Computer Laboratory

Technical reports

Theorem proving with the real numbers

John Robert Harrison

November 1996, 147 pages

This technical report is based on a dissertation submitted June 1996 by the author for the degree of Doctor of Philosophy to the University of Cambridge, Churchill College.

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.

Full text

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BibTeX record

@TechReport{UCAM-CL-TR-408,
  author =	 {Harrison, John Robert},
  title = 	 {{Theorem proving with the real numbers}},
  year = 	 1996,
  month = 	 nov,
  url = 	 {http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-408.ps.gz},
  institution =  {University of Cambridge, Computer Laboratory},
  number = 	 {UCAM-CL-TR-408}
}