We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.
@article{bwmeta1.element.doi-10_1515_forma-2015-0013,
author = {Roland Coghetto},
title = {Groups -- Additive Notation},
journal = {Formalized Mathematics},
volume = {23},
year = {2015},
pages = {127-160},
zbl = {1318.20001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0013}
}
Roland Coghetto. Groups – Additive Notation. Formalized Mathematics, Tome 23 (2015) pp. 127-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_forma-2015-0013/
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