Summary. In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear” https://en.wikipedia.org/wiki/Pappus’s_hexagon_theorem. More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leończuk and Krzysztof Prażmowski. Eugeniusz Kusak and Wojciech Leończuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover9 https://www.cs.unm.edu/~mccune/prover9/, the successor of the Otter prover and ott2miz by Josef Urban See its homepage https://github.com/JUrban/ott2miz [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].
{"title":"Pappus’s Hexagon Theorem in Real Projective Plane","authors":"Roland Coghetto","doi":"10.2478/forma-2021-0007","DOIUrl":"https://doi.org/10.2478/forma-2021-0007","url":null,"abstract":"Summary. In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear” https://en.wikipedia.org/wiki/Pappus’s_hexagon_theorem. More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leończuk and Krzysztof Prażmowski. Eugeniusz Kusak and Wojciech Leończuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover9 https://www.cs.unm.edu/~mccune/prover9/, the successor of the Otter prover and ott2miz by Josef Urban See its homepage https://github.com/JUrban/ott2miz [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"57 1","pages":"69 - 76"},"PeriodicalIF":0.3,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86445657","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. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].
{"title":"Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions","authors":"T. Mitsuishi","doi":"10.2478/forma-2021-0011","DOIUrl":"https://doi.org/10.2478/forma-2021-0011","url":null,"abstract":"Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"59 2 1","pages":"103 - 115"},"PeriodicalIF":0.3,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75989624","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 article contains many auxiliary theorems which were missing in the Mizar Mathematical Library to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series and are needed in upcoming articles.
{"title":"Miscellaneous Graph Preliminaries. Part I","authors":"Sebastian Koch","doi":"10.2478/forma-2021-0003","DOIUrl":"https://doi.org/10.2478/forma-2021-0003","url":null,"abstract":"Summary This article contains many auxiliary theorems which were missing in the Mizar Mathematical Library to the best of the author’s knowledge. Most of them regard graph theory as formalized in the GLIB series and are needed in upcoming articles.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"41 1 1","pages":"21 - 38"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90900159","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 in Mizar [1], [2] the inverse function theorem for the class of C1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.
{"title":"Inverse Function Theorem. Part I1","authors":"Kazuhisa Nakasho, Yuichi Futa","doi":"10.2478/forma-2021-0002","DOIUrl":"https://doi.org/10.2478/forma-2021-0002","url":null,"abstract":"Summary In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely 𝔼 ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"26 1","pages":"9 - 19"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89744514","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 [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.
{"title":"Functional Space Consisted by Continuous Functions on Topological Space","authors":"Hiroshi Yamazaki, K. Miyajima, Y. Shidama","doi":"10.2478/forma-2021-0005","DOIUrl":"https://doi.org/10.2478/forma-2021-0005","url":null,"abstract":"Summary In this article, using the Mizar system [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"5 1","pages":"49 - 62"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699591","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 in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say D, is defined generally as a map from a commutative ring A to A-Module M with specific conditions. However we start with simpler case, namely dom D = rng D. This allows to define a derivation in other rings such as a polynomial ring. A derivation is a map D : A → A satisfying the following conditions: (i) D(x + y) = Dx + Dy, (ii) D(xy) = xDy + yDx, ∀x, y ∈ A. Typical properties are formalized such as: D(∑i=1nxi)=∑i=1nDxi Dleft( {sumlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {D{x_i}} and D(∏i=1nxi)=∑i=1nx1x2⋯Dxi⋯xn(∀xi∈A). Dleft( {prodlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {{x_1}{x_2} cdots D{x_i} cdots {x_n}} left( {forall {x_i} in A} right). We also formalized the Leibniz Formula for power of derivation D : Dn(xy)=∑i=0n(in)DixDn-iy. {D^n}left( {xy} right) = sumlimits_{i = 0}^n {left( {_i^n} right){D^i}x{D^{n - i}}y.} Lastly applying the definition to the polynomial ring of A and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].
