{"title":"Elementary Number Theory Problems. Part VIII","authors":"Artur Korniłowicz","doi":"10.2478/forma-2023-0009","DOIUrl":"https://doi.org/10.2478/forma-2023-0009","url":null,"abstract":"Abstract In this paper problems 25, 86, 88, 105, 111, 137–142, and 184–185 from [12] are formalized, using the Mizar formalism [3], [1], [4]. This is a continuation of the work from [5], [6], and [2] as suggested in [8]. The automatization of selected lemmas from [11] proven in this paper as proposed in [9] could be an interesting future work.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"30 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":"135738202","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":"Formalization of Orthogonal Decomposition for Hilbert Spaces","authors":"Hiroyuki Okazaki","doi":"10.2478/forma-2022-0023","DOIUrl":"https://doi.org/10.2478/forma-2022-0023","url":null,"abstract":"Summary In this article, we formalize the theorems about orthogonal decomposition of Hilbert spaces, using the Mizar system [1], [2]. For any subspace S of a Hilbert space H, any vector can be represented by the sum of a vector in S and a vector orthogonal to S. The formalization of orthogonal complements of Hilbert spaces has been stored in the Mizar Mathematical Library [4]. We referred to [5] and [6] in the formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"56 1","pages":"295 - 299"},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83161792","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":"Prime Representing Polynomial with 10 Unknowns – Introduction. Part II","authors":"Karol Pąk","doi":"10.2478/forma-2022-0020","DOIUrl":"https://doi.org/10.2478/forma-2022-0020","url":null,"abstract":"Summary In our previous work [7] we prove that the set of prime numbers is diophantine using the 26-variable polynomial proposed in [4]. In this paper, we focus on the reduction of the number of variables to 10 and it is the smallest variables number known today [5], [10]. Using the Mizar [3], [2] system, we formalize the first step in this direction by proving Theorem 1 [5] formulated as follows: Let k ∈ ℕ. Then k is prime if and only if there exists f, i, j, m, u ∈ ℕ+, r, s, t ∈ ℕ unknowns such that DFI is square ∧ (M2-1)S2+1 is square ∧((MU)2-1)T2+1 is square∧(4f2-1)(r-mSTU)2+4u2S2T2<8fuST(r-mSTU)FL|(H-C)Z+F(f+1)Q+F(k+1)((W2-1)Su-W2u2+1) matrix{ {DFI,is,square,,,{Lambda},left( {{M^2} - 1} right){S^2} + 1,,is,,square,,{Lambda}} hfill cr {left( {{{left( {MU} right)}^2} - 1} right){T^2} + 1,,is,,square{Lambda}} hfill cr {left( {4{f^2} - 1} right){{left( {r - mSTU} right)}^2} + 4{u^2}{S^2}{T^2} < 8fuSTleft( {r - mSTU} right)} hfill cr {FL|left( {H - C} right)Z + Fleft( {f + 1} right)Q + Fleft( {k + 1} right)left( {left( {{W^2} - 1} right)Su - {W^2}{u^2} + 1} right)} hfill cr } where auxiliary variables A − I, L, M, S − W, Q ∈ ℤ are simply abbreviations defined as follows W = 100fk(k + 1), U = 100u3W 3 + 1, M = 100mUW + 1, S = (M −1)s+k+1, T = (MU −1)t+W −k+1, Q = 2MW −W 2−1, L = (k+1)Q, A = M(U +1), B = W +1, C = r +W +1, D = (A2 −1)C2 +1, E = 2iC2LD, F = (A2 −1)E2 +1, G = A+F (F −A), H = B+2(j −1)C, I = (G2 −1)H2 +1. It is easily see that (0.1) uses 8 unknowns explicitly along with five implicit one for each diophantine relationship: is square, inequality, and divisibility. Together with k this gives a total of 14 variables. This work has been partially presented in [8].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"40 1","pages":"245 - 253"},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77695865","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":"Prime Representing Polynomial with 10 Unknowns","authors":"Karol Pąk","doi":"10.2478/forma-2022-0021","DOIUrl":"https://doi.org/10.2478/forma-2022-0021","url":null,"abstract":"Summary In this article we formalize in Mizar [1], [2] the final step of our attempt to formally construct a prime representing polynomial with 10 variables proposed by Yuri Matiyasevich in [4]. The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomial and substituted ones as i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zero power. We consider both adding unused variables and eliminating them. The second part of the paper deals with the construction of the polynomial proposed by Yuri Matiyasevich. First, we introduce a diophantine polynomial over 4 variables that has roots in integers if and only if indicated variable is the square of a natural number, and another two is the square of an odd natural number. We modify the polynomial by adding two variables in such a way that the root additionally requires the divisibility of these added variables. Then we modify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructing a prime representing polynomial with 10 variables. This work has been partially presented in [8] with the obtained polynomial constructing a prime representing polynomial with 10 variables in Theorem (85).