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Formally Verified Approximations of Definite Integrals
2018
Journal of automated reasoning
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This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. ...
Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. ...
Formally Verified Approximations of Definite Integrals ...
doi:10.1007/s10817-018-9463-7
fatcat:g3jumo65prehlovv7mwnoqjsse
Formally Verified Approximations of Definite Integrals
[chapter]
2016
Lecture Notes in Computer Science
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Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. ...
This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. ...
Conclusion We have presented a method for computing and formally verifying numerical enclosures of univariate definite integrals using the Coq proof assistant. ...
doi:10.1007/978-3-319-43144-4_17
fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa
On the formal analysis of Gaussian optical systems in HOL
2016
Formal Aspects of Computing
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To demonstrate the usefulness of the reported formalization, we utilize it to formally analyze some fractional order systems, namely, a fractional electrical component Resistoductance, a fractional integrator ...
Fractional order systems, which involve integration and differentiation of non integer order, are increasingly being used in the fields of control systems, robotics, signal processing and circuit theory ...
We verified the continuity by differentiability using the classical definition of derivative formalized in HOL [15] . ...
doi:10.1007/s00165-016-0367-1
fatcat:env7aa6ivzbojly4pr4rwcgj2e
Formalization of Transform Methods Using HOL Light
[chapter]
2017
Lecture Notes in Computer Science
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In particular, we present the motivation of the formalization, which is followed by the related work. ...
Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. ...
Acknowledgements This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan. ...
doi:10.1007/978-3-319-62075-6_22
fatcat:nh6h44hfkbc5dh6kk7aaz4cf4y
Formalization of Transform Methods using HOL Light
[article]
2017
arXiv
pre-print
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In particular, we present the motivation of the formalization, which is followed by the related work. ...
Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. ...
Acknowledgements This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan. ...
arXiv:1705.10050v1
fatcat:wpigu4et6bgp5m3m7xmtqicuxm
On the Formalization of Gamma Function in HOL
2014
Journal of automated reasoning
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Firstly, we extend the existing integration theory of HOL4 by formalizing a variant of improper integrals using sequential limits. ...
In order to illustrate the practical effectiveness and utilization of our work, we formally verify some properties of Euler's generalized power rule of differentiation, Mittag-Leffler functions and the ...
The formalization, presented in this section, took around 5500 lines of HOL code and approximately 300 man-hours. ...
doi:10.1007/s10817-014-9311-3
fatcat:utrfokmtdzgwhk3fg7x5jnmzka
Formal verification of the heavy hitter problem
2012
2012 25th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE)
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This model is then utilized to formally verify some interesting probabilistic and statistical properties associated with the heavy hitter problem in HOL. ...
The heavy hitter problem is used to assess the frequency of occurrence of an element in a given data stream. ...
However, Coble's formalization of Lebesgue integral can only consider finite-valued measures, functions and integrals. ...
doi:10.1109/ccece.2012.6335035
dblp:conf/ccece/HelaliTH12
fatcat:cl4fa2ay3fculaupr7m4pjwrou
Towards the Formalization of Fractional Calculus in Higher-Order Logic
[article]
2015
arXiv
pre-print
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Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). ...
Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones. ...
But now the current formalization seems quite stable as most of the classical properties have been formally verified for our definitions. ...
arXiv:1505.02140v1
fatcat:hvyf3xhf2nbh7bj4gijiav7rai
Formal Reasoning about Expectation Properties for Continuous Random Variables
[chapter]
2009
Lecture Notes in Computer Science
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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 ...
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. ...
Besides the formalization of the above definitions, many useful properties regarding the Lebesgue integral have also been verified in [4] as higher-orderlogic theorems. ...
doi:10.1007/978-3-642-05089-3_28
fatcat:4ppnedhwjbf3pp3b7jsc6moski
Towards the Formalization of Fractional Calculus in Higher-Order Logic
[chapter]
2015
Lecture Notes in Computer Science
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Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). ...
Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones. ...
But now the current formalization seems quite stable as most of the classical properties have been formally verified for our definitions. ...
doi:10.1007/978-3-319-20615-8_21
fatcat:lgx5adcufvfenedqbkpuuo4wnu
A Certificate-Based Approach to Formally Verified Approximations
2019
International Conference on Interactive Theorem Proving
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Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations. ...
We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant. ...
I T P 2 0 1 9
8:16 A Certificate-Based Approach to Formally Verified Approximations Error bounding. ...
doi:10.4230/lipics.itp.2019.8
dblp:conf/itp/BrehardMP19
fatcat:b2twcbftqrdihpid6s3fxlw4ha
Formalization of Lerch's Theorem using HOL Light
[article]
2018
arXiv
pre-print
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In this paper, we extend our higher-order logic formalization of the Laplace transform, which includes the formal definition of the Laplace transform and verification of its various classical properties ...
For illustration, we present the formal analysis of a 4-π soft error crosstalk model, which is widely used in nanometer technologies, such as, Integrated Circuits (ICs). ...
We formally verified these using our new definition of the Laplace transform [27] . ...
arXiv:1806.03049v1
fatcat:dxqnz5t2pvhvzaueddrd2hxq7a
Formalization of Laplace Transform Using the Multivariable Calculus Theory of HOL-Light
[chapter]
2013
Lecture Notes in Computer Science
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In order to demonstrate the practical effectiveness of this formalization, we use it to formally verify the transfer function of Linear Transfer Converter (LTC) circuit, which is a commonly used electrical ...
To facilitate formal analysis of these systems, we present the formalization of Laplace transform using the multivariable calculus theories of HOL-Light. ...
This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan. ...
doi:10.1007/978-3-642-45221-5_50
fatcat:fks2jihksrfalktlbopv5weapy
One-parameter extension of the Doi-Peliti formalism and its relation with orthogonal polynomials
2012
Physical Review E
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An extension of the Doi-Peliti formalism for stochastic chemical kinetics is proposed. Using the extension, path-integral expressions consistent with previous studies are obtained. ...
In addition, the extended formalism is naturally connected to orthogonal polynomials. ...
Here, we briefly explain some basic definitions for a coherent-state path-integral expression, which are especially useful in the Doi-Peliti formalism. ...
doi:10.1103/physreve.86.042102
pmid:23214630
fatcat:4d5ytjszkjhe3dvkzij7bsuewq
Formal Probabilistic Analysis: A Higher-Order Logic Based Approach
[chapter]
2010
Lecture Notes in Computer Science
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Some major contributions include the formalization of both discrete and continuous random variables and the verification of some of their corresponding probabilistic and statistical properties. ...
Recently, a significant amount of formalization has been done in higherorder logic that allows us to conduct precise probabilistic analysis using theorem proving and thus overcome the limitations of the ...
In order to verify the correctness of the above definitions of expectation, they are utilized in [20, 21] to formally verify the following classical expectation properties. ...
doi:10.1007/978-3-642-11811-1_2
fatcat:hhk4xlpnevflrf2w3ijc3hweni
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