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A Machine-Checked Proof of the Odd Order Theorem [chapter]

Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot, Stéphane Le Roux, Assia Mahboubi, Russell O'Connor, Sidi Ould Biha, Ioana Pasca, Laurence Rideau (+3 others)
2013 Lecture Notes in Computer Science  
This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant.  ...  The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq.  ...  The authors would like to thank the Coq team for their continuous development, improvement and maintenance of the Coq proof assistant.  ... 
doi:10.1007/978-3-642-39634-2_14 fatcat:43zakbhshbftjfdptlw2fzyskm

Engineering mathematics

Georges Gonthier
2013 SIGPLAN notices  
By combining tools and techniques from type theory, language design, and software engineering we have managed to capture enough of these practices to formalize the proof of the Odd Order theorem, a landmark  ...  Even with the assistance of computer tools, the formalized description and verification of research-level mathematics remains a daunting task, not least because of the talent with which mathematicians  ...  Odd Order theorem.  ... 
doi:10.1145/2480359.2429071 fatcat:kclmi57xebehrn3xlblowglqve

Software engineering for mathematics (keynote)

Georges Gonthier
2013 Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering - ESEC/FSE 2013  
By drawing on tools and techniques from type theory, language design, and software engineering we captured enough of these practices to formalize the proof of the Odd Order theorem, a landmark result in  ...  This library then allowed us to write a formal proof comparable in size and abstraction level to the 250-page textbook proof of the Odd Order theorem.  ...  He now heads the Mathematical Components project at the MSR Inria Joint Center, which recently completed the formalization of the Odd Order theorem, the landmark result that ultimately lead to the Classification  ... 
doi:10.1145/2491411.2505429 dblp:conf/sigsoft/Gonthier13 fatcat:m5bidz3uzbebxn6tnirio32sgy

Mathematical proof: from mathematics to school mathematics

Helena Rocha
2019 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
mathematical proof in the classroom.  ...  This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed  ...  In a mathematical proof, definitions, statements and procedures are intertwined in a suitable way in order to get the desired result.  ... 
doi:10.1098/rsta.2018.0045 pmid:30966974 pmcid:PMC6365843 fatcat:im66ix244nbadczhede43sqbhu

Mathematical Methods for Physics and Engineering

K. F. Riley, M. P. Hobson, S. J. Bence, Donald Spector
1999 American Journal of Physics  
The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics ever likely to be needed for an undergraduate course in any of the physical sciences.  ...  read mathematics at the University of Cambridge and proceeded to a Ph.D. there in theoretical and experimental nuclear physics.  ...  summation orders, etc. on the xxv www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-0-521-67971-8 -Mathematical Methods for Physics and Engineering: Third  ... 
doi:10.1119/1.19216 fatcat:znfudbcikzbwhcg2pye7ytima4

Teaching mathematics to software engineers [chapter]

Jeannette M. Wing
1995 Lecture Notes in Computer Science  
Based on my experience in teaching formal methods to practicing and aspiring software engineers, I present some of the common stumbling blocks faced when writing formal specifications.  ...  The most conspicuous problem is learning to abstract. I address all these problems indirectly by giving a list of hints to specifiers.  ...  and teaching the Models, Methods, and Analysis courses for the MSE program.  ... 
doi:10.1007/3-540-60043-4_44 fatcat:6bfujov3anc43oz56efpop4ibu

Engineering Mathematics and Modern Technology

Qefsere Doko Gjonbalaj
2018 International Journal of Educational Technology and Learning  
Technology has become an essential tool for doing mathematics in today's world. It can be used in a variety of ways to improve and enhance the teaching of mathematics.  ...  Our main goal, of all efforts in this regard, is the connection between mathematics and technology.  ...  Thus intuition and rigour need not be at odds with each other. By providing a suitably powerful context, intuition naturally leads into the rigour of mathematical proof.  ... 
doi:10.20448/2003.21.8.13 fatcat:4ig3rer2o5edpccjfasj2gf26u

A Machine-Checked Proof of the Odd Order Theorem

Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, Francois Garillot, Stephane Le Roux, Assia Mahboubi, Russell O'Connor, Sidi Ould Biha, Ioana Pasca, Laurence Rideau (+2 others)
2018
The authors would like to thank the Coq team for their continuous development, improvement and maintenance of the Coq proof assistant.  ...  We are now able to state the Odd Order Theorem. The Odd Order Theorem Theorem 1 (Odd Order theorem [14] ). Every finite group of odd order is solvable.  ...  Introduction The Odd Order Theorem asserts that every finite group of odd order is solvable.  ... 
doi:10.1184/r1/6490739.v1 fatcat:qkmtitffqzbwlk3iaclergfrni

