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.Institute of Applied Analysis, University of Ulm, 89069 Ulm, GermanyGrzegorz 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 Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.J. H. Conway. On numbers and games. A K Peters Ltd., Natick, MA, second edition, 2001.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 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

Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar

This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems,definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial Intelligence and Symbolic Computation AISC 201

Computational Design of Alloy Nanostructures for Optical Sensing of Hydrogen

Pd nanoalloys show great potential as hysteresis-free, reliable hydrogen sensors. Here, a multiscale modeling approach is employed to determine optimal conditions for optical hydrogen sensing using the Pd-Au-H system. Changes in hydrogen pressure translate to changes in hydrogen content and eventually the optical spectrum. At the single particle level, the shift of the plasmon peak position with hydrogen concentration (i.e., the "optical" sensitivity) is approximately constant at 180 nm/c(H) for nanodisk diameters of greater than or similar to 100 nm. For smaller particles, the optical sensitivity is negative and increases with decreasing diameter, due to the emergence of a second peak originating from coupling between a localized surface plasmon and interband transitions. In addition to tracking peak position, the onset of extinction as well as extinction at fixed wavelengths is considered. We carefully compare the simulation results with experimental data and assess the potential sources for discrepancies. Invariably, the results suggest that there is an upper bound for the optical sensitivity that cannot be overcome by engineering composition and/or geometry. While the alloy composition has a limited impact on optical sensitivity, it can strongly affect H uptake and consequently the "thermodynamic" sensitivity and the detection limit. Here, it is shown how the latter can be improved by compositional engineering and even substantially enhanced via the formation of an ordered phase that can be synthesized at higher hydrogen partial pressures

Formalising Mathematics in Simple Type Theory

Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics. This point is seen by examining formal proofs of a theorem about stereographic projections. A formalisation using the HOL Light proof assistant is contrasted with one using Isabelle/HOL. Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes. However, every formal system can be outgrown, and mathematics should be formalised with a view that it will eventually migrate to a new formalism

The Mizar Mathematical Library in OMDoc: Translation and Applications

Contains fulltext : 111268.pdf (preprint version ) (Open Access

Maintaining a library of formal mathematics

The Lean mathematical library mathlib is developed by a community of users with very different backgrounds and levels of experience. To lower the barrier of entry for contributors and to lessen the burden of reviewing contributions, we have developed a number of tools for the library which check proof developments for subtle mistakes in the code and generate documentation suited for our varied audience.Comment: To appear in Proceedings of CICM 202

Formalizing Arrow's theorem

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