A Machine-Checked Proof of the Odd Order Theorem

Gonthier, Georges; Asperti, Andrea; Avigad, Jeremy; Bertot, Yves; Cohen, Cyril; Garillot, François; Roux, Stéphane; Mahboubi, Assia; O'Connor, Russell; Biha, Sidi Ould; Pasca, Ioana; Rideau, Laurence; Solovyev, Alexey; Tassi, E.; Théry, Laurent

doi:10.1007/978-3-642-39634-2_14

Cited by 237 publications

(181 citation statements)

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“…The largest single piece of formalized mathematics to this day is a proof of the Feit-Thompson Odd Order Theorem, done by collaborative efforts of 15 mathematicians over a period of six years ( [7]). The proof itself which spans 250 pages of mathematics was formalized into more than 150,000 lines of code with roughly 4,000 definitions and 13,000 theorems.…”

Section: Introductionmentioning

confidence: 99%

DeepAlgebra - An Outline of a Program

Chojecki

2017

Lecture Notes in Computer Science

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Abstract. We outline a program in the area of formalization of mathematics to automate theorem proving in algebra and algebraic geometry. We propose a construction of a dictionary between automated theorem provers and (La)TeX exploiting syntactic parsers. We describe its application to a repository of human-written facts and definitions in algebraic geometry (The Stacks Project). We use deep learning techniques.

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Section: Introductionmentioning

confidence: 99%

DeepAlgebra - An Outline of a Program

Chojecki

2017

Lecture Notes in Computer Science

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“…However, the verification of the validity of the proof was highly difficult at 1980s [7] since its total number of pages are about 300. Gonthier et al [8] verified the proof by using SSReflect which is an extension of the proof assistant Coq. The formalized theorem and lemmas which are formalized in the process of the verification are utilized for the formalization of mathematical science.…”

Section: Introductionmentioning

confidence: 99%

Formal Verification of Robertson-Type Uncertainty Relation

Masuhara¹,

Kuriyama²,

Yoshida³

et al. 2015

JQIS

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Formal verification using interactive theorem provers have been noticed as a method of verification of proofs that are too big for humans to check the validity of them. The purpose of this work is to verify the validity of Robertson-type uncertainty relation toward verifying unconditional security of quantum key distributions. We verify the validity of the relation by using proof assistant Coq and it is turned out that the theorem regarding the relation formally holds. The source code for Coq which represents the validity of the theorem is printed in Appendix.

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“…This is foreshadowed by the computer-assisted proofs of Fowler, and of course of Appel-Haken of "Four colors suffice" fame. It has come closer with recent formal (as opposed to computer-assisted) proofs of the Jordan Curve Theorem [22], the Four-Colour Theorem [17], the Feit-Thompson Theorem of finite group theory [18], the Prime Number Theorem (both the elementary [2] and non-elementary [27] proofs) and Kepler's Conjecture [23].…”

mentioning

confidence: 97%

“…Theorems would be discovered and proved, even more so than in the "big machinery" scenario, in large, coordinated groups; the publication [18] describing the proof of Feit-Thompson, for instance, lists no fewer than fifteen co-authors! The possibility of simultaneous discovery, and still less of repeated discovery, would surely be sadly diminished.…”

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confidence: 99%

Some Comments on Multiple Discovery in Mathematics

Whitty¹

2017

jhummath

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SynopsisAmong perhaps many things common to Kuratowski's Theorem in graph theory, Reidemeister's Theorem in topology, and Cook's Theorem in theoretical computer science is this: all belong to the phenomenon of simultaneous discovery in mathematics. We are interested to know whether this phenomenon, and its close cousin repeated discovery, give rise to meaningful questions regarding causes, trends, categories, etc. With this in view we unearth many more examples, find some tenuous connections, and draw some tentative conclusions.The characterisation by forbidden minors of graph planarity, the Reidemiester moves in knot theory, and the NP-completeness of satisfiability are all discoveries of twentieth-century mathematics which occurred twice, more or less simultaneously, on different sides of the Atlantic. One may enumerate similar coincidences to one's heart's content. Michael Deakin wrote about some more in his admirable magazine Function [10], Frank Harary reminisces about a dozen or so in graph theory [25], and there are whole books on individual instances, such as Hall on Newton-Leibniz [24] and Roquette on the Brauer-Hasse-Noether Theorem [44].Mathematics is quintessentially a selfless joint endeavour, in which the pursuit of knowledge is its own reward. This is not a polite fiction; G.H. Hardy's somewhat sour remark [26] "if a mathematician . . . were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think the better of him if I did)"

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A Machine-Checked Proof of the Odd Order Theorem

Cited by 237 publications

References 32 publications

DeepAlgebra - An Outline of a Program

DeepAlgebra - An Outline of a Program

Formal Verification of Robertson-Type Uncertainty Relation

Some Comments on Multiple Discovery in Mathematics

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