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].
@article{bwmeta1.element.doi-10_1515_forma-2016-0005,
author = {Yuichi Futa and Yasunari Shidama},
title = {Lattice of $\mathbb{Z}$-module},
journal = {Formalized Mathematics},
volume = {24},
year = {2016},
pages = {49-68},
zbl = {1343.13017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0005}
}
Yuichi Futa; Yasunari Shidama. Lattice of ℤ-module. Formalized Mathematics, Tome 24 (2016) pp. 49-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2016-0005/
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