5 research outputs found
Continuity of Barycentric Coordinates in Euclidean Topological Spaces
In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an affine closed space of Δn and the set of vectors created from barycentric coordinates of points of this subset.Institute of Informatics, University of BiaĆystok, 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. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.CzesĆaw ByliĆski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.Jing-Chao Chen. The Steinitz theorem and the dimension of a real linear space. Formalized Mathematics, 6(3):411-415, 1997.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata DarmochwaĆ and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Artur KorniĆowicz. The correspondence between n-dimensional Euclidean space and the product of n real lines. Formalized Mathematics, 18(1):81-85, 2010, doi: 10.2478/v10037-010-0011-0.Eugeniusz Kusak, Wojciech LeoĆczuk, and MichaĆ Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol PÄ
k. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Karol PÄ
k. Linear transformations of Euclidean topological spaces. Formalized Mathematics, 19(2):103-108, 2011, doi: 10.2478/v10037-011-0016-3.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 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.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992
Brouwer Fixed Point Theorem for Simplexes
In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its n + 1 affinely indepedent vertices of Δn. First we introduce the Lebesgue number, which for an arbitrary open cover of a compact metric space M is a positive real number so that any ball of about such radius must be completely contained in a member of the cover. Then we introduce the notion of a bounded simplicial complex and the diameter of a bounded simplicial complex. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of Δn. This article is based on [16].Institute of Informatics, University of BiaĆystok, 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.Grzegorz Bancerek and Yasunari Shidama. Introduction to matroids. Formalized Mathematics, 16(4):325-332, 2008, doi:10.2478/v10037-008-0040-0.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.CzesĆaw ByliĆski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Agata DarmochwaĆ. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559-562, 1991.Roman Duda. Wprowadzenie do topologii. PWN, 1986.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.StanisĆawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur KorniĆowicz. The correspondence between n-dimensional Euclidean space and the product of n real lines. Formalized Mathematics, 18(1):81-85, 2010, doi: 10.2478/v10037-010-0011-0.Yatsuka Nakamura, Andrzej Trybulec, and CzesĆaw ByliĆski. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1-13, 1999.Adam Naumowicz. On Segre's product of partial line spaces. Formalized Mathematics, 9(2):383-390, 2001.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol PÄ
k. Small inductive dimension of topological spaces. Formalized Mathematics, 17(3):207-212, 2009, doi: 10.2478/v10037-009-0025-7.Karol PÄ
k. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Karol PÄ
k. Abstract simplicial complexes. Formalized Mathematics, 18(1):95-106, 2010, doi: 10.2478/v10037-010-0013-y.Karol PÄ
k. Sperner's lemma. Formalized Mathematics, 18(4):189-196, 2010, doi: 10.2478/v10037-010-0022-x.Karol PÄ
k. Continuity of barycentric coordinates in Euclidean topological spaces. Formalized Mathematics, 19(3):139-144, 2011, doi: 10.2478/v10037-011-0022-5.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 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 defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Weak Completeness Theorem for Propositional Linear Time Temporal Logic
The author is the winner of the Mizar Prize for Young Researchers in 2012 for this article.I would like to thank Prof. Dr. Stephan Merz for valuable hints which helped me to prove the theorem. I would particularly like to thank Dr. Artur KorniĆowicz who patiently answered a lot of my questions regarding writing this article. I would like to thank Dr. Josef Urban for discussions and encouragement to write the article. I would like to thank Prof. Andrzej Trybulec, Dr. Adam Naumowicz, Dr. Grzegorz Bancerek and Karol Pak for their help in preparation of the article.We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete PNPs which is used to construct the model.This work has been supported by the Polish Ministry of Science and Higher Education project âManaging a Large Repository of Computer-verified Mathematical Knowledgeâ (N N519 385136).Department of Logic, Informatics and Philosophy of Science, University of BiaĆystok, Plac Uniwersytecki 1, 15-420 BiaĆystok, 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. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Königâs lemma. Formalized Mathematics, 2(3):397-402, 1991.Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185-190, 1996.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.CzesĆaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Mariusz Giero. The axiomatization of propositional linear time temporal logic. Formalized Mathematics, 19(2):113-119, 2011, doi: 10.2478/v10037-011-0018-1.Mariusz Giero. The derivations of temporal logic formulas. Formalized Mathematics, 20(3):215-219, 2012, doi: 10.2478/v10037-012-0025-x.Mariusz Giero. The properties of sets of temporal logic subformulas. Formalized Mathematics, 20(3):221-226, 2012, doi: 10.2478/v10037-012-0026-9.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69-72, 1999.Fred Kršoger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Karol Pak. Continuity of barycentric coordinates in Euclidean topological spaces. Formalized Mathematics, 19(3):139-144, 2011, doi: 10.2478/v10037-011-0022-5.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133-137, 1999.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 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
The Formalization of Decision-Free Petri Net
In this article we formalize the definition of Decision-Free Petri Net (DFPN) presented in [19]. Then we formalize the concept of directed path and directed circuit nets in Petri nets to prove properties of DFPN. We also present the definition of firing transitions and transition sequences with natural numbers marking that always check whether transition is enabled or not and after firing it only removes the available tokens (i.e., it does not remove from zero number of tokens). At the end of this article, we show that the total number of tokens in a circuit of decision-free Petri net always remains the same after firing any sequences of the transition.Shah Pratima K. - Shinshu University Nagano, JapanKawamoto Pauline N. - Shinshu University Nagano, JapanGiero Mariusz - University of BiaĆystok 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.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Pauline N. Kawamoto, Yasushi Fuwa, and Yatsuka Nakamura. Basic Petri net concepts. Formalized Mathematics, 3(2):183-187, 1992.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.JarosĆaw Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Robert Milewski. Subsequences of almost, weakly and poorly one-to-one finite sequences. Formalized Mathematics, 13(2):227-233, 2005.Karol Pak. Continuity of barycentric coordinates in Euclidean topological spaces. Formalized Mathematics, 19(3):139-144, 2011. doi:10.2478/v10037-011-0022-5.Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5 (3):317-322, 1996.MichaĆ 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.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Jiacun Wang. Timed Petri Nets, Theory and Application. Kluwer Academic Publishers, 1998.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990