Summary Surreal numbers, a fascinating mathematical concept introduced by John Conway, have attracted considerable interest due to their unique properties. In this article, we formalize the basic concept of surreal numbers close to the original Conway’s convention in the field of combinatorial game theory. We define surreal numbers with the pre-order in the Mizar system which satisfy the following condition: x ⩽ y iff Lx ≪ {y} Λ {x} ≪ Ry.
{"title":"Conway Numbers – Formal Introduction","authors":"Karol Pąk","doi":"10.2478/forma-2023-0018","DOIUrl":"https://doi.org/10.2478/forma-2023-0018","url":null,"abstract":"Summary Surreal numbers, a fascinating mathematical concept introduced by John Conway, have attracted considerable interest due to their unique properties. In this article, we formalize the basic concept of surreal numbers close to the original Conway’s convention in the field of combinatorial game theory. We define surreal numbers with the pre-order in the Mizar system which satisfy the following condition: x ⩽ y iff Lx ≪ {y} Λ {x} ≪ Ry.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"254 1","pages":"193 - 203"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343721","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 A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2]. We treat an affine space as the n -fold Cartesian product k n as the same manner appeared in [4]. Points in this space are identified as n -tuples of elements from the set k . The formalization of points, which are n -tuples of numbers, is described in terms of a mapping from n to k , where the domain n corresponds to the set n = { 0, 1, . . ., n − 1 } , and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n -tuples of numbers [10]. This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].
{"title":"Introduction to Algebraic Geometry","authors":"Yasushige Watase","doi":"10.2478/forma-2023-0007","DOIUrl":"https://doi.org/10.2478/forma-2023-0007","url":null,"abstract":"Summary A classical algebraic geometry is study of zero points of system of multivariate polynomials [3], [7] and those zero points would be corresponding to points, lines, curves, surfaces in an affine space. In this article we give some basic definition of the area of affine algebraic geometry such as algebraic set, ideal of set of points, and those properties according to [4] in the Mizar system[5], [2]. We treat an affine space as the n -fold Cartesian product k n as the same manner appeared in [4]. Points in this space are identified as n -tuples of elements from the set k . The formalization of points, which are n -tuples of numbers, is described in terms of a mapping from n to k , where the domain n corresponds to the set n = { 0, 1, . . ., n − 1 } , and the target domain k is the same as the scalar ring or field of polynomials. The same approach has been applied when evaluating multivariate polynomials using n -tuples of numbers [10]. This formalization aims at providing basic notions of the field which enable to formalize geometric objects such as algebraic curves which is used e.g. in coding theory [11] as well as further formalization of the fields [8] in the Mizar system, including the theory of polynomials [6].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"2 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":"135639861","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 Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13], [14]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [16], [9] and formalized in [11] and [12]. Some propositions of the composition mapping ( f + · g , or f +* g using Mizar formalism, where f , g are a ne mappings), are proved following [3], [15]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized. We proved the agreement of the same function expressed with different parameters and formalized those centroids with parameters. In addition, various properties of membership functions on intervals where the endpoints of the domain are fixed and on general intervals are formalized in Mizar [1], [2]. Our formal development contains also some numerical results which can be potentially useful to encode either fuzzy numbers [7], or even fuzzy implications [5], [6] and extends the possibility of building hybrid rough-fuzzy approach in the future [8].
