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Author:Adam Grabowski
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (2), pp.93-100De Gruyter
Abstract:Summary In the article we present in the Mizar system [ 1 ], [ 8 ] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [ 13 ]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize ...
Author:Peter Jaeger
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (2), pp.101-105De Gruyter
Abstract:Summary We start with the definition of stopping time according to [ 4 ], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [ 6 ...
Author:Noboru Endou
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (1), pp.1-29De Gruyter
Abstract:Summary The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure construc...
Author:Artur Korniłowicz , Karol Pąk
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (2), pp.87-92De Gruyter
Abstract:Summary In the article we formalized in the Mizar system [ 2 ] the Vieta formula about the sum of roots of a polynomial anxn + an− 1 xn− 1 + ··· + a 1 x + a 0 defined over an algebra...
Author:Roland Coghetto
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (2), pp.107-119De Gruyter
Abstract:Summary In 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 coni...
Author:Roland Coghetto
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (1), pp.55-62De Gruyter
Abstract:Summary Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group. Then, we prove that, using the notations of Borsuk and Szmielew in [3] “Consider in space ℝℙ2 points P1, P2...
Author:Adam Grabowski , Artur Korniłowicz
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (1), pp.39-48De Gruyter
Abstract:Summary The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer ...
Author:Sebastian Koch
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (2), pp.121-139De Gruyter
Abstract:Summary In 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 el...
Author:Christoph Schwarzweller
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (1), pp.63-72De Gruyter
Abstract:Summary We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial ring...
Author:Artur Korniłowicz
Data Source:[J].Formalized Mathematics, 2017, Vol.25 (1), pp.31-37De Gruyter
Abstract:Summary In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].

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