Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U0 (FinSETS) and U1 (SETS): Grothendieck Universe ω=Grothendieck Universe U0=U1 {rm{Grothendieck}},{rm{Universe}},omega = {rm{Grothendieck}},{rm{Universe}},{{bf{U}}_0} = {{bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].
宇宙是一个概念，从Mizar数学库(MML)创建之初就以几种形式存在(Universe, Universe_closure, Universe)[25]，然后是后来的the_universe_of，[33]，以及最近的定义GrothendieckUniverse[26]，[11]，[11]。这些定义在许多文章[28,33,8,35]，[19,32,31,15,6]，以及[34,12,20,22,21]，[27,2,3,23,16,7,4,5]中都很有用。本文利用Mizar系统[9][10]，简单地证明了[26]中定义的Grothendieck对宇宙的定义与Artin、Grothendieck和Verdier(第0章Univers et Appendice“Univers”(par N. Bourbaki) de l ' exposeise I.“prefaisse - aux”)对宇宙的原始定义是一致的[1]，以及关于宇宙的MML的不同定义是如何相互关联的。我们还证明了Mac Lane([18])引入的宇宙定义与MML的定义是兼容的。虽然宇宙可能是空的，但我们考虑了非空宇宙的性质，完成了[25]中所证明的性质。根据Robert M. Solovay2的概念，我们引入了“平凡”和“非平凡”宇宙的概念，这取决于它们是否包含集合ω (NAT)。以下结果将宇宙U0 (FinSETS)和U1 (SETS)联系起来:Grothendieck Universe ω=Grothendieck Universe U0=U1 {rm{Grothendieck}}，{rm{Universe}}，omega = {rm{Grothendieck}}，{rm{Universe}}，{{bf{U}}_0} = {{bf{U}}_1}在进入最后一节之前，我们建立了一些微不足道的命题，允许在考虑的宇宙之外构造集合。最后一节致力于塔斯基-格罗滕迪克的宇宙塔的构建，以序数为索引(见8)。示例，Grothendieck宇宙，ncatlab.org[24])。Grothendieck的宇宙在当前的著作中被引用:“假设存在足够的(Grothendieck)宇宙”，Jacob Lurie在“高等拓扑理论”[17]，“附件B -关于Grothendieck宇宙的一些结果”，Olivia Caramello和Riccardo Zanfa在“通过堆的相对拓扑理论”[13]，“注释1.1.5(引用Michael Shulman[30])”，Emily Riehl在“语境中的范畴论”[29]，更具体地说是“Grothendieck拓扑的严格宇宙”[14]。
{"title":"Non-Trivial Universes and Sequences of Universes","authors":"Roland Coghetto","doi":"10.2478/forma-2022-0005","DOIUrl":"https://doi.org/10.2478/forma-2022-0005","url":null,"abstract":"Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U0 (FinSETS) and U1 (SETS): Grothendieck Universe ω=Grothendieck Universe U0=U1 {rm{Grothendieck}},{rm{Universe}},omega = {rm{Grothendieck}},{rm{Universe}},{{bf{U}}_0} = {{bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"61 1","pages":"53 - 66"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86234519","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 [11] the existence (and uniqueness) of splitting fields has been formalized. In this article we apply this result by providing splitting fields for the polynomials X2 − 2, X3 − 1, X2 + X + 1 and X3 − 2 over Q using the Mizar [2], [1] formalism. We also compute the degrees and bases for these splitting fields, which requires some additional registrations to adopt types properly. The main result, however, is that the polynomial X3 − 2 does not split over 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) . Because X3 − 2 obviously has a root over 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) this shows that the field extension 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) is not normal over Q [3], [4], [5] and [7].
