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 Theorem Proving with the Real Numbers written by John Robert Harrison and published by . This book was released on 1996 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "This thesis discusses the use of the real numbers in theorem proving. Typically, 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 floating 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 describe 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. We aim to show that a theory of real numbers is useful in practice and interesting in theory, and that the 'LCF style' of theorem proving is well suited to the kind of work we describe. We hope also to convince the reader that the kind of mathematics needed for applications is well within the abilities of current theorem proving technology."

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 Numbers Generalizations of the Reals and Theories of Continua written by P. Ehrlich and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since their appearance in the late 19th century, the Cantor--Dedekind theory of real numbers and philosophy of the continuum have emerged as pillars of standard mathematical philosophy. On the other hand, this period also witnessed the emergence of a variety of alternative theories of real numbers and corresponding theories of continua, as well as non-Archimedean geometry, non-standard analysis, and a number of important generalizations of the system of real numbers, some of which have been described as arithmetic continua of one type or another. With the exception of E.W. Hobson's essay, which is concerned with the ideas of Cantor and Dedekind and their reception at the turn of the century, the papers in the present collection are either concerned with or are contributions to, the latter groups of studies. All the contributors are outstanding authorities in their respective fields, and the essays, which are directed to historians and philosophers of mathematics as well as to mathematicians who are concerned with the foundations of their subject, are preceded by a lengthy historical introduction.

Download or read book An Introduction to Proof through Real Analysis written by Daniel J. Madden and published by John Wiley & Sons. This book was released on 2017-09-12 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own. An Introduction to Proof through Real Analysis is based on course material developed and refined over thirty years by Professor Daniel J. Madden and was designed to function as a complete text for both first proofs and first analysis courses. Written in an engaging and accessible narrative style, this book systematically covers the basic techniques of proof writing, beginning with real numbers and progressing to logic, set theory, topology, and continuity. The book proceeds from natural numbers to rational numbers in a familiar way, and justifies the need for a rigorous definition of real numbers. The mathematical climax of the story it tells is the Intermediate Value Theorem, which justifies the notion that the real numbers are sufficient for solving all geometric problems. • Concentrates solely on designing proofs by placing instruction on proof writing on top of discussions of specific mathematical subjects • Departs from traditional guides to proofs by incorporating elements of both real analysis and algebraic representation • Written in an engaging narrative style to tell the story of proof and its meaning, function, and construction • Uses a particular mathematical idea as the focus of each type of proof presented • Developed from material that has been class-tested and fine-tuned over thirty years in university introductory courses An Introduction to Proof through Real Analysis is the ideal introductory text to proofs for second and third-year undergraduate mathematics students, especially those who have completed a calculus sequence, students learning real analysis for the first time, and those learning proofs for the first time. Daniel J. Madden, PhD, is an Associate Professor of Mathematics at The University of Arizona, Tucson, Arizona, USA. He has taught a junior level course introducing students to the idea of a rigorous proof based on real analysis almost every semester since 1990. Dr. Madden is the winner of the 2015 Southwest Section of the Mathematical Association of America Distinguished Teacher Award. Jason A. Aubrey, PhD, is Assistant Professor of Mathematics and Director, Mathematics Center of the University of Arizona.

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 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 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 Problems And Proofs In Real Analysis Theory Of Measure And Integration written by James J Yeh and published by World Scientific Publishing Company. This book was released on 2014-01-15 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of the proofs of 391 problems in Real Analysis: Theory of Measure and Integration (3rd Edition).Most of the problems in Real Analysis are not mere applications of theorems proved in the book but rather extensions of the proven theorems or related theorems. Proving these problems tests the depth of understanding of the theorems in the main text.This volume will be especially helpful to those who read Real Analysis in self-study and have no easy access to an instructor or an advisor.

Download or read book The Art of Proof written by Matthias Beck and published by Springer Science & Business Media. This book was released on 2010-08-17 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Download or read book Theorem Proving in Higher Order Logics written by Otmane Ait Mohamed and published by Springer Science & Business Media. This book was released on 2008-07-30 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 Joe Hurd and published by Springer. This book was released on 2005-08-29 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume constitutes the proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005), which was held during22–25August2005inOxford,UK.TPHOLscoversallaspectsoftheorem proving in higher order logics as well as related topics in theorem proving and veri?cation. There were 49 papers submitted to TPHOLs 2005 in the full research c- egory, each of which was refereed by at least three reviewers selected by the programcommittee. Of these submissions, 20 researchpapersand 4 proof pearls were accepted for presentation at the conference and publication in this volume. In keeping with longstanding tradition, TPHOLs 2005 also o?ered a venue for the presentation of work in progress, where researchers invited discussion by means of a brief introductory talk and then discussed their work at a poster session. A supplementary proceedings volume was published as a 2005 technical report of the Oxford University Computing Laboratory. The organizers are grateful to Wolfgang Paul and Andrew Pitts for agreeing to give invited talks at TPHOLs 2005.

Download or read book The Real Numbers written by John Stillwell and published by Springer Science & Business Media. This book was released on 2013-10-16 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Download or read book Theorem Proving in Higher Order Logics written by Stefan Berghofer and published by Springer. This book was released on 2009-08-20 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics, TPHOLs 200, held in Munich, Germany, in August 2009. The 26 revised full papers presented together with 1 proof pearl, 4 tool presentations, and 3 invited papers were carefully reviewed and selected from 55 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, formalization of mathematical theories, advances in theorem prover technology, as well as industrial application of theorem provers.

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.

Download or read book Real Analysis written by Daniel W. Cunningham and published by CRC Press. This book was released on 2021-01-19 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: Typically, undergraduates see real analysis as one of the most difficult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to find a way to bridge the gap between a student’s preparation and the mathematical skills that are required to be successful in such a course. Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies offered in this textbook will be appreciated by all readers. Features Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis Suitable for junior or senior undergraduates majoring 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.