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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.
Highlights
The booklet Six-Cornered Snowflake, which was written by Kepler in 1611, contains the statement of what is known as the Kepler conjecture: no packing of congruent balls in Euclidean three-space has density greater than that of the face-centered cubic packing [27]
Isabelle/HOL supports a form of computational reflection that allows executable terms to be exported as ML and executed, with the results of the computation re-integrated in the proof assistant as theorems
Using basic interval arithmetic and subdivisions, it would theoretically be possible to prove all nonlinear inequalities that arise in the project. This method does not work well in practice because the number of subdivisions required to establish some inequalities would be enormous, especially for multivariate inequalities. Both the C++ informal verification code and our formal verification procedure implemented in OCaml and HOL Light use improved interval extensions based on Taylor approximations
Summary
THOMAS HALES1, MARK ADAMS2,3, GERTRUD BAUER4, TAT DAT DANG5, JOHN HARRISON6, LE TRUONG HOANG7, CEZARY KALISZYK8, VICTOR MAGRON9, SEAN MCLAUGHLIN10, TAT THANG NGUYEN7, QUANG TRUONG NGUYEN1, TOBIAS NIPKOW11, STEVEN OBUA12, JOSEPH PLESO13, JASON RUTE14, ALEXEY SOLOVYEV15, THI HOAI AN TA7, NAM TRUNG TRAN7, THI DIEP TRIEU16, JOSEF URBAN17, KY VU18 and ROLAND ZUMKELLER19.
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