Download or read book Theorem Proving with the Real Numbers written by John Harrison and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the use of the real numbers in theorem proving. Typ ically, theorem provers only support a few 'discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification of float ing point hardware and hybrid systems. It also allows the formalization of many more branches of classical mathematics, which is particularly relevant for attempts to inject more rigour into computer algebra systems. Our work is conducted in a version of the HOL theorem prover. We de scribe the rigorous definitional construction of the real numbers, using a new version of Cantor's method, and the formalization of a significant portion of real analysis. We also describe an advanced derived decision procedure for the 'Tarski subset' of real algebra as well as some more modest but practically useful tools for automating explicit calculations and routine linear arithmetic reasoning. Finally, we consider in more detail two interesting application areas. We discuss the desirability of combining the rigour of theorem provers with the power and convenience of computer algebra systems, and explain a method we have used in practice to achieve this. We then move on to the verification of floating point hardware. After a careful discussion of possible correctness specifications, we report on two case studies, one involving a transcendental function.
Download or read book The Real Numbers and Real Analysis written by Ethan D. Bloch and published by Springer Science & Business Media. This book was released on 2011-05-27 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
Download or read book Real Analysis Classic Version written by Halsey Royden and published by Pearson Modern Classics for Advanced Mathematics Series. This book was released on 2017-02-13 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is designed for graduate-level courses in real analysis. Real Analysis, 4th Edition, covers the basic material that every graduate student should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in undergraduate mathematics and familiarity with the material covered in an undergraduate course on the fundamental concepts of analysis.
Download or read book How to Prove It written by Daniel J. Velleman and published by Cambridge University Press. This book was released on 2006-01-16 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.
Download or read book Proofs from THE BOOK written by Martin Aigner and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.
Download or read book Foundations of Analysis written by Joseph L. Taylor and published by American Mathematical Soc.. This book was released on 2012 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: Foundations of Analysis has two main goals. The first is to develop in students the mathematical maturity and sophistication they will need as they move through the upper division curriculum. The second is to present a rigorous development of both single and several variable calculus, beginning with a study of the properties of the real number system. The presentation is both thorough and concise, with simple, straightforward explanations. The exercises differ widely in level of abstraction and level of difficulty. They vary from the simple to the quite difficult and from the computational to the theoretical. Each section contains a number of examples designed to illustrate the material in the section and to teach students how to approach the exercises for that section. --Book cover.
Download or read book The Fundamental Theorem of Algebra written by Benjamin Fine and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics.
Download or read book Interactive Theorem Proving written by Lennart Beringer and published by Springer. This book was released on 2012-08-10 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed proceedings of the Third International Conference on Interactive Theorem Proving, ITP 2012, held in Princeton, NJ, USA, in August 2012. The 21 revised full papers presented together with 4 rough diamond papers, 3 invited talks, and one invited tutorial were carefully reviewed and selected from 40 submissions. Among the topics covered are formalization of mathematics; program abstraction and logics; data structures and synthesis; security; (non-)termination and automata; program verification; theorem prover development; reasoning about program execution; and prover infrastructure and modeling styles.
Download or read book Theorem Proving in Higher Order Logics written by Otmane Ait Mohamed and published by Springer. This book was released on 2008-10-04 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2008, held in Montreal, Canada, in August 2008. The 17 revised full papers presented together with 1 proof pearl (concise and elegant presentations of interesting examples), 5 tool presentations, and 2 invited papers were carefully reviewed and selected from 40 submissions. The papers cover all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification such as formal semantics of specification, modeling, and programming languages, specification and verification of hardware and software, formalisation of mathematical theories, advances in theorem prover technology, as well as industrial application of theorem provers.
