ATP and Presentation Service for Mizar Formalizations

This paper describes the Automated Reasoning for Mizar (MizAR) service, which integrates several automated reasoning, artificial intelligence, and presentation tools with Mizar and its authoring environment. The service provides ATP assistance to Mizar authors in finding and explaining proofs, and offers generation of Mizar problems as challenges to ATP systems. The service is based on a sound translation from the Mizar language to that of first-order ATP systems, and relies on the recent progress in application of ATP systems in large theories containing tens of thousands of available facts. We present the main features of MizAR services, followed by an account of initial experiments in finding proofs with the ATP assistance. Our initial experience indicates that the tool offers substantial help in exploring the Mizar library and in preparing new Mizar articles

Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].This work has been partially supported by the NSERC grant OGP 9207Rudnicki Piotr - University of Alberta, Edmonton, CanadaStewart Lorna - University of Alberta, Edmonton, CanadaGrzegorz 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. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 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.Frank Harary. Graph theory. Addison-Wesley, 1969.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.J. Mycielski. Sur le coloriage des graphes. Colloquium Mathematicum, 3:161-162, 1955.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Piotr Rudnicki and Lorna Stewart. The Mycielskian of a graph. Formalized Mathematics, 19(1):27-34, 2011, doi: 10.2478/v10037-011-0005-6.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Oswald Veblen. Analysis Situs, volume V. AMS Colloquium Publications, 193

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

I would like to thank Piotr Rudnicki for taking me on as his summer student and being a mentor to me. Piotr was an incredibly caring, intelligent, funny, passionate human being. I am proud to know I was his last student, in a long line of students he has mentored and cared about throughout his life. Thank you Piotr, for the opportunity you gave me, and for the faith, confidence and trust you showed in me. I will miss you.We first provide a modified version of the proof in [3] that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of ℝ1 (that is the real line with the usual topology) are Lindel¨of. We utilize this result in the proof that the Sorgenfrey line is Lindel¨of, which is based on the proof found in [8]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindel¨of, and therefore the product space of two Lindel¨of spaces need not be Lindel¨of. Further, we note that the Sorgenfrey line is regular, following from [3]:59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindel¨of. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in [3], that the Niemytzki plane is not normal. Information was also gathered from [15].Arnaud Adam St. - University of Alberta Edmonton, CanadaRudnicki Piotr - University of Alberta Edmonton, CanadaGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. On constructing topological spaces and Sorgenfrey line. Formalized Mathematics, 13(1):171-179, 2005.Grzegorz Bancerek. Niemytzki plane - an example of Tychonoff space which is not T4. Formalized Mathematics, 13(4):515-524, 2005.Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1): 35-43, 1998.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 and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Ryszard Engelking. Outline of General Topology. North-Holland Publishing Company, 1968.Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13 (1):139-146, 2005.Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pak. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009. doi:10.2478/v10037-009-0024-8.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Lynn Arthur Steen and J. Arthur Jr. Seebach. Counterexamples in Topology. Springer-Verlag, 1978.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library.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.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990

Molecular markers (c-erbB-2, p53) in breast cancer.

The aim of our study was to evaluate the correlation between clinical characteristics, histopatologic features and c-erbB-2 as well as p53 expression in cancer tissues. Breast cancer tissue was obtained from 184 female subjects with primary breast cancer. According to hormonal status patients were divided into two groups - 64 belonged to the premenopausal group and 120 to postmenopausal group. Each patient underwent mammectomy and axillary lymphadenectomy. c-erbB-2 protooncogene was detected in 54% cases, and was correlated with infiltrating type of cancer growth, as well as larger tumor size. The presence of p53 antioncogene was observed only in 33% of cases, mainly in infiltrating duct carcinomas. The incidence of c-erbB-2 and p53 positive cases was higher among subjects, whose ultrasound and mammography revealed malignancy. There was no correlation found between of c-erbB-2 expression and axillary lymph nodes involvement It seems probable, that c-erbB-2 and p53 status of cancer tissue may prove to be useful in assessment of the level of biological aggressiveness in breast carcinomas and hence can be used as a prognostic factor

Representation of the Fibonacci and Lucas Numbers in Terms of Floor and Ceiling

In the paper we show how to express the Fibonacci numbers and Lucas numbers using the floor and ceiling operations.Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Piotr Rudnicki. Two programs for SCM. Part I - preliminaries. Formalized Mathematics, 4(1):69-72, 1993.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.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.Konrad Raczkowski and Andrzej Nędzusiak. 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.Robert M. Solovay. Fibonacci numbers. Formalized Mathematics, 10(2):81-83, 2002.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.Piotr Wojtecki and Adam Grabowski. Lucas numbers and generalized Fibonacci numbers. Formalized Mathematics, 12(3):329-333, 2004