在[11]中，分裂域的存在性(和唯一性)被形式化了。在本文中，我们通过使用Mizar[2]，[1]形式为多项式X2−2,X3−1,X2 + X + 1和X3−2 / Q提供分裂域来应用这一结果。我们还计算这些拆分字段的度数和基数，这需要一些额外的注册才能正确地采用类型。然而，主要的结果是，多项式X3−2不会在𝒬(23)mathcal{Q}left({root 3 of 2} right)上分裂。因为X3−2显然在𝒬(23)mathcal{Q}left({root 3 of 2} right)上有根，这表明域扩展𝒬(23)mathcal{Q}left({root 3 of 2} right)不是正态分布在Q[3]，[4]，[5]和[7]上。
{"title":"Splitting Fields for the Rational Polynomials X2−2, X2+X+1, X3−1, and X3−2","authors":"Christoph Schwarzweller, Sara Burgoa","doi":"10.2478/forma-2022-0003","DOIUrl":"https://doi.org/10.2478/forma-2022-0003","url":null,"abstract":"Summary In [11] the existence (and uniqueness) of splitting fields has been formalized. In this article we apply this result by providing splitting fields for the polynomials X2 − 2, X3 − 1, X2 + X + 1 and X3 − 2 over Q using the Mizar [2], [1] formalism. We also compute the degrees and bases for these splitting fields, which requires some additional registrations to adopt types properly. The main result, however, is that the polynomial X3 − 2 does not split over 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) . Because X3 − 2 obviously has a root over 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) this shows that the field extension 𝒬(23) mathcal{Q}left( {root 3 of 2 } right) is not normal over Q [3], [4], [5] and [7].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"175 1","pages":"23 - 30"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74029675","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 goal of this article is to clarify the relationship between Riemann’s improper integrals and Lebesgue integrals. In previous articles [6], [7], we treated Riemann’s improper integrals [1], [11] and [4] on arbitrary intervals. Therefore, in this article, we will continue to clarify the relationship between improper integrals and Lebesgue integrals [8], using the Mizar [3], [2] formalism.
{"title":"Absolutely Integrable Functions","authors":"N. Endou","doi":"10.2478/forma-2022-0004","DOIUrl":"https://doi.org/10.2478/forma-2022-0004","url":null,"abstract":"Summary The goal of this article is to clarify the relationship between Riemann’s improper integrals and Lebesgue integrals. In previous articles [6], [7], we treated Riemann’s improper integrals [1], [11] and [4] on arbitrary intervals. Therefore, in this article, we will continue to clarify the relationship between improper integrals and Lebesgue integrals [8], using the Mizar [3], [2] formalism.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"1 1","pages":"31 - 52"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88845533","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, Feed-forward Neural Network is formalized in the Mizar system [1], [2]. First, the multilayer perceptron [6], [7], [8] is formalized using functional sequences. Next, we show that a set of functions generated by these neural networks satisfies equicontinuousness and equiboundedness property [10], [5]. At last, we formalized the compactness of the function set of these neural networks by using the Ascoli-Arzela’s theorem according to [4] and [3].
{"title":"Compactness of Neural Networks","authors":"K. Miyajima, Hiroshi Yamazaki","doi":"10.2478/forma-2022-0002","DOIUrl":"https://doi.org/10.2478/forma-2022-0002","url":null,"abstract":"Summary In this article, Feed-forward Neural Network is formalized in the Mizar system [1], [2]. First, the multilayer perceptron [6], [7], [8] is formalized using functional sequences. Next, we show that a set of functions generated by these neural networks satisfies equicontinuousness and equiboundedness property [10], [5]. At last, we formalized the compactness of the function set of these neural networks by using the Ascoli-Arzela’s theorem according to [4] and [3].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"63 1","pages":"13 - 21"},"PeriodicalIF":0.3,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79153560","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 formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system [1], [2], in two independent ways, proving the existence of a polynomial without formulating it explicitly as well as with its indication. First, we reuse nearly all the techniques invented to prove the MRDP-theorem [11]. Applying a trick with Mizar schemes that go beyond first-order logic we give a short sophisticated proof for the existence of such a polynomial but without formulating it explicitly. Then we formulate the polynomial proposed in [6] that has 26 variables in the Mizar language as follows (w·z+h+j−q)2+((g·k+g+k)·(h+j)+h−z)2+(2 · k3·(2·k+2)·(n + 1)2+1−f2)2+ (p + q + z + 2 · n − e)2 + (e3 · (e + 2) · (a + 1)2 + 1 − o2)2 + (x2 − (a2 −′ 1) · y2 − 1)2 + (16 · (a2 − 1) · r2 · y2 · y2 + 1 − u2)2 + (((a + u2 · (u2 − a))2 − 1) · (n + 4 · d · y)2 + 1 − (x + c · u)2)2 + (m2 − (a2 −′ 1) · l2 − 1)2 + (k + i · (a − 1) − l)2 + (n + l + v − y)2 + (p + l · (a − n − 1) + b · (2 · a · (n + 1) − (n + 1)2 − 1) − m)2 + (q + y · (a − p − 1) + s · (2 · a · (p + 1) − (p + 1)2 − 1) − x)2 + (z + p · l · (a − p) + t · (2 · a · p − p2 − 1) − p · m)2 and we prove that that for any positive integer k so that k + 1 is prime it is necessary and sufficient that there exist other natural variables a-z for which the polynomial equals zero. 26 variables is not the best known result in relation to the set of prime numbers, since any diophantine equation over ℕ can be reduced to one in 13 unknowns [8] or even less [5], [13]. The best currently known result for all prime numbers, where the polynomial is explicitly constructed is 10 [7] or even 7 in the case of Fermat as well as Mersenne prime number [4]. We are currently focusing our formalization efforts in this direction.
