A Modular Formalisation of Finite Group Theory

Gonthier, Georges; Mahboubi, Assia; Rideau, Laurence; Tassi, E.; Théry, Laurent

doi:10.1007/978-3-540-74591-4_8

Cited by 40 publications

(51 citation statements)

References 9 publications

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“…One of the key results, the Feit-Thomson (or odd-order) theorem (Feit and Thompson, 1963) takes 255 pages to itself. A formalization of the odd-order theorem has recently been started by the INRIA-Microsoft research project on "mathematical components" lead by G. Gonthier (Gonthier, Mahboubi, Rideau, Tassi and Thery, 2007). As another example, the preprint of F. Almgren's masterpiece in geometric measure theory, familiarly referred to as the "Big Paper", is 1728 pages long.…”

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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“…For our purposes, three classes of types are of particular importance. These are discrete types, countable types, and finite types [16].…”

Section: Type Theory Preliminariesmentioning

confidence: 99%

Regular Language Representations in the Constructive Type Theory of Coq

Doczkal¹,

Smolka²

2018

J Autom Reasoning

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We explore the theory of regular language representations in the constructive type theory of Coq. We cover various forms of automata (deterministic, nondeterministic, oneway, two-way), regular expressions, and the logic WS1S. We give translations between all representations, show decidability results, and provide operations for various closure properties. Our results include a constructive decidability proof for the logic WS1S, a constructive refinement of the Myhill-Nerode characterization of regularity, and translations from twoway automata to one-way automata with verified upper bounds for the increase in size. All results are verified with an accompanying Coq development of about 3000 lines.

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“…We integrated Théry's formalization of elliptic curves [30] in our framework, and showed that the set of points of the elliptic curve E a,b can be construed as a finite cyclic group, as defined in SSREFLECT standard library [19]; 2. We defined Icart function, and showed that it generates points in the curve E a,b .…”

Section: Application To Elliptic Curvesmentioning

confidence: 99%

“…The proof involves the various notions of equivalence we develop in this paper and is thus an excellent testbed for evaluating the applicability of our methods. Additionally, the proof builds on several large developments (including Théry's formalization of elliptic curves [30] and Gonthier et al formalization of finite groups [19]) and demonstrates that CertiCrypt blends well with large and complex mathematical libraries, and is apt to support proofs involving advanced algebraic and number-theoretical reasoning.…”

Section: Introductionmentioning

confidence: 95%

Verified indifferentiable hashing into elliptic curves

Barthe

Grégoire

Heraud

et al. 2013

JCS

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Abstract. Many cryptographic systems based on elliptic curves are proven secure in the Random Oracle Model, assuming there exist probabilistic functions that map elements in some domain (e.g. bitstrings) onto uniformly and independently distributed points in a curve. When implementing such systems, and in order for the proof to carry over to the implementation, those mappings must be instantiated with concrete constructions whose behavior does not deviate significantly from random oracles. In contrast to other approaches to public-key cryptography, where candidates to instantiate random oracles have been known for some time, the first generic construction for hashing into ordinary elliptic curves indifferentiable from a random oracle was put forward only recently by Brier et al. We present a machine-checked proof of this construction. The proof is based on an extension of the CertiCrypt framework with logics and mechanized tools for reasoning about approximate forms of observational equivalence, and integrates mathematical libraries of group theory and elliptic curves.

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A Modular Formalisation of Finite Group Theory

Cited by 40 publications

References 9 publications

Social processes, program verification and all that

Social processes, program verification and all that

Regular Language Representations in the Constructive Type Theory of Coq

Verified indifferentiable hashing into elliptic curves

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