Summary In this paper problems 48, 80, 87, 89, and 124 from [7] are formalized, using the Mizar formalism [1], [2], [4]. The work is natural continuation of [5] and [3] as suggested in [6].
{"title":"Elementary Number Theory Problems. Part VII","authors":"Artur Korniłowicz","doi":"10.2478/forma-2023-0003","DOIUrl":"https://doi.org/10.2478/forma-2023-0003","url":null,"abstract":"Summary In this paper problems 48, 80, 87, 89, and 124 from [7] are formalized, using the Mizar formalism [1], [2], [4]. The work is natural continuation of [5] and [3] as suggested in [6].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995373","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}
Summary In this paper we present the Mizar formalization of the 36th problem from W. Sierpiński’s book “250 Problems in Elementary Number Theory” [10].
{"title":"Elementary Number Theory Problems. Part XI","authors":"Adam Naumowicz","doi":"10.2478/forma-2023-0021","DOIUrl":"https://doi.org/10.2478/forma-2023-0021","url":null,"abstract":"Summary In this paper we present the Mizar formalization of the 36th problem from W. Sierpiński’s book “250 Problems in Elementary Number Theory” [10].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"71 1","pages":"229 - 275"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346222","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}
Abstract This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements. The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors. The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups. The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].
{"title":"Internal Direct Products and the Universal Property of Direct Product Groups","authors":"Alexander M. Nelson","doi":"10.2478/forma-2023-0010","DOIUrl":"https://doi.org/10.2478/forma-2023-0010","url":null,"abstract":"Abstract This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements. The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors. The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups. The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135736687","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}
Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.
{"title":"Differentiation on Interval","authors":"Noboru Endou","doi":"10.2478/forma-2023-0002","DOIUrl":"https://doi.org/10.2478/forma-2023-0002","url":null,"abstract":"Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135891241","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}
Summary In this article sets of certain subgraphs of a graph are formalized in the Mizar system [7], [1], based on the formalization of graphs in [11] briefly sketched in [12]. The main result is the spanning subgraph theorem.
{"title":"Introduction to Graph Enumerations","authors":"Sebastian Koch","doi":"10.2478/forma-2023-0004","DOIUrl":"https://doi.org/10.2478/forma-2023-0004","url":null,"abstract":"Summary In this article sets of certain subgraphs of a graph are formalized in the Mizar system [7], [1], based on the formalization of graphs in [11] briefly sketched in [12]. The main result is the spanning subgraph theorem.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995205","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}
Summary In this article, we develop our formalised concept of Conway numbers as outlined in [9]. We focus mainly pre-order properties, birthday arithmetic contained in the Chapter 1, Properties of Order and Equality of John Conway’s seminal book. We also propose a method for the selection of class representatives respecting the relation defined by the pre-ordering in order to facilitate combining the results obtained for the original and tree-theoretic definitions of Conway numbers.
{"title":"Integration of Game Theoretic and Tree Theoretic Approaches to Conway Numbers","authors":"Karol Pąk","doi":"10.2478/forma-2023-0019","DOIUrl":"https://doi.org/10.2478/forma-2023-0019","url":null,"abstract":"Summary In this article, we develop our formalised concept of Conway numbers as outlined in [9]. We focus mainly pre-order properties, birthday arithmetic contained in the Chapter 1, Properties of Order and Equality of John Conway’s seminal book. We also propose a method for the selection of class representatives respecting the relation defined by the pre-ordering in order to facilitate combining the results obtained for the original and tree-theoretic definitions of Conway numbers.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"38 1","pages":"205 - 213"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345055","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}
{"title":"Elementary Number Theory Problems. Part X – Diophantine Equations","authors":"Artur Korniłowicz","doi":"10.2478/forma-2023-0016","DOIUrl":"https://doi.org/10.2478/forma-2023-0016","url":null,"abstract":"Summary This paper continues the formalization of problems defined in the book “250 Problems in Elementary Number Theory” by Wacław Sierpiński.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"80 1","pages":"171 - 180"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345917","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}
Summary The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable in terms of “bag” (as described in detail in [9]), the same notion as multiset over a finite set, in the Mizar system [1], [2]. Polynomial rings and ring of formal power series, both in one variable, have been formalized in [6], [5] respectively, and elements of these rings are represented by infinite sequences of scalars. On the other hand, formalization of a multivariate polynomial requires extra techniques of using “bag” to represent monomials of variables, and polynomials are formalized as a function from bags of variables to the scalar ring. This means the way of construction of the rings are different between single variable and multi variables case (which implies some tedious constructions, e.g. in the case of ten variables in [8], or generally in the problem of prime representing polynomial [7]). Introducing bag-based construction to one variable polynomial ring provides straight way to apply mathematical induction to polynomial rings with respect to the number of variables. Another consequence from the article, a polynomial ring is a subring of an algebra [4] over the same scalar ring, namely a corresponding formal power series. A sketch of actual formalization of the article is consists of the following four steps: 1. translation between Bags 1 (the set of all bags of a singleton) and N; 2. formalization of a bag-based formal power series in multivariable case over a commutative ring denoted by Formal-Series ( n, R ); 3. formalization of a polynomial ring in one variable by restricting one variable case denoted by Polynom-Ring (1, R ). A formal proof of the fact that polynomial rings are a subring of Formal-Series ( n, R ), that is R -Algebra, is included as well; 4. formalization of a ring isomorphism to the existing polynomial ring in one variable given by sequence: Polynom-Ring (1, R ) →˜ Polynom-Ring .
