In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].
@article{bwmeta1.element.doi-10_1515_forma-2017-0007,
author = {Yuichi Futa and Yasunari Shidama},
title = {Embedded Lattice and Properties of Gram Matrix},
journal = {Formalized Mathematics},
volume = {25},
year = {2017},
pages = {73-86},
zbl = {06726498},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2017-0007}
}
Yuichi Futa; Yasunari Shidama. Embedded Lattice and Properties of Gram Matrix. Formalized Mathematics, Tome 25 (2017) pp. 73-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2017-0007/