A formally verified proof of the prime number theorem

Avigad, Jeremy; Donnelly, Kevin; Gray, David; Raff, Paul

doi:10.1145/1297658.1297660

Cited by 53 publications

(47 citation statements)

References 12 publications

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“…In fact, automation of formal reasoning has recently gone far beyond the elementary facts of arithmetic, permitting the formalization and automatic verification of complex results such as the asymptotic distribution of prime numbers (Avigad, Donnelly, Gray and Raff, 2007), the four color theorem (Gonthier, December 15-17, 2007;Gonthier, 2008) or the Jordan curve theorem (Hales, 2007). All these developments are conspicuous (spanning from 30 to 75 thousand lines of code), but their complexity is still negligible when compared with, say, the size of a modern operating system.…”

Section: Proofs and Programsmentioning

confidence: 99%

Social processes, program verification and all that

Asperti

Geuvers

Natarajan

2009

Math. Struct. Comp. Sci.

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In a controversial paper at the end of 1970's, R.A. De Millo, R.J. Lipton and A.J. Perlis (DeMillo, Lipton and Perlis, 1979) argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and in particular with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of those theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable but formal verification is not antithetical to it. Formal verification should strive not only to cope, but to ease and enhance the collaborative, organic nature of this process, eventually helping to master the growing complexity of scientific knowledge. IntroductionHeavier than air flying machines are impossible.-S.P. Langley (Langley, 1891) Formal verification of programs, no matter how obtained, will not play the same role in the development of computer science and software engineering as proofs do in mathematics.-R.A. De Millo, R.J. Lipton, A.J. Perlis (DeMillo et al., 1979) Samuel Pierpont Langley was a professor of astronomy and physics, and a world-expert in aerodynamics during the late nineteenth and early twentieth century. The esteem with which he is held can be seen from the fact that one of the research centers of NASA is named after Langley. At the height of his research career, Samuel Langley published a result (Langley, 1891) which came to be known as "Langley's Law." According to this erroneous law -higher the speed, lower the drag -more power was required in order to make an aircraft fly slower, and if this were to be true then heavier than air flying machines would certainly have been an impossibility. Fortunately the Wright Brothers Andrea Asperti, Herman Geuvers and Raja Natarajan 2 had not read Langley's book, and they went on to develop the first manned aircrafts which could be controlled in-flight from the aircraft itself.In It is precisely in view of these achievements, however, that we can look back at (DeMillo et al., 1979) with a less passionate and more objective spirit, making a more stringent analysis of their thesis and arguments, without focusing on the polemic frame intentionally chosen by its authors, viz., in Lamport's words † , as a debate between a reasonable engineering approach that completely ignores verification and a completely unrealistic view of verification advocated only by its most naive proponents.In fact, some of the thesis advocated by De Millo, Lipton and Perlis are sharable, very pertinent and still relevant; on the other hand, most of their arguments, after thirty years of research, sounds obsolete and a bit trite, asking for a critical reappraisal. Here is a quick summary of our critique, before we enunciate it in detail.-De Millo et al. state that . . . intuition is the final authority (quoting the logician Rosser).: Intuitions and analogies may help in the explanation a...

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Section: Proofs and Programsmentioning

confidence: 99%

Social processes, program verification and all that

Asperti

Geuvers

Natarajan

2009

Math. Struct. Comp. Sci.

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“…A good amount of work was also spent in the investigation of related fields (Abel summations, properties of the Θ function, upper and lower bounds for Euler's e constant) that at the end have not been used in the main proof, but still have an interest in themselves. The following In Hardy's book [7], the proof of Bertrand's postulate takes 42 lines, while Chebyshev's theorem takes precisely three pages (90 lines): this gives a de Bruijn factor of 20-25, that is in line with other developments in related subjects (see [3,2]). The most interesting datum is however the average time required to formalize a line of mathematical text, that in our case is about 1.5 hours (in [2], on a different arithmetical subject, we gave an estimation of 2 hours per line).…”

Section: Discussionmentioning

confidence: 54%

“…In this paper we address a weaker result, due to Chebyshev around 1850, stating that the order of magnitude of π(n) is n/ log n, meaning that we can find two constants c 1 and c 2 such that, for any n c 1 n log(n) ≤ π(n) ≤ c 2 n log n Even if Chebyshev's theorem is sensibly simpler than the prime number theorem, already formalized by Avigad et al in Isabelle [3] and by Harrison in HOL Light [5], it is far form trivial (in Hardy and Wright's famous textbook [7], it takes pages 340-344 of chapter 22). In particular, our point was to give a fully arithmetical (and constructive) proof of this theorem.…”

Section: Introductionmentioning

confidence: 83%

About the Formalization of Some Results by Chebyshev in Number Theory

Asperti¹,

Ricciotti²

2009

Lecture Notes in Computer Science

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Abstract.We discuss the formalization, in the Matita Interactive Theorem Prover, of a famous result by Chebyshev concerning the distribution of prime numbers, essentially subsuming, as a corollary, Bertrand's postulate. Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. Differently from other recent formalizations of other results in number theory, our proof is entirely arithmetical. It makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, factorization, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base.

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“…Proofs of significant mathematical theorems like the Jordan Curve Theorem or the Prime Number Theorem have been formalized and formally checked in powerful systems like HOL light [10] [1]. Some large scale formalization projects related to current research mathematics are under way [11].…”

Section: Introductionmentioning

confidence: 99%

Parsing and Disambiguation of Symbolic Mathematics in the Naproche System

Cramer

Koepke

Schröder

2011

Lecture Notes in Computer Science

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Abstract. The Naproche system is a system for linguistically analysing and proof-checking mathematical texts written in a controlled natural language. The aim is to have an input language that is as close as possible to the language that mathematicians actually use when writing textbooks or papers. Mathematical texts consist of a combination of natural language and symbolic mathematics, with symbolic mathematics obeying its own syntactic rules. We discuss the difficulties that a program for parsing and disambiguating symbolic mathematics must face and present how these difficulties have been tackled in the Naproche system. One of these difficulties is the fact that information provided in the preceding contextincluding information provided in natural language -can influence the way a symbolic expression has to be disambiguated.

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A formally verified proof of the prime number theorem

Cited by 53 publications

References 12 publications

Social processes, program verification and all that

Social processes, program verification and all that

About the Formalization of Some Results by Chebyshev in Number Theory

Parsing and Disambiguation of Symbolic Mathematics in the Naproche System

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