Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863.

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

Splitting Fields

Summary. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial p ∈ F [X] as the smallest field extension of F, in which p splits into linear factors. From this follows, that for a splitting field E of p we have E = F (A) where A is the set of p’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to F -isomorphims i.e. isomorphisms i with i|F = Id F . We prove that two splitting fields of p ∈ F [X] are F -isomorphic using the well-known technique [4], [3] of extending isomorphisms from F 1 → F 2 to F 1(a) → F 2(b) for a and b being algebraic over F 1 and F 2, respectively.

Algorithm NextFit for the Bin Packing ProblemThis work was supported by JSPS KAKENHI Grant Numbers JP20K11689, JP20K11676, JP16K00033, JP17K00013, JP20K11808, and JP17K00183.

Summary. The bin packing problem is a fundamental and important optimization problem in theoretical computer science [4], [6]. An instance is a sequence of items, each being of positive size at most one. The task is to place all the items into bins so that the total size of items in each bin is at most one and the number of bins that contain at least one item is minimum. Approximation algorithms have been intensively studied. Algorithm NextFit would be the simplest one. The algorithm repeatedly does the following: If the first unprocessed item in the sequence can be placed, in terms of size, additionally to the bin into which the algorithm has placed an item the last time, place the item into that bin; otherwise place the item into an empty bin. Johnson [5] proved that the number of the resulting bins by algorithm NextFit is less than twice the number of the fewest bins that are needed to contain all items. In this article, we formalize in Mizar [1], [2] the bin packing problem as follows: An instance is a sequence of positive real numbers that are each at most one. The task is to find a function that maps the indices of the sequence to positive integers such that the sum of the subsequence for each of the inverse images is at most one and the size of the image is minimum. We then formalize algorithm NextFit, its feasibility, its approximation guarantee, and the tightness of the approximation guarantee.

Pappus’s Hexagon Theorem in Real Projective PlaneThis work has been supported by the “Centre autonome de formation et de recherche en mathématiques et sciences avec assistants de preuve” ASBL (non-profit organization). Enterprise number: 0777.779.751. Belgium.

Summary. In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear” https://en.wikipedia.org/wiki/Pappus’s_hexagon_theorem . More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leończuk and Krzysztof Prażmowski. Eugeniusz Kusak and Wojciech Leończuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover9 https://www.cs.unm.edu/~mccune/prover9/ , the successor of the Otter prover and ott2miz by Josef Urban See its homepage https://github.com/JUrban/ott2miz [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].

Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear FunctionsThis work has been partially supported in 2019-2020 by the domestic research grant of University of Marketing and Distribution Sciences in Kobe (Japan).

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].

On Primary Ideals. Part I

Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.

Ascoli-Arzelà TheoremThis work was supported by JSPS KAKENHI Grant Numbers JP17K00182.

Summary. In this article we formalize the Ascoli-Arzelà theorem [5], [6], [8] in Mizar [1], [2]. First, we gave definitions of equicontinuousness and equiboundedness of a set of continuous functions [12], [7], [3], [9]. Next, we formalized the Ascoli-Arzelà theorem using those definitions, and proved this theorem.

On Weakly Associative Lattices and Near Lattices

Summary. The main aim of this article is to introduce formally two generalizations of lattices, namely weakly associative lattices and near lattices, which can be obtained from the former by certain weakening of the usual well-known axioms. We show selected propositions devoted to weakly associative lattices and near lattices from Chapter 6 of [15], dealing also with alternative versions of classical axiomatizations. Some of the results were proven in the Mizar [1], [2] system with the help of Prover9 [14] proof assistant.

Algebraic Extensions

Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.

Miscellaneous Graph Preliminaries. Part I

Summary This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series and are needed in upcoming articles.