Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ab” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab). 2. A pair of an Abelian group Mab and a ring homomorphism R→ρ Rmathop to limits^rho End (Mab) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_ModR(M, N). 4. The set R-module homomorphisms of M to N, denoted by HomR(M, N), forms R-module. 5. A formal proof of HomR(¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself as a left R-module. 6. A formal proof of AbGrLMod(M′ab, ρ′) ≅ M where M′ab is an abelian group obtained by removing the scalar multiplication from M, and ρ′ is obtained by currying the scalar multiplication of M. The removal of the multiplication from M has been done by the forgettable functor defined as AbGr in the article.
{"title":"Ring of Endomorphisms and Modules over a Ring","authors":"Yasushige Watase","doi":"10.2478/forma-2022-0016","DOIUrl":"https://doi.org/10.2478/forma-2022-0016","url":null,"abstract":"Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ab” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab). 2. A pair of an Abelian group Mab and a ring homomorphism R→ρ Rmathop to limits^rho End (Mab) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_ModR(M, N). 4. The set R-module homomorphisms of M to N, denoted by HomR(M, N), forms R-module. 5. A formal proof of HomR(¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself as a left R-module. 6. A formal proof of AbGrLMod(M′ab, ρ′) ≅ M where M′ab is an abelian group obtained by removing the scalar multiplication from M, and ρ′ is obtained by currying the scalar multiplication of M. The removal of the multiplication from M has been done by the forgettable functor defined as AbGr in the article.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"164 1","pages":"211 - 221"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75989002","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 paper reports on the formalization of ten selected problems from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [5] using the Mizar system [4], [1], [2]. Problems 12, 13, 31, 32, 33, 35 and 40 belong to the chapter devoted to the divisibility of numbers, problem 47 concerns relatively prime numbers, whereas problems 76 and 79 are taken from the chapter on prime and composite numbers.
{"title":"Elementary Number Theory Problems. Part V","authors":"Artur Korniłowicz, Adam Naumowicz","doi":"10.2478/forma-2022-0018","DOIUrl":"https://doi.org/10.2478/forma-2022-0018","url":null,"abstract":"Summary This paper reports on the formalization of ten selected problems from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [5] using the Mizar system [4], [1], [2]. Problems 12, 13, 31, 32, 33, 35 and 40 belong to the chapter devoted to the divisibility of numbers, problem 47 concerns relatively prime numbers, whereas problems 76 and 79 are taken from the chapter on prime and composite numbers.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"606 1","pages":"229 - 234"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76795328","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 main purpose of the article is to construct a sophisticated polynomial proposed by Matiyasevich and Robinson [5] that is often used to reduce the number of unknowns in diophantine representations, using the Mizar [1], [2] formalism. The polynomial Jk(a1,…,ak,x)=∏ɛ1,…,ɛk∈{ ±1 }(x+ɛ1a1+ɛ2a2W)+…+ɛkakWk-1 {J_k}left( {{a_1}, ldots ,{a_k},x} right) = prodlimits_{{varepsilon _1}, ldots ,{varepsilon _k} in left{ { pm 1} right}} {left( {x + {varepsilon _1}sqrt {{a_1}} + {varepsilon _2}sqrt {{a_2}} W} right) + ldots + {varepsilon _k}sqrt {{a_k}} {W^{k - 1}}} with W=∑i=1kx i2 W = sumnolimits_{i = 1}^k {x_i^2} has integer coefficients and Jk(a1, . . ., ak, x) = 0 for some a1, . . ., ak, x ∈ ℤ if and only if a1, . . ., ak are all squares. However although it is nontrivial to observe that this expression is a polynomial, i.e., eliminating similar elements in the product of all combinations of signs we obtain an expression where every square root will occur with an even power. This work has been partially presented in [7].
