =		{10.4230/LIPIcs.ITP.2024.8},
  annote =	{Keywords: Lean, Directed Topology, Van Kampen Theorem, Directed Homotopy Theory, Formalised Mathematics}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.35
Formal Verification of the Empty Hexagon Number

Authors: Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule

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

Abstract
A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position (i.e., no three points on a common line) contains an empty convex hexagon. Heule and Scheucher solved this problem with a combination of geometric insights and automated reasoning techniques by constructing CNF formulas ϕ_n, with O(n⁴) clauses, such that if ϕ_n is unsatisfiable then every set of n points in general position must contain an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17 300 CPU hours of parallel computation. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers ideas in discrete computational geometry and SAT encoding techniques by introducing a framework that connects geometric objects to propositional assignments. We see this as a key step towards the formal verification of other SAT-based results in geometry, since the abstractions we use have been successfully applied to similar problems. Overall, we hope that our work sets a new standard for the verification of geometry problems relying on extensive computation, and that it increases the trust the mathematical community places in computer-assisted proofs.
Cite as

Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule. Formal Verification of the Empty Hexagon Number. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{subercaseaux_et_al:LIPIcs.ITP.2024.35,
  author =	{Subercaseaux, Bernardo and Nawrocki, Wojciech and Gallicchio, James and Codel, Cayden and Carneiro, Mario and Heule, Marijn J. H.},
  title =	{{Formal Verification of the Empty Hexagon Number}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{35:1--35:19},
  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.2024.35},
  URN =		{urn:nbn:de:0030-drops-207633},
  doi =		{10.4230/LIPIcs.ITP.2024.35},
  annote =	{Keywords: Empty Hexagon Number, Discrete Computational Geometry, Erd\H{o}s-Szekeres}
}
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.TYPES.2023.1
Classification of Covering Spaces and Canonical Change of Basepoint

Authors: Jelle Wemmenhove, Cosmin Manea, and Jim Portegies

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Using the language of homotopy type theory (HoTT), we 1) prove a synthetic version of the classification theorem for covering spaces, and 2) explore the existence of canonical change-of-basepoint isomorphisms between homotopy groups. There is some freedom in choosing how to translate concepts from classical algebraic topology into HoTT. The final translations we ended up with are easier to work with than the ones we started with. We discuss some earlier attempts to shed light on this translation process. The proofs are mechanized using the Coq proof assistant and closely follow classical treatments like those by Hatcher [Allen Hatcher, 2002].
Cite as

Jelle Wemmenhove, Cosmin Manea, and Jim Portegies. Classification of Covering Spaces and Canonical Change of Basepoint. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wemmenhove_et_al:LIPIcs.TYPES.2023.1,
  author =	{Wemmenhove, Jelle and Manea, Cosmin and Portegies, Jim},
  title =	{{Classification of Covering Spaces and Canonical Change of Basepoint}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo 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.2023.1},
  URN =		{urn:nbn:de:0030-drops-204795},
  doi =		{10.4230/LIPIcs.TYPES.2023.1},
  annote =	{Keywords: Synthetic Homotopy Theory, Homotopy Type Theory, Covering Spaces, Change-of-Basepoint Isomorphism}
}
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.
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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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