Download or read book Theorem Proving in Higher Order Logics written by Richard J. Boulton and published by Springer. This book was released on 2003-06-30 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume constitutes the proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2001) held 3–6 September 2001 in Edinburgh, Scotland. TPHOLs covers all aspects of theorem proving in higher order logics, as well as related topics in theorem proving and veri?cation. TPHOLs 2001 was collocated with the 11th Advanced Research Working Conference on Correct Hardware Design and Veri?cation Methods (CHARME 2001). This was held 4–7 September 2001 in nearby Livingston, Scotland at the Institute for System Level Integration, and a joint half-day session of talks was arranged for the 5th September in Edinburgh. An excursion to Traquair House and a banquet in the Playfair Library of Old College, University of Edinburgh were also jointly organized. The proceedings of CHARME 2001 have been p- lished as volume 2144 of Springer-Verlag’s Lecture Notes in Computer Science series, with Tiziana Margaria and Tom Melham as editors. Each of the 47 papers submitted in the full research category was refereed by at least 3 reviewers who were selected by the Program Committee. Of these submissions, 23 were accepted for presentation at the conference and publication in this volume. In keeping with tradition, TPHOLs 2001 also o?ered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief preliminary talk and then discuss their work at a poster session. A supplementary proceedings containing associated papers for work in progress was published by the Division of Informatics at the University of Edinburgh.
Download or read book Theorem Proving in Higher Order Logics written by Victor A. Carreño and published by . This book was released on 2002 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Higher Order Logic Theorem Proving and its Applications written by L.J.M. Claesen and published by Elsevier. This book was released on 2014-05-23 with total page 588 pages. Available in PDF, EPUB and Kindle. Book excerpt: The HOL system is a higher order logic theorem proving system implemented at Edinburgh University, Cambridge University and INRIA. Its many applications, from the verification of hardware designs at all levels to the verification of programs and communication protocols are considered in depth in this volume. Other systems based on higher order logic, namely Nuprl and LAMBDA are also discussed. Features given particular consideration are: novel developments in higher order logic and its implementations in HOL; formal design and verification methodologies for hardware and software; public domain availability of the HOL system. Papers addressing these issues have been divided as follows: Mathematical Logic; Induction; General Modelling and Proofs; Formalizing and Modelling of Automata; Program Verification; Hardware Description Language Semantics; Hardware Verification Methodologies; Simulation in Higher Order Logic; Extended Uses of Higher Order Logic. Academic and industrial researchers involved in formal hardware and software design and verification methods should find the publication especially interesting and it is hoped it will also provide a useful reference tool for those working at software institutes and within the electronics industries.
Download or read book Understanding Mathematical Proof written by John Taylor and published by CRC Press. This book was released on 2016-04-19 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn
Download or read book Formalized Probability Theory and Applications Using Theorem Proving written by Hasan, Osman and published by IGI Global. This book was released on 2015-03-31 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scientists and engineers often have to deal with systems that exhibit random or unpredictable elements and must effectively evaluate probabilities in each situation. Computer simulations, while the traditional tool used to solve such problems, are limited in the scale and complexity of the problems they can solve. Formalized Probability Theory and Applications Using Theorem Proving discusses some of the limitations inherent in computer systems when applied to problems of probabilistic analysis, and presents a novel solution to these limitations, combining higher-order logic with computer-based theorem proving. Combining practical application with theoretical discussion, this book is an important reference tool for mathematicians, scientists, engineers, and researchers in all STEM fields.
Download or read book A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton s Principia written by Jacques Fleuriot and published by Springer Science & Business Media. This book was released on 2012-09-30 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.
Download or read book How to Prove It written by Daniel J. Velleman and published by Cambridge University Press. This book was released on 2019-07-18 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: Helps students transition from problem solving to proving theorems, with a new chapter on number theory and over 150 new exercises.
Download or read book Interactive Theorem Proving written by Marko Van Eekelen and published by Springer. This book was released on 2011-08-02 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Second International Conference on Interactive Theorem proving, ITP 2011, held in Berg en Dal, The Netherlands, in August 2011. The 25 revised full papers presented were carefully reviewed and selected from 50 submissions. Among the topics covered are counterexample generation, verification, validation, term rewriting, theorem proving, computability theory, translations from one formalism to another, and cooperation between tools. Several verification case studies were presented, with applications to computational geometry, unification, real analysis, etc.
