Constructing the real numbers in HOL

Harrison, John

doi:10.1007/bf01384233

Cited by 25 publications

(12 citation statements)

References 5 publications

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“…Until that date, many theorem provers did not even support negative numbers; it was suddenly urgent to deal with floating-point arithmetic and numerical algorithms. Harrison tackled this (Harrison 1994), and went on to accomplish great things in formalized mathematics, including verifying a floatingpoint exponential function (Harrison 2000) and (much later) playing a major role in verifying the celebrated Kepler conjecture (Hales et al 2015).…”

Section: Cambridge and The Emergence Of Hardware Verificationmentioning

confidence: 99%

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Michael John Caldwell Gordon. 28 February 1948—22 August 2017

Paulson

2018

Biogr. Mems Fell. R. Soc.

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Michael Gordon was a pioneer in the field of interactive theorem proving and hardware verification. In the 1970s, he had the vision of formally verifying system designs, proving their correctness using mathematics and logic. He demonstrated his ideas on real-world computer designs. His students extended the work to such diverse areas as the verification of floating-point algorithms, the verification of probabilistic algorithms and the verified translation of source code to correct machine code. He was elected to the Royal Society in 1994, and he continued to produce outstanding research until retirement. His achievements include his work at Edinburgh University helping to create Edinburgh LCF, the first interactive theorem prover of its kind, and the ML family of functional programming languages. He adopted higher-order logic as a general formalism for verification, showing that it could specify hardware designs from the gate level right up to the processor level. It turned out to be an ideal formalism for many problems in computer science and mathematics. His tools and techniques have exerted a huge influence across the field of formal verification.

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Section: Cambridge and The Emergence Of Hardware Verificationmentioning

confidence: 99%

“…Russell was referring to the tedious construction of the real numbers from the rationals using Dedekind cuts, which was formalized by Harrison (Harrison 1994). While other verification groups preferred theft, Mike and his students were firmly committed to rigour.…”

Section: Software Verification Arm6 and Verified Compilersmentioning

confidence: 99%

Michael John Caldwell Gordon. 28 February 1948—22 August 2017

Paulson

2018

Biogr. Mems Fell. R. Soc.

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“…Various mechanizations of standard analysis (see, for example, Harrison's work using the HOL-Light system [13,14]) have developed theories of limits, derivatives, continuity of functions and so on, taking as their foundations the real numbers. Our work, however, will now go one step further, and show how the reals can be used to build a richer number system.…”

Section: Theorem 41 (Completeness Of the Reals)mentioning

confidence: 99%

Mechanizing Nonstandard Real Analysis

Fleuriot

Paulson²

2000

LMS J. Comput. Math.

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This paper first describes the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic (HOL). The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis (NSA). The construction of the hyperreal number system has been carried out strictly through the use of definitions to ensure that the foundations of NSA in Isabelle are sound. Mechanizing the construction has required that various number systems including the rationals and the reals be built up first. Moreover, to construct the hyperreals from the reals has required developing a theory of filters and ultrafilters and proving Zorn's lemma, an equivalent form of the axiom of choice.This paper also describes the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis. The current work provides both standard and nonstandard definitions for the various notions, and proves their equivalence in each case. To achieve this aim, systematic methods, through which sets and functions are extended to the hyperreals, are developed in the framework. The merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem-proving are highlighted throughout the exposition.

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“…Fueled by applications in automated theorem proving and verification, where one must represent the real numbers in a computer, nuances of the differences between various constructions of the reals become very pronounce. We refer the reader to [6] and [17,18] for more details on the constructive reals and on theorem proving with the real numbers, respectively.…”

Section: Introductionmentioning

confidence: 99%