{"title":"Isosceles Triangular and Isosceles Trapezoidal Membership Functions Using Centroid Method","authors":"Takashi Mitsuishi","doi":"10.2478/forma-2023-0006","DOIUrl":"https://doi.org/10.2478/forma-2023-0006","url":null,"abstract":"Summary Since isosceles triangular and trapezoidal membership functions [4] are easy to manage, they were applied to various fuzzy approximate reasoning [10], [13], [14]. The centroids of isosceles triangular and trapezoidal membership functions are mentioned in this article [16], [9] and formalized in [11] and [12]. Some propositions of the composition mapping ( f + · g , or f +* g using Mizar formalism, where f , g are a ne mappings), are proved following [3], [15]. Then different notations for the same isosceles triangular and trapezoidal membership function are formalized. We proved the agreement of the same function expressed with different parameters and formalized those centroids with parameters. In addition, various properties of membership functions on intervals where the endpoints of the domain are fixed and on general intervals are formalized in Mizar [1], [2]. Our formal development contains also some numerical results which can be potentially useful to encode either fuzzy numbers [7], or even fuzzy implications [5], [6] and extends the possibility of building hybrid rough-fuzzy approach in the future [8].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"57 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":"134995207","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, we introduce indefinite integrals [8] (antiderivatives) and proof integration by substitution in the Mizar system [2], [3]. In our previous article [15], we have introduced an indefinite-like integral, but it is inadequate because it must be an integral over the whole set of real numbers and in some sense it causes some duplication in the Mizar Mathematical Library [13]. For this reason, to define the antiderivative for a function, we use the derivative of an arbitrary interval as defined recently in [7]. Furthermore, antiderivatives are also used to modify the integration by substitution and integration by parts. In the first section, we summarize the basic theorems on continuity and derivativity (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [12], Isabelle/HOL [11], Coq [4], see [5]). In the second section, we generalize some theorems that were noticed during the formalization process. In the last section, we define the antiderivatives and formalize the integration by substitution and the integration by parts. We referred to [1] and [6] in our development.
{"title":"Antiderivatives and Integration","authors":"Noboru Endou","doi":"10.2478/forma-2023-0012","DOIUrl":"https://doi.org/10.2478/forma-2023-0012","url":null,"abstract":"Summary In this paper, we introduce indefinite integrals [8] (antiderivatives) and proof integration by substitution in the Mizar system [2], [3]. In our previous article [15], we have introduced an indefinite-like integral, but it is inadequate because it must be an integral over the whole set of real numbers and in some sense it causes some duplication in the Mizar Mathematical Library [13]. For this reason, to define the antiderivative for a function, we use the derivative of an arbitrary interval as defined recently in [7]. Furthermore, antiderivatives are also used to modify the integration by substitution and integration by parts. In the first section, we summarize the basic theorems on continuity and derivativity (for interesting survey of formalizations of real analysis in another proof-assistants like ACL2 [12], Isabelle/HOL [11], Coq [4], see [5]). In the second section, we generalize some theorems that were noticed during the formalization process. In the last section, we define the antiderivatives and formalize the integration by substitution and the integration by parts. We referred to [1] and [6] in our development.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"94 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":"135737183","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 continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F [ T ] = { p ( a 1 , . . . a n ) | p ∈ F [ X ], a i ∈ T } and F ( T ) = F [ T ] for finite algebraic T ⊆ E . We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).
在本文中，我们继续Mizar[1]，[2]，[4]，[3]中场论的形式化。我们引入正规扩展:如果F的每个多项式在E中有一个根已经在E中分裂，那么F的(代数)扩展E是正规的。我们通过极小多项式[7]、分裂域和固定单态[6]、[5]证明了表征(有限扩展)。这需要扩展[11]和[12]的结果，特别是F [T] = {p (a 1，…)。an) | p∈F [X]， ai∈T}， F (T) = F [T]我们还提供了反例，𝒬(∛2)在𝒬上不正常(比较[13])。
{"title":"Normal Extensions","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2023-0011","DOIUrl":"https://doi.org/10.2478/forma-2023-0011","url":null,"abstract":"Summary In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F [ T ] = { p ( a 1 , . . . a n ) | p ∈ F [ X ], a i ∈ T } and F ( T ) = F [ T ] for finite algebraic T ⊆ E . We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"15 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":"135737790","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 introduces multidimensional measure spaces and the integration of functions on these spaces in Mizar. Integrals on the multidimensional Cartesian product measure space are defined and appropriate formal apparatus to deal with this notion is provided as well.