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"30 1","pages":"255 - 279"},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85646399","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":"Existence and Uniqueness of Algebraic Closures","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2022-0022","DOIUrl":"https://doi.org/10.2478/forma-2022-0022","url":null,"abstract":"Summary This is the second part of a two-part article formalizing existence and uniqueness of algebraic closures, using the Mizar [2], [1] formalism. Our proof follows Artin’s classical one as presented by Lang in [3]. In the first part we proved that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]F simultaneously. To do so we needed the polynomial ring F [X1, X2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]F. The desired field extension E then is F [X1, X2, …]I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In this second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F. To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F. In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"138 1","pages":"281 - 294"},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77114598","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":"Artin’s Theorem Towards the Existence of Algebraic Closures","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2022-0014","DOIUrl":"https://doi.org/10.2478/forma-2022-0014","url":null,"abstract":"Summary This is the first part of a two-part article formalizing existence and uniqueness of algebraic closures using the Mizar system [1], [2]. Our proof follows Artin’s classical one as presented by Lang in [3]. In this first part we prove that for a given field F there exists a field extension E such that every non-constant polynomial p ∈ F [X] has a root in E. Artin’s proof applies Kronecker’s construction to each polynomial p ∈ F [X]F simultaneously. To do so we need the polynomial ring F [X1, X2, ...] with infinitely many variables, one for each polynomal p ∈ F [X]F . The desired field extension E then is F [X1, X2, ...]I, where I is a maximal ideal generated by all non-constant polynomials p ∈ F [X]. Note, that to show that I is maximal Zorn’s lemma has to be applied. In the second part this construction is iterated giving an infinite sequence of fields, whose union establishes a field extension A of F, in which every non-constant polynomial p ∈ A[X] has a root. The field of algebraic elements of A then is an algebraic closure of F . To prove uniqueness of algebraic closures, e.g. that two algebraic closures of F are isomorphic over F, the technique of extending monomorphisms is applied: a monomorphism F → A, where A is an algebraic closure of F can be extended to a monomorphism E → A, where E is any algebraic extension of F . In case that E is algebraically closed this monomorphism is an isomorphism. Note that the existence of the extended monomorphism again relies on Zorn’s lemma.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"44 1","pages":"199 - 207"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80256973","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":"On Implicit and Inverse Function Theorems on Euclidean Spaces","authors":"Kazuhisa Nakasho, Y. Shidama","doi":"10.2478/forma-2022-0012","DOIUrl":"https://doi.org/10.2478/forma-2022-0012","url":null,"abstract":"Summary Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using the Mizar system [1], [2], we formalize these theorems on Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in this formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"21 1","pages":"159 - 168"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74779611","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 IV","authors":"Artur Korniłowicz","doi":"10.2478/forma-2022-0017","DOIUrl":"https://doi.org/10.2478/forma-2022-0017","url":null,"abstract":"Summary In this paper problems 17, 18, 26, 27, 28, and 98 from [9] are formalized, using the Mizar formalism [8], [2], [3], [6].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"5 1","pages":"223 - 228"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89399191","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 VI","authors":"Adam Grabowski","doi":"10.2478/forma-2022-0019","DOIUrl":"https://doi.org/10.2478/forma-2022-0019","url":null,"abstract":"Summary This paper reports on the formalization in Mizar system [1], [2] of ten selected problems from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [7] (see [6] for details of this concrete dataset). This article is devoted mainly to arithmetic progressions: problems 52, 54, 55, 56, 60, 64, 70, 71, and 73 belong to the chapter “Arithmetic Progressions”, and problem 50 is from “Relatively Prime Numbers”.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"278 1","pages":"235 - 244"},"PeriodicalIF":0.3,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77147540","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":"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}
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