{"title":"Prime Representing Polynomial","authors":"Karol Pąk","doi":"10.2478/forma-2021-0020","DOIUrl":"https://doi.org/10.2478/forma-2021-0020","url":null,"abstract":"Summary The main purpose of formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system [1], [2], in two independent ways, proving the existence of a polynomial without formulating it explicitly as well as with its indication. First, we reuse nearly all the techniques invented to prove the MRDP-theorem [11]. Applying a trick with Mizar schemes that go beyond first-order logic we give a short sophisticated proof for the existence of such a polynomial but without formulating it explicitly. Then we formulate the polynomial proposed in [6] that has 26 variables in the Mizar language as follows (w·z+h+j−q)2+((g·k+g+k)·(h+j)+h−z)2+(2 · k3·(2·k+2)·(n + 1)2+1−f2)2+ (p + q + z + 2 · n − e)2 + (e3 · (e + 2) · (a + 1)2 + 1 − o2)2 + (x2 − (a2 −′ 1) · y2 − 1)2 + (16 · (a2 − 1) · r2 · y2 · y2 + 1 − u2)2 + (((a + u2 · (u2 − a))2 − 1) · (n + 4 · d · y)2 + 1 − (x + c · u)2)2 + (m2 − (a2 −′ 1) · l2 − 1)2 + (k + i · (a − 1) − l)2 + (n + l + v − y)2 + (p + l · (a − n − 1) + b · (2 · a · (n + 1) − (n + 1)2 − 1) − m)2 + (q + y · (a − p − 1) + s · (2 · a · (p + 1) − (p + 1)2 − 1) − x)2 + (z + p · l · (a − p) + t · (2 · a · p − p2 − 1) − p · m)2 and we prove that that for any positive integer k so that k + 1 is prime it is necessary and sufficient that there exist other natural variables a-z for which the polynomial equals zero. 26 variables is not the best known result in relation to the set of prime numbers, since any diophantine equation over ℕ can be reduced to one in 13 unknowns [8] or even less [5], [13]. The best currently known result for all prime numbers, where the polynomial is explicitly constructed is 10 [7] or even 7 in the case of Fermat as well as Mersenne prime number [4]. We are currently focusing our formalization efforts in this direction.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"30 2","pages":"221 - 228"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72600790","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, using the Mizar system [2], [3], we deal with Riemann’s improper integral on infinite interval [1]. As with [4], we referred to [6], which discusses improper integrals of finite values.
{"title":"Improper Integral. Part II","authors":"N. Endou","doi":"10.2478/forma-2021-0024","DOIUrl":"https://doi.org/10.2478/forma-2021-0024","url":null,"abstract":"Summary In this article, using the Mizar system [2], [3], we deal with Riemann’s improper integral on infinite interval [1]. As with [4], we referred to [6], which discusses improper integrals of finite values.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"35 1","pages":"279 - 294"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76543289","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 deal with Riemann’s improper integral [1], using the Mizar system [2], [3]. Improper integrals with finite values are discussed in [5] by Yamazaki et al., but in general, improper integrals do not assume that they are finite. Therefore, we have formalized general improper integrals that does not limit the integral value to a finite value. In addition, each theorem in [5] assumes that the domain of integrand includes a closed interval, but since the improper integral should be discusses based on the half-open interval, we also corrected it.
