Three Chapters of Measure Theory in Isabelle/HOL

Hölzl, Johannes; Heller, Armin

doi:10.1007/978-3-642-22863-6_12

Cited by 63 publications

(46 citation statements)

References 9 publications

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“…Obviously some amount of probability theory. Different fragments of probability theory are now formalized in many theorem provers, including HOL4, HOL-light, PVS, Mizar and Isabelle [12,17,18,21,23]. Surprisingly, for the proof presented here, not much more than Markov's Inequality is required.…”

Section: Discussionmentioning

confidence: 99%

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Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

Noschinski

2012

Interactive Theorem Proving

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Section: Discussionmentioning

confidence: 99%

“…This construction is supported by Isabelle's extensive library on probability theory [17]. However, the elements of the product space of probability spaces are functions 2 2 N → {0, 1} which are only specified on E n .…”

Section: Probability Spacementioning

confidence: 99%

Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

Noschinski

2012

Interactive Theorem Proving

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“…Various higher-order-logic formalizations of probability theory can be found in the literature, e.g., [19,20,21]. The formalizations by Mhamdi [20] and Hölzl [21] are based on extended real numbers (including ±∞) and also include the formalization of Lebesgue integral for reasoning about statistical properties.…”

Section: Related Workmentioning

confidence: 99%

“…The formalizations by Mhamdi [20] and Hölzl [21] are based on extended real numbers (including ±∞) and also include the formalization of Lebesgue integral for reasoning about statistical properties. This way, they are more mature than Hurd's [19] formalization of measure and probability theories, which is based on simple real numbers and offers a limited support for reasoning about statistical properties [22].…”

Section: Related Workmentioning

confidence: 99%

Using Probabilistic Analysis for the Certification of Machine Control Systems

Mashkoor

Hasan

Beer

2013

Security Engineering and Intelligence Informatics

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Abstract. Traditional testing techniques often reach their limits when employed for the assessment of critical Machine Control Systems as they contain a large amount of random and unpredictable components. The probabilistic analysis approach can assist in their evaluation by providing a subjective evidence of their safety and reliability. The synergy of probabilistic analysis and expressiveness of higher-order logic theorem proving results into convincing modelling and reasoning of several stringent safety cases that contribute towards the certification of high-assurance systems.

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“…Let us define the complete lattice Rbar = R ∪ {−∞, +∞}, inspired by [14]. Since we are able to decide whether a set is bounded, we can define supremum and infimum functions for nonempty subset of Rbar.…”

Section: Bounds and Limitsmentioning

confidence: 99%

Improving Real Analysis in Coq: A User-Friendly Approach to Integrals and Derivatives

Boldo

Lelay

Melquiond

2012

Certified Programs and Proofs

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Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, which may be used for this purpose. Unfortunately, its definitions of derivative and integral are unpractical as they are partial functions that demand a proof term. This proof term makes the handling of mathematical formulas cumbersome and does not conform to traditional analysis. Other proof assistants usually do not suffer from this issue; for instance, they may rely on Hilbert's epsilon to get total operators. In this paper, we propose a way to define total operators for derivative and integral without having to extend Coq's standard axiomatization of real numbers. We proved the compatibility of our definitions with the standard library's in order to leverage existing results. We also greatly improved automation for real analysis proofs that use Coq standard definitions. We exercised our approach on lemmas involving iterated partial derivatives and differentiation under the integral sign, that were missing from the formal proof of a numerical program solving the wave equation.

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Three Chapters of Measure Theory in Isabelle/HOL

Cited by 63 publications

References 9 publications

Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

Using Probabilistic Analysis for the Certification of Machine Control Systems

Improving Real Analysis in Coq: A User-Friendly Approach to Integrals and Derivatives

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