{"title":"Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation","authors":"Yasushige Watase","doi":"10.2478/forma-2021-0001","DOIUrl":"https://doi.org/10.2478/forma-2021-0001","url":null,"abstract":"Summary In this article we formalize in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say D, is defined generally as a map from a commutative ring A to A-Module M with specific conditions. However we start with simpler case, namely dom D = rng D. This allows to define a derivation in other rings such as a polynomial ring. A derivation is a map D : A → A satisfying the following conditions: (i) D(x + y) = Dx + Dy, (ii) D(xy) = xDy + yDx, ∀x, y ∈ A. Typical properties are formalized such as: D(∑i=1nxi)=∑i=1nDxi Dleft( {sumlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {D{x_i}} and D(∏i=1nxi)=∑i=1nx1x2⋯Dxi⋯xn(∀xi∈A). Dleft( {prodlimits_{i = 1}^n {{x_i}} } right) = sumlimits_{i = 1}^n {{x_1}{x_2} cdots D{x_i} cdots {x_n}} left( {forall {x_i} in A} right). We also formalized the Leibniz Formula for power of derivation D : Dn(xy)=∑i=0n(in)DixDn-iy. {D^n}left( {xy} right) = sumlimits_{i = 0}^n {left( {_i^n} right){D^i}x{D^{n - i}}y.} Lastly applying the definition to the polynomial ring of A and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"47 1","pages":"1 - 8"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81477809","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 problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p2 + 1 = q2 + r2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22n + k (n = 1, 2, . . . ) are composite.
本文利用Mizar形式主义[1]，[2]，[3]，对[6]中的问题14、15、29、30、34、78、83、97和116进行形式化。证明了素数可除性的一些性质。证明了p2 + 1 = q2 + r2式(其中p, q, r为质数)至少有四个解，并证明了至少有五个质数可以表示为两个整数的四次幂的和。我们还证明了对于至少一个正整数，这个数及其后继数的四次方之和是合数。最后，证明了有无穷多个大于零的奇数k，使得所有形式为22n + k (n = 1,2，…)的数是合成的。
{"title":"Elementary Number Theory Problems. Part II","authors":"Artur Korniłowicz, Dariusz Surowik","doi":"10.2478/forma-2021-0006","DOIUrl":"https://doi.org/10.2478/forma-2021-0006","url":null,"abstract":"Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p2 + 1 = q2 + r2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22n + k (n = 1, 2, . . . ) are composite.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"1 1","pages":"63 - 68"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89615122","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 in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.
{"title":"Algebraic Extensions","authors":"Christoph Schwarzweller, Agnieszka Rowinska-Schwarzweller","doi":"10.2478/forma-2021-0004","DOIUrl":"https://doi.org/10.2478/forma-2021-0004","url":null,"abstract":"Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"104 1","pages":"39 - 47"},"PeriodicalIF":0.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74448802","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 some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future. This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and postconditions [17], [19], [7], [5].
{"title":"General Theory and Tools for Proving Algorithms in Nominative Data Systems","authors":"Adrian Jaszczak","doi":"10.2478/forma-2020-0024","DOIUrl":"https://doi.org/10.2478/forma-2020-0024","url":null,"abstract":"Summary In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future. This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and postconditions [17], [19], [7], [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"405 1","pages":"269 - 278"},"PeriodicalIF":0.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75016281","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 define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1]. i := 0 s := x b := y c := x while (i <> n) c := s s := b ps := p*s qc := q*c b := ps − qc i := i + j return s This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [18], [21], [7], [5].
本文定义了有限序列的一些性质，并验证了在给定P和Q系数以及两个第一元素(x和y)的情况下计算Lucas序列[23]，[20]的第n个元素的算法的部分正确性。该算法在Mizar系统[3]，[1]中用指示数据语言[22]进行了编码。i:= 0 s:= x b:= y c:= x while (i <> n) c:= s s:= b ps:= p*s qc:= q*c b:= ps−qc i:= i + j return s本文继续验证用简单命名复值标称数据[6]、[8]、[19]、[11]、[16]、[17]编写的算法[10]、[14]、[12]、[15]、[13]。该算法的有效性以此类数据上的语义Floyd-Hoare三元组的形式呈现[9]。正确性的证明是基于一个扩展的Floyd-Hoare逻辑[2]，[4]的推理系统，该推理系统具有部分前置和后置条件[18]，[21]，[7]，[5]。
{"title":"Partial Correctness of an Algorithm Computing Lucas Sequences","authors":"Adrian Jaszczak","doi":"10.2478/forma-2020-0025","DOIUrl":"https://doi.org/10.2478/forma-2020-0025","url":null,"abstract":"Summary In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1]. i := 0 s := x b := y c := x while (i <> n) c := s s := b ps := p*s qc := q*c b := ps − qc i := i + j return s This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [18], [21], [7], [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"34 1","pages":"279 - 288"},"PeriodicalIF":0.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78744126","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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