Formalized Mathematics最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}