The Axiomatization of Propositional Linear Time Temporal Logic

The article introduces propositional linear time temporal logic as a formal system. Axioms and rules of derivation are defined. Soundness Theorem and Deduction Theorem are proved [9].Institute of Sociology, University of Białystok, PolandGrzegorz 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. 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.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69-72, 1999.Fred Kröger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 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

Propositional Linear Temporal Logic with Initial Validity Semantics

In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators.In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article.Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].This work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification , and BST225 Database of mathematical texts checked by computer.Faculty of Economics and Informatics, University of Białystok, Kalvariju 135, LT-08221 Vilnius, Lithuaniais work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification, and BST225 Database of mathematical texts checked by computer.↩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, 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:10.1007/978-3-319-20615-8 17. [Crossref]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.Agata Darmochwał. 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. [Crossref]Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1): 69–72, 1999.Fred Kröger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 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

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 Korniłowicz 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 “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Department of Logic, Informatics and Philosophy of Science, University of Białystok, Plac Uniwersytecki 1, 15-420 Białystok, 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. König’s 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.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.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. 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.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 Kr¨oger 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 Properties of Sets of Temporal Logic Subformulas

I would like to thank Prof. Andrzej Trybulec, Dr. Artur Korniłowicz, Dr. Adam Naumowicz and Karol Pak for their help in preparation of the article.This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an ordered positive-negative pair of finite sequences of formulas (PNP). PNPs represent states of a temporal model.This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Department of Logic, Informatics and Philosophy of Science, University of Białystok, Plac Uniwersytecki 1, 15-420 Białystok, 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. König’s lemma. Formalized Mathematics, 2(3):397-402, 1991.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 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.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.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69-72, 1999.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Fred Kröger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.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 Derivations of Temporal Logic Formulas

I would like to thank Prof. Andrzej Trybulec, Dr. Artur Korniłowicz, Dr. Adam Naumowicz and Karol Pak for their help in preparation of the article.This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [12]. We introduce n-ary connectives and prove their properties. We derive temporal logic formulas.This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Department of Logic, Informatics and Philosophy of Science, University of Białystok, Plac Uniwersytecki 1, 15-420 Białystok, 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. 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. Partial functions. Formalized Mathematics, 1(2):357-367, 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.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69-72, 1999.Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.Fred Kr¨oger and Stephan Merz. Temporal Logic and State Systems. Springer-Verlag, 2008.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. 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 Axiomatization of Propositional Logic

This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α),(α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)),(¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.Faculty of Economics and Informatics, University of Białystok, Kalvariju 135, LT-08221 Vilnius, LithuaniaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Leszek Borys. On paracompactness of metrizable spaces. Formalized Mathematics, 3(1): 81–84, 1992.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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.Mariusz Giero. Propositional linear temporal logic with initial validity semantics. Formalized Mathematics, 23(4):379–386, 2015. doi:10.1515/forma-2015-0030.Witold Pogorzelski. Dictionary of Formal Logic. Wydawnictwo UwB - Bialystok, 1992.Witold Pogorzelski. Notions and theorems of elementary formal logic. Wydawnictwo UwB - Bialystok, 1994.Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Formalized Mathematics, 6(3):339–343, 1997.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133–137, 1999.Anita Wasilewska. An Introduction to Classical and Non-Classical Logics. SUNY Stony Brook, 2005.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 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 Białystok 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. 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.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.Jarosław 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.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.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

On the General Position of Special Polygons 1

Summary. In this paper we introduce the notion of general position. We also show some auxiliary theorems for proving Jordan curve theorem. The following main theorems are proved: 1. End points of a polygon are in the same component of a complement of another polygon if number of common points of these polygons is even; 2. Two points of polygon L are in the same component of a complement of polygon M if two points of polygon M are in the same component of polygon L

1. TREE AND CLASSIFICATION OF A SET

Summary. This article is a continuation of [3] article. Further properties of classification of sets are proved. The notion of hierarchy of a set is introduced. Properties of partitions and hierarchies are shown. The main theorem says that for each hierarchy there exists a classification which union equals to the considered hierarchy