本文关注的是在Mizar系统[1]，[2]中，用“bag”(如[9]中详细描述的)来形式化一个变量中的多变量形式幂级数和多项式[3]，与有限集合上的多集的概念相同。一元多项式环和形式幂级数环分别形式化在[6]和[5]中，这两个环的元素用无穷数列标量表示。另一方面，多元多项式的形式化需要额外的技术，即使用“袋”来表示变量的单项式，多项式被形式化为从变量袋到标量环的函数。这意味着环的构造方式在单变量和多变量情况下是不同的(这意味着一些繁琐的构造，例如在[8]中有十个变量的情况下，或者通常在素数表示多项式[7]的问题中)。在单变量多项式环中引入基于袋的构造，为多项式环在变量数上的数学归纳法应用提供了直接的途径。本文的另一个结论是，多项式环是同一标量环上代数[4]的子环，即相应的形式幂级数。一个草图的实际形式化的文章是由以下四个步骤:1。袋子1(所有袋子的集合)和N之间的转换;2. 交换环上基于袋的多变量形式幂级数的形式化，表示为formal - series (n, R)3.用多项式环(1,R)来限制一个变量的情况，从而形式化了一个变量多项式环。多项式环是形式级数(n, R)的子级数，即R -代数的一个形式证明;4. 多项式环在单变量上的同构的形式化:多项式环(1,R)→→多项式环。
{"title":"On Bag of 1. Part I","authors":"Yasushige Watase","doi":"10.2478/forma-2023-0001","DOIUrl":"https://doi.org/10.2478/forma-2023-0001","url":null,"abstract":"Summary The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable in terms of “bag” (as described in detail in [9]), the same notion as multiset over a finite set, in the Mizar system [1], [2]. Polynomial rings and ring of formal power series, both in one variable, have been formalized in [6], [5] respectively, and elements of these rings are represented by infinite sequences of scalars. On the other hand, formalization of a multivariate polynomial requires extra techniques of using “bag” to represent monomials of variables, and polynomials are formalized as a function from bags of variables to the scalar ring. This means the way of construction of the rings are different between single variable and multi variables case (which implies some tedious constructions, e.g. in the case of ten variables in [8], or generally in the problem of prime representing polynomial [7]). Introducing bag-based construction to one variable polynomial ring provides straight way to apply mathematical induction to polynomial rings with respect to the number of variables. Another consequence from the article, a polynomial ring is a subring of an algebra [4] over the same scalar ring, namely a corresponding formal power series. A sketch of actual formalization of the article is consists of the following four steps: 1. translation between Bags 1 (the set of all bags of a singleton) and N; 2. formalization of a bag-based formal power series in multivariable case over a commutative ring denoted by Formal-Series ( n, R ); 3. formalization of a polynomial ring in one variable by restricting one variable case denoted by Polynom-Ring (1, R ). A formal proof of the fact that polynomial rings are a subring of Formal-Series ( n, R ), that is R -Algebra, is included as well; 4. formalization of a ring isomorphism to the existing polynomial ring in one variable given by sequence: Polynom-Ring (1, R ) →˜ Polynom-Ring .","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995200","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}
{"title":"Elementary Number Theory Problems. Part IX","authors":"Artur Korniłowicz","doi":"10.2478/forma-2023-0015","DOIUrl":"https://doi.org/10.2478/forma-2023-0015","url":null,"abstract":"Summary This paper continues the formalization of chosen problems defined in the book “250 Problems in Elementary Number Theory” by Wacław Sierpiński.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"239 1","pages":"161 - 169"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346884","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}
Summary Conway’s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway’s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally. This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.
{"title":"The Ring of Conway Numbers in Mizar","authors":"Karol Pąk","doi":"10.2478/forma-2023-0020","DOIUrl":"https://doi.org/10.2478/forma-2023-0020","url":null,"abstract":"Summary Conway’s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway’s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally. This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"23 1","pages":"215 - 228"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139344652","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}
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