{"title":"Prime Representing Polynomial with 10 Unknowns – Introduction","authors":"Karol Pąk","doi":"10.2478/forma-2022-0013","DOIUrl":"https://doi.org/10.2478/forma-2022-0013","url":null,"abstract":"Summary The main purpose of the article is to construct a sophisticated polynomial proposed by Matiyasevich and Robinson [5] that is often used to reduce the number of unknowns in diophantine representations, using the Mizar [1], [2] formalism. The polynomial Jk(a1,…,ak,x)=∏ɛ1,…,ɛk∈{ ±1 }(x+ɛ1a1+ɛ2a2W)+…+ɛkakWk-1 {J_k}left( {{a_1}, ldots ,{a_k},x} right) = prodlimits_{{varepsilon _1}, ldots ,{varepsilon _k} in left{ { pm 1} right}} {left( {x + {varepsilon _1}sqrt {{a_1}} + {varepsilon _2}sqrt {{a_2}} W} right) + ldots + {varepsilon _k}sqrt {{a_k}} {W^{k - 1}}} with W=∑i=1kx i2 W = sumnolimits_{i = 1}^k {x_i^2} has integer coefficients and Jk(a1, . . ., ak, x) = 0 for some a1, . . ., ak, x ∈ ℤ if and only if a1, . . ., ak are all squares. However although it is nontrivial to observe that this expression is a polynomial, i.e., eliminating similar elements in the product of all combinations of signs we obtain an expression where every square root will occur with an even power. This work has been partially presented in [7].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"5 1","pages":"169 - 198"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86167834","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 paper, using the Mizar system [1], [2], provides useful tools for working with real linear spaces and real normed spaces. These include the identification of a real number set with a one-dimensional real normed space, the relationships between real linear spaces and real Euclidean spaces, the transformation from a real linear space to a real vector space, and the properties of basis and dimensions of real linear spaces. We referred to [6], [10], [8], [9] in this formalization.
{"title":"Transformation Tools for Real Linear Spaces","authors":"Kazuhisa Nakasho","doi":"10.2478/forma-2022-0008","DOIUrl":"https://doi.org/10.2478/forma-2022-0008","url":null,"abstract":"Summary This paper, using the Mizar system [1], [2], provides useful tools for working with real linear spaces and real normed spaces. These include the identification of a real number set with a one-dimensional real normed space, the relationships between real linear spaces and real Euclidean spaces, the transformation from a real linear space to a real vector space, and the properties of basis and dimensions of real linear spaces. We referred to [6], [10], [8], [9] in this formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"47 1","pages":"93 - 98"},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86070907","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 vertex, edge and total colorings of graphs are formalized in the Mizar system [4] and [1], based on the formalization of graphs in [5].
本文在文献[5]形式化图的基础上，在Mizar系统[4]和[1]中形式化了图的顶点、边和总着色。
{"title":"Introduction to Graph Colorings","authors":"Sebastian Koch","doi":"10.2478/forma-2022-0009","DOIUrl":"https://doi.org/10.2478/forma-2022-0009","url":null,"abstract":"Summary In this article vertex, edge and total colorings of graphs are formalized in the Mizar system [4] and [1], based on the formalization of graphs in [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"49 1","pages":"99 - 124"},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74161536","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 study, using the Mizar system [1], [2], we reuse formalization e orts in fuzzy sets described in [5] and [6]. This time the centroid method which is one of the fuzzy inference processes is formulated [10]. It is the most popular of all defuzzied methods ([11], [13], [7]) – here, defuzzified crisp value is obtained from domain of membership function as weighted average [8]. Since the integral is used in centroid method, the integrability and bounded properties of membership functions are also mentioned to fill the formalization gaps present in the Mizar Mathematical Library, as in the case of another fuzzy operators [4]. In this paper, the properties of piecewise linear functions consisting of two straight lines are mainly described.
{"title":"Definition of Centroid Method as Defuzzification","authors":"T. Mitsuishi","doi":"10.2478/forma-2022-0010","DOIUrl":"https://doi.org/10.2478/forma-2022-0010","url":null,"abstract":"Summary In this study, using the Mizar system [1], [2], we reuse formalization e orts in fuzzy sets described in [5] and [6]. This time the centroid method which is one of the fuzzy inference processes is formulated [10]. It is the most popular of all defuzzied methods ([11], [13], [7]) – here, defuzzified crisp value is obtained from domain of membership function as weighted average [8]. Since the integral is used in centroid method, the integrability and bounded properties of membership functions are also mentioned to fill the formalization gaps present in the Mizar Mathematical Library, as in the case of another fuzzy operators [4]. In this paper, the properties of piecewise linear functions consisting of two straight lines are mainly described.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"122 1","pages":"125 - 134"},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79457245","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 We formalize in Mizar [1], [2] the notion of characteristic subgroups using the definition found in Dummit and Foote [3], as subgroups invariant under automorphisms from its parent group. Along the way, we formalize notions of Automorphism and results concerning centralizers. Much of what we formalize may be found sprinkled throughout the literature, in particular Gorenstein [4] and Isaacs [5]. We show all our favorite subgroups turn out to be characteristic: the center, the derived subgroup, the commutator subgroup generated by characteristic subgroups, and the intersection of all subgroups satisfying a generic group property.