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

Fixpoint Theorem for Continuous Functions on Chain-Complete Posets

This text includes the definition of chain-complete poset, fix-point theorem on it, and the definition of the function space of continuous functions on chain-complete posets [10].Ishida Kazuhisa - Neyagawa-shi, Osaka, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.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.Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501-505, 1996. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=000258624500003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.Glynn Winskel. The Formal Semantics of Programming Languages. The MIT Press, 1993.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.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.Mariusz Żynel and Czesław Byliński. Properties of relational structures, posets, lattices and maps. Formalized Mathematics, 6(1):123-130, 1997

Counting Derangements, Non Bijective Functions and the Birthday Problem

The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].Institut für Informatik I4, Technische Universität München, Boltzmannstraße 3 85748 Garching, GermanyGrzegorz 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.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.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.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.Karol Pąk. Cardinal numbers and finite sets. Formalized Mathematics, 13(3):399-406, 2005.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.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.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Pascal’s Theorem in Real Projective Plane

SummaryIn this article we check, with the Mizar system [2], Pascal’s theorem in the real projective plane (in projective geometry Pascal’s theorem is also known as the Hexagrammum Mysticum Theorem)1. Pappus’ theorem is a special case of a degenerate conic of two lines. For proving Pascal’s theorem, we use the techniques developed in the section “Projective Proofs of Pappus’ Theorem” in the chapter “Pappus’ Theorem: Nine proofs and three variations” [11]. We also follow some ideas from Harrison’s work. With HOL Light, he has the proof of Pascal’s theorem2. For a lemma, we use PROVER93 and OTT2MIZ by Josef Urban4 [12, 6, 7]. We note, that we don’t use Skolem/Herbrand functions (see “Skolemization” in [1]).Rue de la Brasserie 5, 7100 La Louvière, BelgiumJesse Alama. Escape to Mizar for ATPs. arXiv preprint arXiv:1204.6615, 2012.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.Roland Coghetto. Homography in ℝ ℙ2. Formalized Mathematics, 24(4):239–251, 2016. doi: 10.1515/forma-2016-0020.Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55–62, 2017. doi: 10.1515/forma-2017-0005.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Adam Grabowski. Solving two problems in general topology via types. In Types for Proofs and Programs, International Workshop, TYPES 2004, Jouy-en-Josas, France, December 15-18, 2004, Revised Selected Papers, pages 138–153, 2004. doi: 10.1007/116179909.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi: 10.1007/s10817-015-9333-5.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces – part I. Formalized Mathematics, 1(4):767–776, 1990.Jürgen Richter-Gebert. Pappos’s Theorem: Nine Proofs and Three Variations, pages 3–31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi: 10.1007/978-3-642-17286-11.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127–136, 2007. doi: 10.2478/v10037-007-0014-7.25210711

About Quotient Orders and Ordering Sequences

SummaryIn preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σf : D → ℝ, Σf (X)= ∑x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article can then be formulated as: ∑x∈Af(x)=∑X∈DΣf(X)(=∑X∈D∑x∈Xf(x)) After that (weakly) ascending/descending finite sequences (based on [3]) are introduced, in analogous notation to their infinite counterparts introduced in [18] and [13].The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from [16], where finite sequence of real numbers were sorted.The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. [10]).Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences s, t on a set X, such that rng t ⊆ rng s, and a function f : X → ℝ such that f is zero for every x ∈ rng s \ rng t, we have ∑ f o s = ∑ f o t.Johannes Gutenberg University, Mainz, GermanyGrzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 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. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest. Introduction to algorithms. MIT Press, 3. ed. edition, 2009. ISBN 0-262-53305-7, 978-0-262-53305-8, 978-0-262-03384-8. http://scans.hebis.de/HEBCGI/show.pl?21502893_toc.pdf.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Gilbert Lee and Piotr Rudnicki. Dickson’s lemma. Formalized Mathematics, 10(1):29–37, 2002.Michael Maschler, Eilon Solan, and Shmuel Zamir. Game theory. Cambridge Univ. Press, 2013. ISBN 978-1-107-00548-8. doi: 10.1017/CBO9780511794216.Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179–184, 2001.Yatsuka Nakamura. Sorting operators for finite sequences. Formalized Mathematics, 12 (1):1–4, 2004.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Formalized Mathematics, 6(3):339–343, 1997.Bernd S. W. Schröder. Ordered Sets: An Introduction. Birkhäuser Boston, 2003. ISBN 978-1-4612-6591-7. https://books.google.de/books?id=hg8GCAAAQBAJ.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 and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.25212113