Formalized Mathematics最新文献

{"title":"Functional Sequence in Norm Space","authors":"Hiroshi Yamazaki","doi":"10.2478/forma-2020-0023","DOIUrl":"https://doi.org/10.2478/forma-2020-0023","url":null,"abstract":"Summary In this article, we formalize in Mizar [1], [2] functional sequences and basic operations on functional sequences in norm space based on [5]. In the first section, we define functional sequence in norm space. In the second section, we define pointwise convergence and prove some related theorems. In the last section we define uniform convergence and limit of functional sequence.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78841197","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 and Field Adjunctions, Algebraic Elements and Minimal Polynomials","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2020-0022","DOIUrl":"https://doi.org/10.2478/forma-2020-0022","url":null,"abstract":"Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a0, . . ., an−1, where n is the degree of the minimal polynomial of a over F .","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72933935","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":"Extended Natural Numbers and Counters","authors":"Sebastian Koch","doi":"10.2478/forma-2020-0021","DOIUrl":"https://doi.org/10.2478/forma-2020-0021","url":null,"abstract":"Summary This article introduces extended natural numbers, i.e. the set ℕ ∪ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph theory.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81601371","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":"A Case Study of Transporting Urysohn’s Lemma from Topology via Open Sets into Topology via Neighborhoods","authors":"Roland Coghetto","doi":"10.2478/forma-2020-0020","DOIUrl":"https://doi.org/10.2478/forma-2020-0020","url":null,"abstract":"Summary Józef Białas and Yatsuka Nakamura has completely formalized a proof of Urysohn’s lemma in the article [4], in the context of a topological space defined via open sets. In the Mizar Mathematical Library (MML), the topological space is defined in this way by Beata Padlewska and Agata Darmochwał in the article [18]. In [7] the topological space is defined via neighborhoods. It is well known that these definitions are equivalent [5, 6]. In the definitions, an abstract structure (i.e. the article [17, STRUCT 0] and its descendants, all of them directly or indirectly using Mizar structures [3]) have been used (see [10], [9]). The first topological definition is based on the Mizar structure TopStruct and the topological space defined via neighborhoods with the Mizar structure: FMT Space Str. To emphasize the notion of a neighborhood, we rename FMT TopSpace (topology from neighbourhoods) to NTopSpace (a neighborhood topological space). Using Mizar [2], we transport the Urysohn’s lemma from TopSpace to NTop-Space. In some cases, Mizar allows certain techniques for transporting proofs, definitions or theorems. Generally speaking, there is no such automatic translating. In Coq, Isabelle/HOL or homotopy type theory transport is also studied, sometimes with a more systematic aim [14], [21], [11], [12], [8], [19]. In [1], two co-existing Isabelle libraries: Isabelle/HOL and Isabelle/Mizar, have been aligned in a single foundation in the Isabelle logical framework. In the MML, they have been used since the beginning: reconsider, registration, cluster, others were later implemented [13]: identify. In some proofs, it is possible to define particular functors between different structures, mainly useful when results are already obtained in a given structure. This technique is used, for example, in [15] to define two functors MXR2MXF and MXF2MXF between Matrix of REAL and Matrix of F-Real and to transport the definition of the addition from one structure to the other: [...] A + B -> Matrix of REAL equals MXF2MXR ((MXR2MXF A) + (MXR2MXF B)) [...]. In this paper, first we align the necessary topological concepts. For the formalization, we were inspired by the works of Claude Wagschal [20]. It allows us to transport more naturally the Urysohn’s lemma ([4, URYSOHN3:20]) to the topological space defined via neighborhoods. Nakasho and Shidama have developed a solution to explore the notions introduced in various ways https://mimosa-project.github.io/mmlreference/current/ [16]. The definitions can be directly linked in the HTML version of the Mizar library (example: Urysohn’s lemma http://mizar.org/version/current/html/urysohn3.html#T20).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76012978","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":"About Graph Unions and Intersections","authors":"Sebastian Koch","doi":"10.2478/forma-2020-0014","DOIUrl":"https://doi.org/10.2478/forma-2020-0014","url":null,"abstract":"Summary In this article the union and intersection of a set of graphs are formalized in the Mizar system [5], based on the formalization of graphs in [7].