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Author:Kazuhisa Ishida , Yasunari Shidama , Adam Grabowski
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.1-10De Gruyter
Abstract:Summary This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a ...
Author:Karol Pąk
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.11-19De Gruyter
Abstract:Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of ε...
Author:Karol Pąk
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.21-28De Gruyter
Abstract:Summary In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us A, B be a subsets of εn, and f : A → B be a homeomorphic. We prove that, if A is closed then f transform the boundary of A to the boundary of B; and if B is clos...
Author:Pratima K. Shah , Pauline N. Kawamoto , Mariusz Giero
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.29-35De Gruyter
Abstract:Summary In this article we formalize the definition of Decision-Free Petri Net (DFPN) presented in [19]. Then we formalize the concept of directed path and directed circuit nets in Petri nets to prove properties of DFPN. We also present the definition of firing transitions and tr...
Author:Grzegorz Bancerek
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.37-56De Gruyter
Abstract:Summary Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church- Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on [1].
Author:Noboru Endou
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.57-68De Gruyter
Abstract:Summary In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show...
Author:Keiko Narita , Noboru Endou , Yasunari Shidama
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.69-77De Gruyter
Abstract:Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of rea...
Author:Roland Coghetto
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.79-84De Gruyter
Abstract:Summary Schmets [22] has developed a measure theory from a generalized notion of a semiring of sets. Goguadze [15] has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved....
Author:Roland Coghetto
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.85-88De Gruyter
Abstract:Summary This article proposes the formalization of some examples of semiring of sets proposed by Goguadze [8] and Schmets [13].
Author:Adam Grabowski
Data Source:[J].Formalized Mathematics, 2014, Vol.22 (1), pp.89-97De Gruyter
Abstract:Summary Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the...

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