###### Conway's Games and Some of their Basic Properties

## Conway's Games and Some of their Basic Properties

We formulate a few basic concepts of J. H. Conway's theory of games based on his book [6]. This is a first step towards formalizing Conway's theory of numbers into Mizar, which is an approach to proving the existence of a FIELD (i.e., a proper class that satisfies the axioms of a real-closed field) that includes the reals and ordinals, thus providing a uniform, independent and simple approach to these two constructions that does not go via the rational numbers and hence does for example not need the notion of a quotient field.

In this first article on Conway's games, we provide a definition of games, their birthdays (or ranks), their trees (a notion which is not in Conway's book, but is useful as a tool), their negates and their signs, together with some elementary properties of these notions. If one is interested only in Conway's numbers, it would have been easier to define them directly, but going via the notion of a game is a more general approach in the sense that a number is a special instance of a game and that there is a rich theory of games that are not numbers.

The main obstacle in formulating these topics in Mizar is that all definitions are highly recursive, which is not entirely simple to translate into the Mizar language. For example, according to Conway's definition, a game is an object consisting of left and right options which are themselves games, and this is by definition the only way to construct a game. This cannot directly be translated into Mizar, but a theorem is included in the article which proves that our definition is equivalent to Conway's.

###### About Quotient Orders and Ordering Sequences

sets. Formalized Mathematics , 1( 1 ):165–167, 1990. [12] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics , 1( 4 ):703–709, 1990. [13] Gilbert Lee and Piotr Rudnicki. Dickson’s lemma. Formalized Mathematics , 10( 1 ):29–37, 2002. [14] Michael Maschler, Eilon Solan, and Shmuel Zamir. Game theory . Cambridge Univ. Press, 2013. ISBN 978-1-107-00548-8. doi: 10.1017/CBO9780511794216. [15] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized

###### Congruences and Trajectories in Planar Semimodular Lattices

lattice diagrams, Order 33 (2016) 231-237. doi: 10.1007/s11083-015-9361-0 [5] G. Czédli, Quasiplanar diagrams and slim semimodular lattices, Order 33 (2016) 239-262. [6] G. Czédli, Diagrams and rectangular extensions of planar semimodular lattices, Algebra Universalis 77 (2017) 443-498. [7] G. Czédli, T. Dékány, L. Ozsvárt, N. Szakács and B. Udvari, On the number of slim, semimodular lattices, Math. Slovaca 66 (2016) 5-18. [8] G. Czédli and G. Makay, Swing lattice game and a short proof of the swing lemma

###### New seventh and eighth order derivative free methods for solving nonlinear equations

free iterative method for solving nonlinear equations with third order convergence, Int. J. Non. Sci., 13 (2012), 505-512. [5] G. F. Torres, C. Soria and I. A. Badillo, Fifth and sixth order iterative algorithms without derivatives for solving nonlinear equations, Int. J. Pure Appl. Math., 83 (2013), 111-119. [6] A. Hajjah, M. Imran amd M. D. H. Gamal, A two step iterative methods free from derivative for solving nonlinear equations, Appl. Math. Sci., 8 (2014), 8021-8027. [7] F. Zafar, N. Yasmin, S. Akram and M. D