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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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SCImago Journal Rank (SJR) 2015: 0.134
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1898-9934
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In This Section
Volume 24, Issue 1 (Mar 2016)

Issues

Lattice of ℤ-module

Yuichi Futa
  • Corresponding author
  • Japan Advanced Institute of Science and Technology Ishikawa, Japan
/ Yasunari Shidama
  • Shinshu University Nagano, Japan
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0005

Summary

In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

MSC: 15A03; 15A63; 11E39; 03B35

Keywords: ℤ-lattice; Gram matrix; integral ℤ-lattice; positive definite ℤ-lattice

MML : identifier: ZMODLAT1; version: 8.1.04 5.36.1267

References

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  • [12] Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free ℤ-module. Formalized Mathematics, 22(4):277-289, 2014. doi:10.2478/forma-2014-0028. [Crossref]

  • [13] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29-49, 2015. doi:10.2478/forma-2015-0003. [Crossref]

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About the article

Received: 2015-12-30

Published Online: 2016-08-31

Published in Print: 2016-03-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0005. Export Citation

© by Yuichi Futa. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

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