Theorem Proving with the Real Numbers

The Distinguished Dissertation series is published on behalf of the Conference of Professors and Heads of Computing and the British Computer Society, who annually select the best British PhD dissertations in computer science for publication. The dissertations are selected on behalf of the CPHC by a panel of eight academics. Each dissertation chosen makes a noteworthy contribution to the subject and reaches a high standard of exposition, placing all results clearly in the context of computer science as a whole. In this way computer scientists with significantly different interests are able to grasp the essentials - or even find a means of entry - to an unfamiliar research topic. Theorem Proving with the Real Numbers discusses the formal development of classical mathematics using a computer. It combines traditional lines of research in theorem proving and computer algebra and shows the usefulness of real numbers in verification.
TOC: Introduction.- Symbolic Computation.- Verification.- Higher Order Logic.- Theorem Proving v Model Checking.- Automated vs Interactive Theorem Proving.- The Real Numbers.- Concluding Remarks.- Constructing the Real Numbers.- Properties of the Real Numbers.- Uniqueness of the Real Numbers.- Constructing the Real Numbers.- Positional Expansions.- Cantor's Method.- Dedekind's Method.- What Choice?- Lemmas about Nearly-Multiplicative Functions.- Details of the Construction.- Adding Negative Numbers.- Handling Equivalence Classes.- Formalized Analysis.- Explicit Calculations.- A Decision Procedure for Real Algebra.- Computer Algebra Systems.- Floating Point Verification.- Conclusions.- Logical Foundations of HOL.- Recent Developments.