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###### Polish Notation

, Logic, Reasoning and Argumentation. Springer, 2015. [14] Andrzej Grzegorczyk. Filozofia logiki i formalna logika niesymplifikacyjna. Zagadnienia Naukoznawstwa, XLVII(4), 2012. In Polish. [15] Jan Łukasiewicz. Uwagi o aksjomacie Nicoda i ‘dedukcji uogólniajacej’. In Ksiega pamiatkowa Polskiego Towarzystwa Filozoficznego, Lwów, 1931. In Polish. [16] Andrzej Nedzusiak. Probability. Formalized Mathematics , 1(4):745-749, 1990. [17] Beata Padlewska. Families of sets. Formalized Mathematics , 1

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## Summary

Two construction functors: simple term with a variable and compound term with an operation and argument terms and schemes of term induction are introduced. The degree of construction as a number of used operation symbols is defined. Next, the term context is investigated. An x-context is a term which includes a variable x once only. The compound term is x-context iff the argument terms include an x-context once only. The context induction is shown and used many times. As a key concept, the context substitution is introduced. Finally, the translations and endomorphisms are expressed by context substitution.

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## Summary

The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.

In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine.

The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].

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## Towards the Construction of a Model of Mizar Concepts

The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variable-dependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependence-closed set of variables and the sup-semilattice of widening of Mizar types (smooth type widening).

In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasi-loci, quasi-terms, quasi-adjectives, and quasi-types. The structural induction and operation of substitution are also introduced. The prefix quasi is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasi-loci, and quasi-terms have no assigned types and, in result, there is no possibility to conduct type-checking of arguments. The other gaps concern inconsistent and out-of-context clusters of adjectives in types. Those rules are required in the Mizar identification process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like equational classes).

MML identifier: ABCMIZ 1, version: 7.9.01 4.101.1015

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## Summary

In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

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###### The Axiomatization of Propositional Linear Time Temporal Logic

. Temporal Logic and State Systems . Springer-Verlag, 2008. [10] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ):115-122, 1990. [11] Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics , 8( 1 ):133-137, 1999. [12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1( 1 ):67-71, 1990. [13] Edmund Woronowicz. Many-argument relations. Formalized

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###### Model Checking. Part II

. Model Checking. MIT Press, 2000. [10] Kazuhisa Ishida. Model checking. Part I. Formalized Mathematics , 14(4):171-186, 2006. [11] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1(1):67-71, 1990. [12] Edmund Woronowicz. Many-argument relations. Formalized Mathematics , 1(4):733-737, 1990.

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###### Stability of the 4-2 Binary Addition Circuit Cells. Part I

. [9] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics , 5(1):37-42, 1996. [10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1(1):67-71, 1990. [11] E. Jean Vuillemin. A very fast multiplication algorithm for VLSI implementation, integration. The VLSI Journal , 1(1):39-52, 1983. [12] Katsumi Wasaki and Pauline N. Kawamoto. 2's complement circuit. Formalized Mathematics , 6(2):189-197, 1997. [13] Edmund Woronowicz. Many-argument

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###### The Derivations of Temporal Logic Formulas

. [14] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1( 1 ):67-71, 1990. [15] Edmund Woronowicz. Many argument relations. Formalized Mathematics , 1( 4 ):733-737, 1990. [16] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics , 1( 1 ):73-83, 1990. [17] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics , 1( 1 ):181-186, 1990.

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###### Stability of n-Bit Generalized Full Adder Circuits (GFAs). Part II

, projections and Cartesian products. Formalized Mathematics , 1(1):97-105, 1990. [13] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics , 4(1):15-22, 1993. [14] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics , 5(1):37-42, 1996. [15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1(1):67-71, 1990. [16] Edmund Woronowicz. Many-argument relations. Formalized Mathematics , 1(4):733-737, 1990. [17

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