<abstract xmlns="http://www.w3.org/1999/xhtml">Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].</abstract>

Morley’s Trisector Theorem

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article we introduce necessary notation and definitions to prove the Euler’s Partition Theorem according to H.S. Wilf’s lecture notes [31]. Our aim is to create an environment which allows to formalize the theorem in a way that is as similar as possible to the original informal proof. Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [30].</abstract>

Flexary Operations

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article we prove the Euler’s Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf’s lecture notes [28] (see also [1]). Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27].</abstract>

Euler’s Partition Theorem

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1]. </abstract>

Introduction to Diophantine Approximation

<abstract xmlns="http://www.w3.org/1999/xhtml">We formalize that the image of a semiring of sets [17] by an injective function is a semiring of sets.We offer a non-trivial example of a semiring of sets in a topological space [21]. Finally, we show that the finite product of a semiring of sets is also a semiring of sets [21] and that the finite product of a classical semiring of sets [8] is a classical semiring of sets. In this case, we use here the notation from the book of Aliprantis and Border [1].</abstract>

Finite Product of Semiring of Sets

<abstract xmlns="http://www.w3.org/1999/xhtml">Nelson algebras were first studied by Rasiowa and Białynicki- Birula [1] under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole [3, 4], and [2] the name “Nelson algebras” was adopted - which is now commonly used to show the correspondence with Nelson’s paper [14] on constructive logic with strong negation. By a Nelson algebra we mean an abstract algebra 〈L, T, -, ¬, →, ⇒, ⊔, ⊓〉where L is the carrier, − is a quasi-complementation (Rasiowa used the sign ~, but in Mizar “−” should be used to follow the approach described in [12] and [10]), ¬ is a weak pseudo-complementation → is weak relative pseudocomplementation and ⇒ is implicative operation. ⊔ and ⊓ are ordinary lattice binary operations of supremum and infimum. In this article we give the definition and basic properties of these algebras according to [16] and [15]. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in [15]. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets [9] in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 [16]. </abstract>

Two Axiomatizations of Nelson Algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">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.</abstract>

Groups – Additive Notation

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Formalized Mathematics

  Pioneering Journal in the field of formalized and mechanically checked mathematics. Excellent articles on latest research in formal methods and computer-checked mathematics. Full-length original research articles in English. Archival issues available at the journal's home page fm.mizar.org/.   To build a uniform and extensible computer-checked repository of mathematical knowledge.  Reconstruction of mathematics based on mechanization of mathematical reasoning.  To check a wide range of mathematical theorems using a computer in an automatic fashion.  General mathematics: especially topology, lattice theory, functional analysis.  Applications: computer-aided verification of software and hardware, computer-aided teaching of mathematics.   Why subscribe and read   Pioneering source of high quality research in formalized and mechanically checked mathematics.  The oldest journal in the field, a medium for the publication of research reporting pioneering findings of broad mathematical significance that are likely to advance the direction and thinking of the field.  Long-term experience in publishing original articles on mechanizing mathematics and various aspects of automated reasoning, since 1990.   Why submit   Fair and constructive peer review by international experts in the field (three referees are designated for the assessment of each manuscript) - double-blind review process.  New publishing technology: articles automatically translated from a formal language into English and automatically typeset, available in both forms - electronic and printed.  Articles abstracted by online indexing services and databases.   Rejection Rate   59% of submitted articles.  The rejection percentage is described in detail in the following paper, pp.239-240: dx.doi.org/10.1007/978-3-540-73086-6_20 Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. Back Issues fm.mizar.org/   Other Files  Addenda 2008Formalized Mathematics Bibliography File.Formalized Mathematics External Bibliography File.  "In years: 2016-2017, the English version of Formalized Mathematics was financed under agreements 548/P-DUN/2016 and 548/1/P-DUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science." 