{"title":"Characteristic Subgroups","authors":"Alexander M. Nelson","doi":"10.2478/forma-2022-0007","DOIUrl":"https://doi.org/10.2478/forma-2022-0007","url":null,"abstract":"Summary We formalize in Mizar [1], [2] the notion of characteristic subgroups using the definition found in Dummit and Foote [3], as subgroups invariant under automorphisms from its parent group. Along the way, we formalize notions of Automorphism and results concerning centralizers. Much of what we formalize may be found sprinkled throughout the literature, in particular Gorenstein [4] and Isaacs [5]. We show all our favorite subgroups turn out to be characteristic: the center, the derived subgroup, the commutator subgroup generated by characteristic subgroups, and the intersection of all subgroups satisfying a generic group property.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"6 21 1","pages":"79 - 91"},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78509108","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 problems 11, 16, 19–24, 39, 44, 46, 74, 75, 77, 82, and 176 from [10] are formalized as described in [6], using the Mizar formalism [1], [2], [4]. Problems 11 and 16 from the book are formulated as several independent theorems. Problem 46 is formulated with a given example of required properties. Problem 77 is not formulated using triangles as in the book is.
{"title":"Elementary Number Theory Problems. Part III","authors":"Artur Korniłowicz","doi":"10.2478/forma-2022-0011","DOIUrl":"https://doi.org/10.2478/forma-2022-0011","url":null,"abstract":"Summary In this paper problems 11, 16, 19–24, 39, 44, 46, 74, 75, 77, 82, and 176 from [10] are formalized as described in [6], using the Mizar formalism [1], [2], [4]. Problems 11 and 16 from the book are formulated as several independent theorems. Problem 46 is formulated with a given example of required properties. Problem 77 is not formulated using triangles as in the book is.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"11 1","pages":"135 - 158"},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80223686","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 paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.
{"title":"Isomorphism between Spaces of Multilinear Maps and Nested Compositions over Real Normed Vector Spaces","authors":"Kazuhisa Nakasho, Yuichi Futa","doi":"10.2478/forma-2022-0006","DOIUrl":"https://doi.org/10.2478/forma-2022-0006","url":null,"abstract":"Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"26 1","pages":"67 - 77"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84709027","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 paper is a continuation of Inoué [5]. As already mentioned in the paper, a number of intuitionistic provable formulas are given with a Hilbert-style proof. For that, we make use of a family of intuitionistic deduction theorems, which are also presented in this paper by means of Mizar system [2], [1]. Our axiom system of intuitionistic propositional logic IPC is based on the propositional subsystem of H1-IQC in Troelstra and van Dalen [6, p. 68]. We also owe Heyting [4] and van Dalen [7]. Our treatment of a set-theoretic intuitionistic deduction theorem is due to Agata Darmochwał’s Mizar article “Calculus of Quantifiers. Deduction Theorem” [3].
本文是inou[5]的延续。正如文中已经提到的，我们用hilbert式证明给出了一些直观可证明的公式。为此，我们利用了一系列直觉演绎定理，这些定理在本文中也通过Mizar系统[2]，[1]给出。我们的直觉命题逻辑IPC公理系统是基于Troelstra和van Dalen [6, p. 68]的H1-IQC命题子系统。我们还欠何亭[4]和范达伦[7]。我们对集合论直觉演绎定理的处理是由于Agata darmochwaov的Mizar文章“量词演算”。演绎定理”[3]。
{"title":"Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part II","authors":"Takao Inoué","doi":"10.2478/forma-2022-0001","DOIUrl":"https://doi.org/10.2478/forma-2022-0001","url":null,"abstract":"Summary This paper is a continuation of Inoué [5]. As already mentioned in the paper, a number of intuitionistic provable formulas are given with a Hilbert-style proof. For that, we make use of a family of intuitionistic deduction theorems, which are also presented in this paper by means of Mizar system [2], [1]. Our axiom system of intuitionistic propositional logic IPC is based on the propositional subsystem of H1-IQC in Troelstra and van Dalen [6, p. 68]. We also owe Heyting [4] and van Dalen [7]. Our treatment of a set-theoretic intuitionistic deduction theorem is due to Agata Darmochwał’s Mizar article “Calculus of Quantifiers. Deduction Theorem” [3].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"336 1","pages":"1 - 12"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76380488","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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