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On Weakly Associative Lattices and Near Lattices 关于弱结合格和近格
IF 0.3
Formalized Mathematics Pub Date : 2021-07-01 DOI: 10.2478/forma-2021-0008
Damian Sawicki, Adam Grabowski
{"title":"On Weakly Associative Lattices and Near Lattices","authors":"Damian Sawicki, Adam Grabowski","doi":"10.2478/forma-2021-0008","DOIUrl":"https://doi.org/10.2478/forma-2021-0008","url":null,"abstract":"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.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78090267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions 由三角形函数和分段线性函数组成的隶属函数的一些性质
IF 0.3
Formalized Mathematics Pub Date : 2021-07-01 DOI: 10.2478/forma-2021-0011
T. Mitsuishi
{"title":"Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions","authors":"T. Mitsuishi","doi":"10.2478/forma-2021-0011","DOIUrl":"https://doi.org/10.2478/forma-2021-0011","url":null,"abstract":"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].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75989624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Miscellaneous Graph Preliminaries. Part I 杂项图初步。第一部分
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0003
Sebastian Koch
{"title":"Miscellaneous Graph Preliminaries. Part I","authors":"Sebastian Koch","doi":"10.2478/forma-2021-0003","DOIUrl":"https://doi.org/10.2478/forma-2021-0003","url":null,"abstract":"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.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90900159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Inverse Function Theorem. Part I1 反函数定理。I1一部分
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0002
Kazuhisa Nakasho, Yuichi Futa
{"title":"Inverse Function Theorem. Part I1","authors":"Kazuhisa Nakasho, Yuichi Futa","doi":"10.2478/forma-2021-0002","DOIUrl":"https://doi.org/10.2478/forma-2021-0002","url":null,"abstract":"Summary In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89744514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Functional Space Consisted by Continuous Functions on Topological Space 拓扑空间上由连续函数组成的泛函空间
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0005
Hiroshi Yamazaki, K. Miyajima, Y. Shidama
{"title":"Functional Space Consisted by Continuous Functions on Topological Space","authors":"Hiroshi Yamazaki, K. Miyajima, Y. Shidama","doi":"10.2478/forma-2021-0005","DOIUrl":"https://doi.org/10.2478/forma-2021-0005","url":null,"abstract":"Summary In this article, using the Mizar system [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation 交换环的求导与求导幂的莱布尼茨公式
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0001
Yasushige Watase
{"title":"Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation","authors":"Yasushige Watase","doi":"10.2478/forma-2021-0001","DOIUrl":"https://doi.org/10.2478/forma-2021-0001","url":null,"abstract":"Summary In this article we formalize in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say D, is defined generally as a map from a commutative ring A to A-Module M with specific conditions. However we start with simpler case, namely dom D = rng D. This allows to define a derivation in other rings such as a polynomial ring. A derivation is a map D : A → A satisfying the following conditions: (i) D(x + y) = Dx + Dy, (ii) D(xy) = xDy + yDx, ∀x, y ∈ A. Typical properties are formalized such as: D(∑i=1nxi)=∑i=1nDxi Dleft( {sumlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {D{x_i}} and D(∏i=1nxi)=∑i=1nx1x2⋯Dxi⋯xn(∀xi∈A). Dleft( {prodlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {{x_1}{x_2} cdots D{x_i} cdots {x_n}} left( {forall {x_i} in A} right). We also formalized the Leibniz Formula for power of derivation D : Dn(xy)=∑i=0n(in)DixDn-iy. {D^n}left( {xy} right) = sumlimits_{i = 0}^n {left( {_i^n} right){D^i}x{D^{n - i}}y.} Lastly applying the definition to the polynomial ring of A and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81477809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Elementary Number Theory Problems. Part II 初等数论问题。第二部分
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0006
Artur Korniłowicz, Dariusz Surowik
{"title":"Elementary Number Theory Problems. Part II","authors":"Artur Korniłowicz, Dariusz Surowik","doi":"10.2478/forma-2021-0006","DOIUrl":"https://doi.org/10.2478/forma-2021-0006","url":null,"abstract":"Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p2 + 1 = q2 + r2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22n + k (n = 1, 2, . . . ) are composite.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89615122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Algebraic Extensions 代数扩展
IF 0.3
Formalized Mathematics Pub Date : 2021-04-01 DOI: 10.2478/forma-2021-0004
Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller
{"title":"Algebraic Extensions","authors":"Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller","doi":"10.2478/forma-2021-0004","DOIUrl":"https://doi.org/10.2478/forma-2021-0004","url":null,"abstract":"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.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74448802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
General Theory and Tools for Proving Algorithms in Nominative Data Systems 标称数据系统中证明算法的一般理论和工具
IF 0.3
Formalized Mathematics Pub Date : 2020-12-01 DOI: 10.2478/forma-2020-0024
Adrian Jaszczak
{"title":"General Theory and Tools for Proving Algorithms in Nominative Data Systems","authors":"Adrian Jaszczak","doi":"10.2478/forma-2020-0024","DOIUrl":"https://doi.org/10.2478/forma-2020-0024","url":null,"abstract":"Summary In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future. This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and postconditions [17], [19], [7], [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75016281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Partial Correctness of an Algorithm Computing Lucas Sequences 卢卡斯序列计算算法的部分正确性
IF 0.3
Formalized Mathematics Pub Date : 2020-12-01 DOI: 10.2478/forma-2020-0025
Adrian Jaszczak
{"title":"Partial Correctness of an Algorithm Computing Lucas Sequences","authors":"Adrian Jaszczak","doi":"10.2478/forma-2020-0025","DOIUrl":"https://doi.org/10.2478/forma-2020-0025","url":null,"abstract":"Summary In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1]. i := 0 s := x b := y c := x while (i <> n) c := s s := b ps := p*s qc := q*c b := ps − qc i := i + j return s This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [18], [21], [7], [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78744126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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