Logical correctness of vector calculation programs

Summary. In C-program, vectors of n-dimension are sometimes represented by arrays, where the dimension n is saved in the 0-th element of each array. If we write the program in non-overwriting type, we can give Logical-Model to each program. Here, we give a program calculating inner product of 2 vectors, as an example of such a type, and its Logical-Model. If the Logical-Model is well defined, and theorems tying the model with previous definitions are given, we can say that the program is logically correct. In case the program is given as implicit function form (i.e., the result of calculation is given by a variable of one of arguments of a function), its Logical-Model is given by a definition of a new predicate form. Logical correctness of such a program is shown by theorems following the definition. As examples of such programs, we presented vector calculation of add, sub, minus and scalar product. and [7] provide the terminology and notation for this paper. In this paper m, n, i are natural numbers and D is a set. The following proposition is true (1) For all n, m holds n ∈ m iff n < m. Let D be a non empty set. One can check that there exists a finite 0-sequence of D which is non empty. The following proposition is true (2) For every non empty set D and for every non empty finite 0-sequence f of D holds len f > 0. Let D be a set and let q be a finite sequence of elements of D. The functor FS2XFS(q) yields a finite 0-sequence of D and is defined by: 37

The Mikabu and Chichibu belts in the Iya-gawa area, Tokushima Prefecture, Shikoku, SW Japan : An integrated research of geology and geomorphology

The Mikabu and Chichibu belts in the Iya-gawa area are studied with respect to their composition, boundaries and the relation with landslides. The Kyobashira-Sugeoi Fault bound the Mikabu Greenstone Belt from the Sambagawa Belt. The Kubo Greenstone body, a part of the Mikabu Greenstone, distributes as an axial part of antiform in the southern margin of the Sambagawa Belt. The Mikabu Greenstone of the area is composed of tholeiitic and alkaline rocks. The Mikabu Greenstone thrust to the south with the Okoyatoko-Nagoro Fault upon the Jurassic accretionary complex of the North Chichibu Belt. Even the phillite generally dipping towards the north in the northern part of the North Chichibu Belt, the oceanic-plate successions of chert-mudstone sequences are stratigraphically upper towards the south. The radiolarian age, estimated at the lithological transition from chert to mudstone in the oceanic-plate sequences, is Early Jurassic. The landslides concentrate in the areas of the Mikabu Greenstone and valley walls along the Iya-gawa River. The linear depressions and ponds, arrange on the mountain ridge between the Miune and Tengu-Zuka along the southern border of the Mikabu Greenstone, are regarded as a sign of large-scale collapse by long-period gravitational mass rock deformation of the mountain body. The weathered zone of the Mikabu Greenstone often includes swelling clay that causes landslides in the area

Implicit Function Theorem. Part I

In this article, we formalize in Mizar [1], [3] the existence and uniqueness part of the implicit function theorem. In the first section, some composition properties of Lipschitz continuous linear function are discussed. In the second section, a definition of closed ball and theorems of several properties of open and closed sets in Banach space are described. In the last section, we formalized the existence and uniqueness of continuous implicit function in Banach space using Banach fixed point theorem. We referred to [7], [8], and [2] in this formalization.This study was supported in part by JSPS KAKENHI Grant Number JP17K00182Nakasho Kazuhisa - Osaka University, Osaka, JapanFuta Yuichi - Tokyo University of Technology, Tokyo, JapanShidama Yasunari - Shinshu University, Nagano, 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.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.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.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. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Google ScholarYasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.25426928

More on the Continuity of Real Functions

In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.Narita Keiko - Hirosaki-city, Aomori, JapanKornilowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandShidama 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 Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Artur Korniłowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.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. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 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

The Differentiable Functions from R into Rⁿ

In control engineering, differentiable partial functions from R into Rⁿ play a very important role. In this article, we formalized basic properties of such functions.Narita Keiko - Hirosaki-city, Aomori, JapanKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandShidama Yasunari - Shinshu University, Nagano, 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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.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. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 19(1):1-9, 2011, doi: 10.2478/v10037-011-0001-x.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.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, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, e2011, doi: 10.2478/v10037-011-0012-7.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 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

The Perfect Number Theorem and Wilson's Theorem

This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson's theorem (that n is prime iff n > 1 and (n - 1)! ≅ -1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function Φ, proves that Φ is multiplicative and that Σ k|n Φ(k) = n.Casella Postale 49, 54038 Montignoso, ItalyM. Aigner and G. M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin Heidelberg New York, 2004.Grzegorz 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. König's theorem. Formalized Mathematics, 1(3):589-593, 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.Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.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. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Yoshinori Fujisawa and Yasushi Fuwa. The Euler's function. Formalized Mathematics, 6(4):549-551, 1997.Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin's test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrzebska and Adam Grabowski. On the properties of the Möbius function. Formalized Mathematics, 14(1):29-36, 2006, doi:10.2478/v10037-006-0005-0.Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279-288, 1992.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.W. J. LeVeque. Fundamentals of Number Theory. Dover Publication, New York, 1996.Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83-86, 1993.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Piotr Rudnicki. Little Bezout theorem (factor theorem). Formalized Mathematics, 12(1):49-58, 2004.Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 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, Yasunari Shidama, and Yatsuka Nakamura. Bessel's inequality. Formalized Mathematics, 11(2):169-173, 2003

More on Continuous Functions on Normed Linear Spaces

In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].Okazaki Hiroyuki - Shinshu University, Nagano, JapanEndou Noboru - Nagano National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 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. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Laurent Schwartz. Cours d'analyse, vol. 1. Hermann Paris, 1967. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech 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

Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Nagano National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.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. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).Inoué Takao - Inaba 2205, Wing-Minamikan Nagano, Nagano, JapanNaumowicz Adam - Institute of Computer Science, University of Białystok, Akademicka 2, 15-267 Białystok, PolandEndou Noboru - Gifu National College of Technology, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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 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. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces n. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Takao Inoué, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 18(4):207-212, 2010, doi: 10.2478/v10037-010-0025-7.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Yatsuka Nakamura, Artur Korniłowicz, Nagato Oya, and Yasunari Shidama. The real vector spaces of finite sequences are finite dimensional. Formalized Mathematics, 17(1):1-9, 2009, doi:10.2478/v10037-009-0001-2.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Walter Rudin. Principles of Mathematical Analysis. MacGraw-Hill, 1976.Laurent Schwartz. Cours d'analyse. Hermann, 1981. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000271006300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Wojciech 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 defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471-474, 2001

Binary Arithmetics

this article some theorems concerning these operators. We also introduce the concept of an n-bit register. Such registers are used in the definition of binary unsigned arithmetic presented in this article. Theorems on the relationships of such concepts to the operations of natural numbers are also given. MML Identifier: BINARITH