Mathematical principles for a first course in software engineering

H.D. Mills, V.R. Basili, J.D. Gannon, R.G. Hamlet
1989 IEEE Transactions on Software Engineering  
The discipline of software engineering has transferred the cummonsense methods of good programming and management to large software projects.  ...  We have developed an introductory computer science course, much as calculus is a basic course for mathematics and the physical sciences, concerned primarily with theoretical foundations and methodology  ...  A simplifying condition for demonstrating that a program satisfies its specification is given in the following theorem, called the Correctness Theorem G.  ... 
doi:10.1109/32.24704 fatcat:boqplornxrbkrn6mtcgy4cmyuq

Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs?

Michael Aristidou
2019 Athens Journal of Education  
As it is already observed by mathematicians and educators, there is a discrepancy between the formal techniques of mathematical logic and the informal techniques of mathematics in regards to proof.  ...  Finally, we argue that even though mathematical logic is central in mathematics, its formal methods are not really necessary in doing and teaching mathematical proofs and the role of those formalities  ...  for his comments on the book Mathematics B' Lyceum, and Babis Stergiou and Christos Lefkonitziatis for communicating to me some of the curriculum historicals related to the paper.  ... 
doi:10.30958/aje.7-1-5 fatcat:rtox7vhbdjcelc4s7u5jovge5a

Engineering Design by Geometric Programming

Chia-Hui Huang
2013 Mathematical Problems in Engineering  
In addition, the engineering design problems are illustrated to evaluate the usefulness of the proposed methods.  ...  A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions, where all functions are of signomial form.  ...  The author also thanks the anonymous referees for their careful reading and fruitful comments on the paper.  ... 
doi:10.1155/2013/568098 fatcat:tqe2p5x4d5cmjfznpdtsviangm

Mathematical Notes

Herbert Busemann, T. J. Rivlin‡, T. H. Slook, Curtis M. Fulton, Sherman K. Stein, F. A. Lewis, R. G. Buschman, C. A. Grimm, Ernest S. Elyash, George Laush, Norman Levine
1960 The American mathematical monthly  
This contradicts the lemma. THEOREM. The order of G is n(n) tf n is odd and (n/2)p(n) if n ts even. Coro.iary 1. A circulant of prime order p belongs to the metacyclic [3] group of degree p.  ...  The following theorem, analogous to Theorem 3.1, characterizes the class K**. THEOREM 3.2. f(x) EG K** if and only if x*f(x)EK. Proof.  ... 
doi:10.1080/00029890.1960.11989486 fatcat:aekydzab2nhpze4juz5dc6k3qy

A reverse engineering approach to the Weil representation

Anne-Marie Aubert, Tomasz Przebinda
2014 Open Mathematics  
Thomas, skipping most of the literature on which the latter is based.  ...  AbstractWe describe a new approach to the Weil representation attached to a symplectic group over a finite or a local field.  ...  Also, we express our gratitude to the two referees for their time and guidance through some related literature. The second author was partially supported by the NSA grant H98230-13-1-0205.  ... 
doi:10.2478/s11533-014-0428-8 fatcat:yfe4armz6vb3roaqizmibzdqxe

MATHEMATICAL EXAMPLES OF SPACE-TIME PHASES

M. DIAKONOVA, R. S. MACKAY
2011 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
The space-time phases of a complex dynamic system are the probability distributions for state as a function of space and time which arise by evolving initial probability distributions from the distant  ...  These examples are expected to serve as useful steps on the way to a general theory of space-time phases.  ...  Acknowledgements We are grateful to Andrei Toom for teaching us the state of the art in the analysis of PCA during his visit to Warwick in January 2010 supported by the EPSRC Mathematics Symposium year  ... 
doi:10.1142/s0218127411029793 fatcat:4hwntgmt4vgwjpukqksylic4wq

Mechanized mathematics

D. H. Lehmer
1966 Bulletin of the American Mathematical Society  
Theorems of this kind are not easy to find in those drab branches of mathematics where elaborate proofs are not the rule. However, there are infinitely many such theorems in number theory alone.  ...  Other theorem proving projects have dealt with parts of plane geometry and first order predicate calculus.  ... 
doi:10.1090/s0002-9904-1966-11553-7 fatcat:jxdywlqlkbd5zovaywfua5d4te
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