7 research outputs found
Multiplication-Related Classes of Complex Numbers
The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced.Department of Carbohydrate Technology, University of Agriculture, Krakow, PolandMarco B. Caminati and Giuseppe Rosolini. Custom automations in Mizar. Journal of Automated Reasoning, 50(2):147–160, 2013.Artur Korniłowicz. On rewriting rules in Mizar. Journal of Automated Reasoning, 50(2): 203–210, 2013.Karol Pąk. Improving legibility of natural deduction proofs is not trivial. Logical Methods in Computer Science, 10, 2014.28219721
A simplified framework for first-order languages and its formalization in Mizar
A strictly formal, set-theoretical treatment of classical first-order logic
is given. Since this is done with the goal of a concrete Mizar formalization of
basic results (Lindenbaum lemma; Henkin, satisfiability, completeness and
Lowenheim-Skolem theorems) in mind, it turns into a systematic pursue of
simplification: we give up the notions of free occurrence, of derivation tree,
and study what inference rules are strictly needed to prove the mentioned
results. Afterwards, we discuss details of the actual Mizar implementation, and
give general techniques developed therein.Comment: Ph.D. thesis, defended on January, 20th, 201
Hammering towards QED
This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “one-stroke” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchette’s Sledgehammer research was supported by the Deutsche Forschungs-
gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and
Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian
Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the
UK’s Engineering and Physical Sciences Research Council (grant GR/S57198/01).
Urban’s work was supported by the Marie-Curie Outgoing International Fellowship
project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands
Organisation for Scientific Research (NWO) project Knowledge-based Automated
Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1
Fermat’s Little Theorem via Divisibility of Newton’s Binomial
AbstractSolving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in [12]), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs [9], [17].Department of Carbohydrate Technology, University of Agriculture,
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