Dual Lattice of ℤ-module Lattice

SummaryIn this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20], [10] and [19].Futa Yuichi - Tokyo University of Technology, Tokyo, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Yuichi Futa and Yasunari Shidama. Lattice of ℤ-module. Formalized Mathematics, 24 (1):49–68, 2016. doi: 10.1515/forma-2016-0005.Yuichi Futa and Yasunari Shidama. Embedded lattice and properties of Gram matrix. Formalized Mathematics, 25(1):73–86, 2017. doi: 10.1515/forma-2017-0007.Yuichi Futa and Yasunari Shidama. Divisible ℤ-modules. Formalized Mathematics, 24 (1):37–47, 2016. doi: 10.1515/forma-2016-0004.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi: 10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205–214, 2012. doi: 10.2478/v10037-012-0024-y.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29–49, 2015. doi: 10.2478/forma-2015-0003.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982. doi: 10.1007/BF01457454.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495–500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877–882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.25215716

Isometric Differentiable Functions on Real Normed Space

In this article, we formalize isometric differentiable functions on real normed space [17], and their properties.Futa Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanEndou Noboru - Gifu National College of Technology Gifu, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81-85, 2007. doi:10.2478/v10037-007-0010-y.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Hiroshi Imura, Yuji Sakai, and Yasunari Shidama. Differentiable functions on normed linear spaces. Part II. Formalized Mathematics, 12(3):371-374, 2004.Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697-702, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51-59, 2011. doi:10.2478/v10037-011-0009-2.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Laurent Schwartz. Cours d’analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Yasunari Shidama. Differentiable functions on normed linear spaces. Formalized Mathematics, 20(1):31-40, 2012. doi:10.2478/v10037-012-0005-1.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Quotient Module of Z-module

In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module.This work was supported by JSPS KAKENHI 22300285.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012, doi: 10.2478/v10037-012-0007-z.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective (the international series in engineering and computer science). 2002.Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Formal definition of probability on finite and discrete sample space for proving security of cryptographic systems using Mizar

Security proofs for cryptographic systems are very important. The ultimate objective of our study is to prove the security of cryptographic systems using the Mizar proof checker. In this study, we formalize the probability on a finite and discrete sample space to achieve our aim. Therefore, we introduce a formalization of the probability distribution and prove the correctness of the formalization using the Mizar proof checking system as a formal verification tool.ArticleArtificial Intelligence Research. 2(4):37-48 (2013)journal articl

Constructing Binary Huffman Tree

This research was presented during the 2013 International Conference on Foundations of Computer Science FCS’13 in Las Vegas, USA.Huffman coding is one of a most famous entropy encoding methods for lossless data compression [16]. JPEG and ZIP formats employ variants of Huffman encoding as lossless compression algorithms. Huffman coding is a bijective map from source letters into leaves of the Huffman tree constructed by the algorithm. In this article we formalize an algorithm constructing a binary code tree, Huffman tree.Hiroyuki Okazaki - This work was supported by JSPS KAKENHI 21240001.Yasunari Shidama - This work was supported by JSPS KAKENHI 22300285.Okazaki Hiroyuki - Shinshu University Nagano, JapanFuta Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397-402, 1991.Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195-204, 1992.Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Piotr Rudnicki. On defining functions on binary trees. Formalized Mathematics, 5(1):9-13, 1996.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.D. A. Huffman. A method for the construction of minimum-redundancy codes. Proceedings of the I.R.E, 1952.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Hiroyuki Okazaki and Yasunari Shidama. Probability on finite set and real-valued random variables. Formalized Mathematics, 17(2):129-136, 2009. doi:10.2478/v10037-009-0014-x.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Embedded Lattice and Properties of Gram Matrix

In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [16] and cryptographic systems with lattice [17].Futa Yuichi - Tokyo University of Technology, Tokyo, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Yuichi Futa and Yasunari Shidama. Lattice of Z-module. Formalized Mathematics, 24 (1):49-68, 2016. doi: 10.1515/forma-2016-0005.Yuichi Futa and Yasunari Shidama. Divisible Z-modules. Formalized Mathematics, 24 (1):37-47, 2016. doi: 10.1515/forma-2016-0004.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathe matics, 20(1):47-59, 2012. doi: 10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205-214, 2012.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29-49, 2015.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lov´asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515-534, 1982.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.Karol Pak. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007.Karol Pak and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143-150, 2007.Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127-136, 2007.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Formalization of Definitions and Theorems Related to an Elliptic Curve Over a Finite Prime Field by Using Mizar

In this paper, we introduce our formalization of the definitions and theorems related to an elliptic curve over a finite prime field. The elliptic curve is important in an elliptic curve cryptosystem whose security is based on the computational complexity of the elliptic curve discrete logarithm problem.ArticleJOURNAL OF AUTOMATED REASONING. 50(2):161-172 (2013)journal articl

Polynomially Bounded Sequences and Polynomial Sequences

AbstractIn this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].Hiroyuki Okazaki - Shinshu University, Nagano, JapanYuichi Futa - Japan Advanced Institute of Science and Technology, Ishikawa, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek and Piotr Rudnicki. Two programs for SCM. Part I - preliminaries. Formalized Mathematics, 4(1):69-72, 1993.E.J. Barbeau. Polynomials. Springer, 2003.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Jon Kleinberg and Eva Tardos. Algorithm Design. Addison-Wesley, 2005.Donald E. Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2 (1):17-28, 1991.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part I: Theory. Formalized Mathematics, 9(1):135-142, 2001.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143-154, 2001.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y. [Crossref]Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195-200, 2004.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Inverse Function Theorem. Part I

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 ↶ ≂ (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.This study has been supported in part by JSPS KAKENHI Grant Numbers JP20K19863 and JP17K00182.Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, JapanYuichi Futa - Tokyo University of Technology, Tokyo, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004.Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27(2):107–115, 2019. doi:10.2478/forma-2019-0012.Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117–131, 2019. doi:10.2478/forma-2019-0013.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach’s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271–274, 2012. doi:10.2478/v10037-012-0032-y.Laurent Schwartz. Th´eorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul diff´erentiel, tome 2. Analyse. Hermann, 1997.29191