{"title":"Multidimensional Measure Space and Integration","authors":"N. Endou, Y. Shidama","doi":"10.2478/forma-2023-0017","DOIUrl":"https://doi.org/10.2478/forma-2023-0017","url":null,"abstract":"Summary This paper introduces multidimensional measure spaces and the integration of functions on these spaces in Mizar. Integrals on the multidimensional Cartesian product measure space are defined and appropriate formal apparatus to deal with this notion is provided as well.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"16 1","pages":"181 - 192"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139344431","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}
Abstract In this article regular graphs, both directed and undirected, are formalized in the Mizar system [7], [2], based on the formalization of graphs as described in [10]. The handshaking lemma is also proven.
{"title":"About Regular Graphs","authors":"Sebastian Koch","doi":"10.2478/forma-2023-0008","DOIUrl":"https://doi.org/10.2478/forma-2023-0008","url":null,"abstract":"Abstract In this article regular graphs, both directed and undirected, are formalized in the Mizar system [7], [2], based on the formalization of graphs as described in [10]. The handshaking lemma is also proven.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"74 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":"135639860","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 the next article in the series formalizing the book of Baczyński and Jayaram “Fuzzy Implications”. We define the laws of contraposition connected with various fuzzy negations, and in order to make the cluster registration mechanism fully working, we construct some more non-classical examples of fuzzy implications. Finally, as the testbed of the reuse of lattice-theoretical approach, we introduce the lattice of fuzzy negations and show its basic properties.
{"title":"On Fuzzy Negations and Laws of Contraposition. Lattice of Fuzzy Negations","authors":"Adam Grabowski","doi":"10.2478/forma-2023-0014","DOIUrl":"https://doi.org/10.2478/forma-2023-0014","url":null,"abstract":"Summary This the next article in the series formalizing the book of Baczyński and Jayaram “Fuzzy Implications”. We define the laws of contraposition connected with various fuzzy negations, and in order to make the cluster registration mechanism fully working, we construct some more non-classical examples of fuzzy implications. Finally, as the testbed of the reuse of lattice-theoretical approach, we introduce the lattice of fuzzy negations and show its basic properties.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"54 1","pages":"151 - 159"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345678","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 article concerns about formalizing a certain lemma on embedding of algebraic structures in the Mizar system, claiming that if a ring A is embedded in a ring B then there exists a ring C which is isomorphic to B and includes A as a subring. This construction applies to algebraic structures such as Abelian groups and rings.
摘要 文章关注在米扎（Mizar）系统中形式化关于代数结构嵌入的某个 Lemma，声称如果一个环 A 嵌入一个环 B，那么存在一个与 B 同构并包含 A 作为子环的环 C。这一构造适用于代数结构，如阿贝尔群和环。
{"title":"Embedding Principle for Rings and Abelian Groups","authors":"Yasushige Watase","doi":"10.2478/forma-2023-0013","DOIUrl":"https://doi.org/10.2478/forma-2023-0013","url":null,"abstract":"Summary The article concerns about formalizing a certain lemma on embedding of algebraic structures in the Mizar system, claiming that if a ring A is embedded in a ring B then there exists a ring C which is isomorphic to B and includes A as a subring. This construction applies to algebraic structures such as Abelian groups and rings.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"6 1","pages":"143 - 150"},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346958","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 formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1]). This process is one of the most famous methods for orthonormalizing a set of vectors. The method is named after Jørgen Pedersen Gram and Erhard Schmidt [4]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [8]. First, we prove some preliminary theorems about real unitary space. Next, we formalize the definition of the Gram-Schmidt process that finds orthonormal basis. We followed [5] in the formalization, continuing work developed in [7], [6].
{"title":"On the Formalization of Gram-Schmidt Process for Orthonormalizing a Set of Vectors","authors":"Hiroyuki Okazaki","doi":"10.2478/forma-2023-0005","DOIUrl":"https://doi.org/10.2478/forma-2023-0005","url":null,"abstract":"Summary In this article, we formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1]). This process is one of the most famous methods for orthonormalizing a set of vectors. The method is named after Jørgen Pedersen Gram and Erhard Schmidt [4]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [8]. First, we prove some preliminary theorems about real unitary space. Next, we formalize the definition of the Gram-Schmidt process that finds orthonormal basis. We followed [5] in the formalization, continuing work developed in [7], [6].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"57 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":"134995194","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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