{"title":"Improper Integral. Part I","authors":"N. Endou","doi":"10.2478/forma-2021-0019","DOIUrl":"https://doi.org/10.2478/forma-2021-0019","url":null,"abstract":"Summary In this article, we deal with Riemann’s improper integral [1], using the Mizar system [2], [3]. Improper integrals with finite values are discussed in [5] by Yamazaki et al., but in general, improper integrals do not assume that they are finite. Therefore, we have formalized general improper integrals that does not limit the integral value to a finite value. In addition, each theorem in [5] assumes that the domain of integrand includes a closed interval, but since the improper integral should be discusses based on the half-open interval, we also corrected it.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"21 1","pages":"201 - 220"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72517453","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}
Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller
Summary In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of p being non square - adjoining a root of p’s discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a ∈ F such that E and ( Fa Fsqrt a ) are isomorphic over F.
{"title":"Quadratic Extensions","authors":"Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller","doi":"10.2478/forma-2021-0021","DOIUrl":"https://doi.org/10.2478/forma-2021-0021","url":null,"abstract":"Summary In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of p being non square - adjoining a root of p’s discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a ∈ F such that E and ( Fa Fsqrt a ) are isomorphic over F.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"89 1","pages":"229 - 240"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86845394","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 aim of this article is to introduce formally ternary Boolean algebras (TBAs) in terms of an abstract ternary operation, and to show their connection with the ordinary notion of a Boolean algebra, already present in the Mizar Mathematical Library [2]. Essentially, the core of this Mizar [1] formalization is based on the paper of A.A. Grau “Ternary Boolean Algebras” [7]. The main result is the single axiom for this class of lattices [12]. This is the continuation of the articles devoted to various equivalent axiomatizations of Boolean algebras: following Huntington [8] in terms of the binary sum and the complementation useful in the formalization of the Robbins problem [5], in terms of Sheffer stroke [9]. The classical definition ([6], [3]) can be found in [15] and its formalization is described in [4]. Similarly as in the case of recent formalizations of WA-lattices [14] and quasilattices [10], some of the results were proven in the Mizar system with the help of Prover9 [13], [11] proof assistant, so proofs are quite lengthy. They can be subject for subsequent revisions to make them more compact.
{"title":"Automatization of Ternary Boolean Algebras","authors":"Wojciech Kuśmierowski, Adam Grabowski","doi":"10.2478/forma-2021-0015","DOIUrl":"https://doi.org/10.2478/forma-2021-0015","url":null,"abstract":"Summary The main aim of this article is to introduce formally ternary Boolean algebras (TBAs) in terms of an abstract ternary operation, and to show their connection with the ordinary notion of a Boolean algebra, already present in the Mizar Mathematical Library [2]. Essentially, the core of this Mizar [1] formalization is based on the paper of A.A. Grau “Ternary Boolean Algebras” [7]. The main result is the single axiom for this class of lattices [12]. This is the continuation of the articles devoted to various equivalent axiomatizations of Boolean algebras: following Huntington [8] in terms of the binary sum and the complementation useful in the formalization of the Robbins problem [5], in terms of Sheffer stroke [9]. The classical definition ([6], [3]) can be found in [15] and its formalization is described in [4]. Similarly as in the case of recent formalizations of WA-lattices [14] and quasilattices [10], some of the results were proven in the Mizar system with the help of Prover9 [13], [11] proof assistant, so proofs are quite lengthy. They can be subject for subsequent revisions to make them more compact.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"26 1","pages":"153 - 159"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81301224","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 check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).
{"title":"Duality Notions in Real Projective Plane","authors":"Roland Coghetto","doi":"10.2478/forma-2021-0016","DOIUrl":"https://doi.org/10.2478/forma-2021-0016","url":null,"abstract":"Summary In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"77 1","pages":"161 - 173"},"PeriodicalIF":0.3,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88733577","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：