Theorem Proving with the Real Numbers

Harrison, John

doi:10.1007/978-1-4471-1591-5

Cited by 140 publications

(128 citation statements)

References 91 publications

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“…But he did not include theorem lifting. Harrison's quotient package [5] is the first one that is able to automatically lift theorems, however only first-order theorems (that is theorems where abstractions, quantifiers and variables do not involve functions that include the quotient type). There is also some work on quotient types in non-HOL based systems and logical frameworks, including theory interpretations in PVS [8], new types in MetaPRL [7], and setoids in Coq [3].…”

Section: Conclusion and Related Workmentioning

confidence: 99%

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Quotients revisited for Isabelle/HOL

Kaliszyk

Urban

2011

Proceedings of the 2011 ACM Symposium on Applied Computing

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Higher-Order Logic (HOL) is based on a small logic kernel, whose only mechanism for extension is the introduction of safe definitions and of non-empty types. Both extensions are often performed in quotient constructions. To ease the work involved with such quotient constructions, we re-implemented in the Isabelle/HOL theorem prover the quotient package by Homeier. In doing so we extended his work in order to deal with compositions of quotients and also specified completely the procedure of lifting theorems from the raw level to the quotient level. The importance for theorem proving is that many formal verifications, in order to be feasible, require a convenient reasoning infrastructure for quotient constructions.

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Section: Conclusion and Related Workmentioning

confidence: 99%

“…In the context of HOL, there have been a few quotient packages already [5,10]. The most notable one is by Homeier [6] …”

Section: Introductionmentioning

confidence: 99%

Quotients revisited for Isabelle/HOL

Kaliszyk

Urban

2011

Proceedings of the 2011 ACM Symposium on Applied Computing

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“…This allows us to use the convergence of a positive measurable function to the Lebesgue integral property, given in Equation (10), to reason about Theorem 1. Using Modus Ponens (MP) rule, we can split the proof goal of Theorem 1 to the following five subgoals, corresponding to the monotonicity and positive simple-function requirement on f n and the three assumptions of Equation (10):…”

Section: Theorem 1 Expectation Of Bounded Random Variablesmentioning

confidence: 99%

“…The obvious advantage of using Equation (1) is the user familiarity with Reimann integral that facilitates the reasoning process regarding the expectation properties in the theorem proving based probabilistic analysis approach. On the other hand, it requires extended real numbers, R = R ∪ {−∞, +∞}, whereas all the foundational work regarding theorem proving based probabilistic analysis has been built upon the standard real numbers R, formalized by Harrison [10]. Thus, the formalization of the expectation definition, given in Equation (1), and making it compatible with the available formal probabilistic analysis infrastructure would require creating a new data type R, and re-verifying the already proven results in a theorem prover for this new data-type, which is a considerable amount of work.…”

Section: Introductionmentioning

confidence: 99%

“…The main motivation behind this choice is the fact that most of the work that we build upon is developed in HOL, such as the formalization of the real number theory [10], probability theory [15], continuous random variables [11] and Lebesgue integration [4]. Though, it is important to note here that the ideas presented in this paper are not specific to the HOL theorem prover and can be adapted to any other higher-order-logic theorem prover as well, such as Isabelle, Coq or PVS.…”

Section: Introductionmentioning

confidence: 99%

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Formal Reasoning about Expectation Properties for Continuous Random Variables

Hasan

Abbasi

Akbarpour

et al. 2009

FM 2009: Formal Methods

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Abstract. Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higher-order-logic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation of a continuous random variable in terms of simple arithmetic operations. In order to illustrate the practical effectiveness and utilization of our approach, we also present the formal verification of expectation properties of the commonly used continuous random variables: Uniform, Triangular and Exponential.

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Formal verification of tail distribution bounds in the HOL theorem prover

Hasan

Tahar

2008

Math Methods in App Sciences

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SUMMARYTail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher-order-logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli( p), Geometric( p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety-critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well-known commercially used algorithm.

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Theorem Proving with the Real Numbers

Cited by 140 publications

References 91 publications

Quotients revisited for Isabelle/HOL

Quotients revisited for Isabelle/HOL

Formal Reasoning about Expectation Properties for Continuous Random Variables

Formal verification of tail distribution bounds in the HOL theorem prover

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