53 research outputs found
Preventing Exceptions to Robins InEquality
For sufficiently large n Ramanujan gave a sufficient condition for the truth
Robin's InEquality (RIE). The
largest known violation of RIE is . In this paper Robin's multipliers
are split into logarithmic terms and relative divisor sums
. A violation of RIE above is proposed to imply
oscillations that cause to exceed . To this aim
Alaoglu and Erd\H{o}s's conjecture for the CA numbers algorithm is used and the
paper's key points are in section 4.2Comment: 22 pages (12 without tables and directories), 6 figures, 2 tables. V2
addresses some typographical bugs, V3 added outlin
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
Tarski Geometry Axioms â Part II
In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation),of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1),congruence equivalence relation (A2),congruence identity (A3),segment construction (A4),SAS (A5),betweenness identity (A6),Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8),the upper dimension axiom (A9),the Euclid axiom (A10),the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS).In order to show that the structure which satisfies all eleven Tarskiâs axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes.Although the tradition of the mechanization of Tarskiâs geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned â we had some doubts about the proof of the Euclidâs axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].Coghetto Roland - Rue de la Brasserie 5, 7100 La LouviĂšre, BelgiumGrabowski Adam - Institute of Informatics, University of BiaĆystok, CioĆkowskiego 1M, 15-245 BiaĆystok, PolandCzesĆaw ByliĆski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99â107, 2005.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47â53, 1990.Roland Coghetto. Circumcenter, circumcircle and centroid of a triangle. Formalized Mathematics, 24(1):17â26, 2016. doi:10.1515/forma-2016-0002.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599â603, 1991.Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371â385, 2014. doi:10.3233/FI-2014-1129.Adam Grabowski. Tarskiâs geometry modelled in Mizar computerized proof assistant. In Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, FedCSIS 2016, GdaĆsk, Poland, September 11â14, 2016, pages 373â381, 2016. doi:10.15439/2016F290.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211â221, 2015. doi:10.1007/s10817-015-9333-5.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5â10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300â314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi:10.1007/s10817-015-9345-1.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarskiâs Euclidean Axiom. 2012. Masterâs thesis.Julien Narboux. Mechanical theorem proving in Tarskiâs geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869 of Lecture Notes in Computer Science, pages 139â156. Springer, 2007.William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22(2):167â176, 2014. doi:10.2478/forma-2014-0017.Wolfram SchwabhĂ€user, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.Alfred Tarski and Steven Givant. Tarskiâs system of geometry. Bulletin of Symbolic Logic, 5(2):175â214, 1999.Andrzej Trybulec and CzesĆaw ByliĆski. Some properties of real numbers. Formalized Mathematics, 1(3):445â449, 1990.Wojciech A. Trybulec. Axioms of incidence. Formalized Mathematics, 1(1):205â213, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291â296, 1990
E. G. K. Lopez-Escobar. A complete, infinitary axiomatization of weak second-order logic. Fundamenta mathematicae, vol. 61 (1967), pp. 93â103.
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