A Formal Proof of the Kepler Conjecture

Hales, Thomas C.; Adams, Mark; Bauer, Gertrud; Dang, Dat Tat; Harrison, John; Trương, Hoàng Lê; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Thang; Nguyen, Truong Q.; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, An Hoai Thi; Tran, Trung; Trieu, Diep Thi; Urban, Josef; Vu, Ky Khac; Zumkeller, Roland

doi:10.1017/fmp.2017.1

Cited by 274 publications

(204 citation statements)

References 27 publications

(31 reference statements)

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“…Referees had objected to this proof because of its reliance on extensive computer calculations. In response, Hales et al [56] launched the Flyspeck project to formally verify his proof. Flyspeck was ultimately successful, confirming and simplifying Hales's argument [57].…”

Section: Formalizing Mathematicsmentioning

confidence: 99%

Computational logic: its origins and applications

Paulson¹

2018

Proc. R. Soc. A.

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Computational logic is the use of computers to establish facts in a logical formalism. Originating in nineteenth century attempts to understand the nature of mathematical reasoning, the subject now comprises a wide variety of formalisms, techniques and technologies. One strand of work follows the ‘logic for computable functions (LCF) approach’ pioneered by Robin Milner, where proofs can be constructed interactively or with the help of users’ code (which does not compromise correctness). A refinement of LCF, called Isabelle, retains these advantages while providing flexibility in the choice of logical formalism and much stronger automation. The main application of these techniques has been to prove the correctness of hardware and software systems, but increasingly researchers have been applying them to mathematics itself.

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Section: Formalizing Mathematicsmentioning

confidence: 99%

Computational logic: its origins and applications

Paulson¹

2018

Proc. R. Soc. A.

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“…Giving a rigorous proof requires a genuine idea, but there exist short, elementary proofs [8]. The three-dimensional problem was solved by Hales [9] via a lengthy and complex computer-assisted proof, which was extraordinarily difficult to check but has since been completely verified using formal logic [10].…”

Section: A Question Is Good Ifmentioning

confidence: 99%

“…In particular, stacking optimal layers from the previous dimension does not always yield an optimal packing. (One can recursively determine the best packings in successive dimensions under such a hypothesis [6], and this procedure yields a suboptimal packing by the time it reaches ℝ 10 .) The sphere packing problem seems to have no simple, systematic solution that works across all dimensions.…”

Section: A Question Is Good Ifmentioning

confidence: 99%

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A Conceptual Breakthrough in Sphere Packing

Cohn¹

2017

Notices Amer. Math. Soc.

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“…They are used to guarantee the correctness of compilers [1], operating systems [2], hardware [3], as well as to certify mathematical results that involve computation [4]. The Mizar system [5] is one of the oldest computer systems used to certify proofs.…”

Section: Introductionmentioning

confidence: 99%

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Kaliszyk

Pąk

2017

Proceedings of the 2017 Federated Conference on Computer Science and Information Systems

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Abstract-The Mizar Mathematical Library is one of the largest collections of machine understandable formal proofs encompassing many areas of today mathematics including results from algebra, analysis, topology, and lattice theory. The Mizar system has so far been the only tool able to completely process, certify, and make use of these developments. In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework. The approach allows rechecking the Mizar formal proofs based on a more succinct and more precisely specified formal infrastructure. Additionally, it necessitates a full formal specification of the mechanisms that ensure the correctness of the defined objects, in particular, the proofs that such mechanisms are correct. The development already covers an important part of the Mizar library foundations. We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects. To demonstrate this, we perform a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.

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A Formal Proof of the Kepler Conjecture

Abstract: This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

Cited by 274 publications

References 27 publications

Computational logic: its origins and applications

Computational logic: its origins and applications

A Conceptual Breakthrough in Sphere Packing

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

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