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Documents authored by Codel, Cayden

Document
DOI: 10.4230/LIPIcs.ITP.2025.9
Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups

Authors: Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer

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

Abstract
Graded unipotent Chevalley groups are an important family of groups on matrices with polynomial entries over a finite field. Using the Lean theorem prover, we verify that three such groups, namely, the A₃- and the two B₃-type groups, satisfy a useful group-theoretic condition. Specifically, these groups are defined by a set of equations called Steinberg relations, and we prove that a certain canonical "smaller" set of Steinberg relations suffices to derive the rest. Our work is motivated by an application for building topologically-interesting objects called higher-dimensional expanders (HDXs). In the past decade, HDXs have formed the basis for many new results in theoretical computer science, such as in quantum error correction and in property testing. Yet despite the increasing prevalence of HDXs, only two methods of constructing them are known. One such method builds an HDX from groups that satisfy the aforementioned property, and the Chevalley groups we use are (essentially) the only ones currently known to satisfy it.
Cite as

Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer. Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wang_et_al:LIPIcs.ITP.2025.9,
  author =	{Wang, Eric and Bhoja, Arohee and Codel, Cayden and Singer, Noah G.},
  title =	{{Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{9:1--9:19},
  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.2025.9},
  URN =		{urn:nbn:de:0030-drops-246071},
  doi =		{10.4230/LIPIcs.ITP.2025.9},
  annote =	{Keywords: Group presentations, term rewriting, metaprogramming, proof automation, the Lean theorem prover}
}
Artifact
Software
DOI: 10.4230/artifacts.22467
EmptyHexagonLean

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

Abstract
Cite as

Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, Marijn J. H. Heule. EmptyHexagonLean (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@misc{dagstuhl-artifact-22467,
   title = {{EmptyHexagonLean}}, 
   author = {Subercaseaux, Bernardo and Nawrocki, Wojciech and Gallicchio, James and Codel, Cayden and Carneiro, Mario and Heule, Marijn J. H.},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:29dc0e7145296997bcb1230b4e03cd14c8d75617;origin=https://github.com/bsubercaseaux/EmptyHexagonLean;visit=swh:1:snp:0e11d6564bd15317306605932e0acd87cf3d7f80;anchor=swh:1:rev:d7f798ffc8deabc2f3ca1ae36e92e0250e57c205}{\texttt{swh:1:dir:29dc0e7145296997bcb1230b4e03cd14c8d75617}} (visited on 2024-11-28)},
   url = {https://github.com/bsubercaseaux/EmptyHexagonLean/tree/itp2024},
   doi = {10.4230/artifacts.22467},
}
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}
}
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