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84069782","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":"Refined Finiteness and Degree Properties in Graphs","authors":"Sebastian Koch","doi":"10.2478/forma-2020-0013","DOIUrl":"https://doi.org/10.2478/forma-2020-0013","url":null,"abstract":"Summary In this article the finiteness of graphs is refined and the minimal and maximal degree of graphs are formalized in the Mizar system [3], based on the formalization of graphs in [4].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82985766","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":"Grothendieck Universes","authors":"Karol Pąk","doi":"10.2478/forma-2020-0018","DOIUrl":"https://doi.org/10.2478/forma-2020-0018","url":null,"abstract":"Summary The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarski’s Axiom A, which states that for every set X there is a Tarski universe U such that X ∈ U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe. First we prove in Theorem (17) that every Grothendieck universe satisfies Tarski’s Axiom A. Then in Theorem (18) we prove that every Grothendieck universe that contains a given set X, even the least (with respect to inclusion) denoted by GrothendieckUniverseX, has as a subset the least (with respect to inclusion) Tarski universe that contains X, denoted by the Tarski-ClassX. Since Tarski universes, as opposed to Grothendieck universes [5], might not be transitive (called epsilon-transitive in the Mizar Mathematical Library [1]) we focused our attention to demonstrate that Tarski-Class X ⊊ GrothendieckUniverse X for some X. Then we show in Theorem (19) that Tarski-ClassX where X is the singleton of any infinite set is a proper subset of GrothendieckUniverseX. Finally we show that Tarski-Class X = GrothendieckUniverse X holds under the assumption that X is a transitive set. The formalisation is an extension of the formalisation used in [4].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78588664","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":"Renamings and a Condition-free Formalization of Kronecker’s Construction","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2020-0012","DOIUrl":"https://doi.org/10.2478/forma-2020-0012","url":null,"abstract":"Summary In [7], [9], [10] we presented a formalization of Kronecker’s construction of a field extension E for a field F in which a given polynomial p ∈ F [X]F has a root [5], [6], [3]. A drawback of our formalization was that it works only for polynomial-disjoint fields, that is for fields F with F ∩ F [X] = ∅. The main purpose of Kronecker’s construction is that by induction one gets a field extension of F in which p splits into linear factors. For our formalization this means that the constructed field extension E again has to be polynomial-disjoint. In this article, by means of Mizar system [2], [1], we first analyze whether our formalization can be extended that way. Using the field of polynomials over F with degree smaller than the degree of p to construct the field extension E does not work: In this case E is polynomial-disjoint if and only if p is linear. Using F [X]/ one can show that for F = ℚ and F = ℤn the constructed field extension E is again polynomial-disjoint, so that in particular algebraic number fields can be handled. For the general case we then introduce renamings of sets X as injective functions f with dom(f) = X and rng(f) ∩ (X ∪ Z) = ∅ for an arbitrary set Z. This, finally, allows to construct a field extension E of an arbitrary field F in which a given polynomial p ∈ F [X]F splits into linear factors. Note, however, that to prove the existence of renamings we had to rely on the axiom of choice.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76170123","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 Quasilattices","authors":"Dominik Kulesza, Adam Grabowski","doi":"10.2478/forma-2020-0019","DOIUrl":"https://doi.org/10.2478/forma-2020-0019","url":null,"abstract":"Summary The main aim of this article is to introduce formally one of the generalizations of lattices, namely quasilattices, which can be obtained from the axiomatization of the former class by certain weakening of ordinary absorption laws. We show propositions QLT-1 to QLT-7 from [15], presenting also some short variants of corresponding axiom systems. Some of the results were proven in the Mizar [1], [2] system with the help of Prover9 [14] proof assistant.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73793816","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":"Multiplication-Related Classes of Complex Numbers","authors":"R. Ziobro","doi":"10.2478/forma-2020-0017","DOIUrl":"https://doi.org/10.2478/forma-2020-0017","url":null,"abstract":"Summary The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74269854","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}