Multiplication-Related Classes of Complex Numbers

The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced.Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandMarco B. Caminati and Giuseppe Rosolini. Custom automations in Mizar. Journal of Automated Reasoning, 50(2):147–160, 2013.Artur Korniłowicz. On rewriting rules in Mizar. Journal of Automated Reasoning, 50(2): 203–210, 2013.Karol Pąk. Improving legibility of natural deduction proofs is not trivial. Logical Methods in Computer Science, 10, 2014.28219721

Improving legibility of natural deduction proofs is not trivial

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page

Fermat’s Little Theorem via Divisibility of Newton’s Binomial

AbstractSolving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in [12]), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs [9], [17].Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandGrzegorz 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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.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. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Marco B. Caminati and Giuseppe Rosolini. Custom automations in Mizar. Journal of Automated Reasoning, 50(2):147-160, 2013.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. Euler’s Theorem and small Fermat’s Theorem. Formalized Mathematics, 7(1):123-126, 1998.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.Jacek Gancarzewicz. Arytmetyka, 2000. In Polish.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997.Artur Korniłowicz. On rewriting rules in Mizar. Journal of Automated Reasoning, 50(2): 203-210, 2013.Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143-154, 2001.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Christoph Schwarzweller. Modular integer arithmetic. Formalized Mathematics, 16(3): 247-252, 2008. doi:10.2478/v10037-008-0029-8. [Crossref]Wacław Sierpinski. Teoria liczb. 1950. In Polish.Wacław Sierpinski. O rozwiazywaniu rownan w liczbach całkowitych, 1956. In Polish.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.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Li Yan, Xiquan Liang, and Junjie Zhao. Gauss lemma and law of quadratic reciprocity. Formalized Mathematics, 16(1):23-28, 2008. doi:10.2478/v10037-008-0004-4. [Crossref]Rafał Ziobro. Some remarkable identities involving numbers. Formalized Mathematics, 22(3):205-208, 2014. doi:10.2478/forma-2014-0023. [Crossref