scholarly journals A FORMAL PROOF OF THE KEPLER CONJECTURE

2017 ◽  
Vol 5 ◽  
Author(s):  
THOMAS HALES ◽  
MARK ADAMS ◽  
GERTRUD BAUER ◽  
TAT DAT DANG ◽  
JOHN HARRISON ◽  
...  
Keyword(s):  
Formal Proof ◽  
Proof Assistants ◽  
Sphere Packings ◽  
Kepler Conjecture ◽  
Hol Light ◽  
Published Account

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.

scholarly journals External Tools for the Formal Proof of the Kepler Conjecture

10.29007/2l48 ◽  
2018 ◽  
Author(s):  
Thomas C. Hales
Keyword(s):  
Linear Programming ◽  
Three Dimensional ◽  
Formal Proof ◽  
Dimensional Euclidean Space ◽  
Proof Assistant ◽  
Original Proof ◽  
Computational Tools ◽  
Kepler Conjecture ◽  
Hol Light ◽  
Computer Calculations

The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the familiar cannonball arrangement. The proof of the Kepler conjecture was announced in 1998, but it went several years without publication because of the lingering doubts of referees about the correctness of the proof. In response to these publication hurdles, the Flyspeck project seeks to give a complete formal proof of the Kepler conjecture using the proof assistant HOL Light.The original proof of the Kepler relies on long computer calculations, and these calculations present special formalization challenges. A major part of the Flyspeck project requires the integration of external computational tools with the proof assistant. Some of these external tools are the GNU linear programming kit, AMPL (a modeling language for mathematical programming), Mathematica calculations, nonlinear optimization, and custom code in C++, C, C#, Java, and Objective Caml.Earlier work by A. Solovyev has implemented efficient linear programming in HOL Light. This talk will include a description of his more recent work that automates the link between linear programming and the Flyspeck project.

scholarly journals TacticToe: Learning to Reason with HOL4 Tactics

10.29007/ntlb ◽  
2018 ◽  
Author(s):  
Thibault Gauthier ◽  
Cezary Kaliszyk ◽  
Josef Urban
Keyword(s):  
Machine Learning ◽  
Automated Reasoning ◽  
Formal Proof ◽  
Small Scale ◽  
Proof Assistant ◽  
Proof Assistants ◽  
A Algorithm ◽  
Learning To Reason ◽  
Traditional Proof ◽  
Selection Of

Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such “hammer” techniques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences combined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39% of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32% in the same amount of time.

scholarly journals Bounds for Local Density of Sphere Packings and the Kepler Conjecture

2011 ◽  
pp. 27-57
Author(s):  
J. C. Lagarias
Keyword(s):  
Local Density ◽  
Sphere Packings ◽  
Kepler Conjecture

scholarly journals Bounds for Local Density of Sphere Packings and the Kepler Conjecture

2002 ◽  
Vol 27 (2) ◽  
pp. 165-193 ◽  
Author(s):  
J. C. Lagarias
Keyword(s):  
Local Density ◽  
Sphere Packings ◽  
Kepler Conjecture

scholarly journals Formalization of real analysis: a survey of proof assistants and libraries

2015 ◽  
Vol 26 (7) ◽  
pp. 1196-1233 ◽  
Author(s):  
SYLVIE BOLDO ◽  
CATHERINE LELAY ◽  
GUILLAUME MELQUIOND
Keyword(s):  
Real Analysis ◽  
Proof Assistants ◽  
Real Numbers ◽  
Similar Role ◽  
Proof Systems ◽  
Hol Light

In the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.

scholarly journals Gauge Integral

2017 ◽  
Vol 25 (3) ◽  
pp. 217-225
Author(s):  
Roland Coghetto
Keyword(s):  
Real Interval ◽  
Proof Assistants ◽  
Lebesgue Integral ◽  
Riemann Integral ◽  
Hol Light ◽  
Mizar Mathematical Library ◽  
System 1

Summary Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korniłowicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12]. A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in [10] and [4]. Using the Mizar system [1], we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval [a, b] (see [2], [3], [15], [14], [11]). In the next section we formalize that the Henstock-Kurzweil integral is linear. In the last section, we verified that a real-valued bounded integrable (in sense Darboux/Riemann [6, 7, 8]) function over a interval a, b is Gauge integrable. Note that, in accordance with the possibilities of the MML [9], we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in [7] (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.

A formal proof of Pick's Theorem

2011 ◽  
Vol 21 (4) ◽  
pp. 715-729 ◽  
Author(s):  
JOHN HARRISON
Keyword(s):  
Simple Polygon ◽  
Formal Proof ◽  
Integer Lattice ◽  
Lattice Points ◽  
Theorem Prover ◽  
Hol Light ◽  
Pick's Theorem ◽  
Arbitrary Polygon

Pick's Theorem relates the area of a simple polygon with vertices at integer lattice points to the number of lattice points in its inside and boundary. We describe a formal proof of this theorem using the HOL Light theorem prover. As sometimes happens for highly geometrical proofs, the formalisation turned out to be more work than initially expected. The difficulties arose mostly from formalising the triangulation process for an arbitrary polygon.

scholarly journals Experiences from Exporting Major Proof Assistant Libraries

2021 ◽  
Author(s):  
Michael Kohlhase ◽  
Florian Rabe
Keyword(s):  
Theorem Proving ◽  
Proof Assistant ◽  
Proof Assistants ◽  
Social Challenges ◽  
Future System ◽  
Hol Light ◽  
System Interoperability

AbstractThe interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented great theoretical, technical, and social challenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution, and our experiences will prove valuable to others undertaking such efforts.

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof

A Formal Proof of the Soundness of the Hybrid CPS Clock Theory

2020 ◽  
Author(s):  
Jianlin Wang ◽  
Chao Peng ◽  
Zhenbing Zeng
Keyword(s):  
Formal Proof
Sign in / Sign up

Export Citation Format

Share Document