Formalization of Lerch's Theorem using HOL Light

TL;DR

This paper formalizes Lerch's theorem within HOL Light to ensure accurate Laplace transform analysis, enabling precise solutions of differential equations in engineering systems, demonstrated through a nanotechnology application.

Contribution

It extends higher-order logic formalization by including Lerch's theorem, ensuring the uniqueness of Laplace transforms for accurate system analysis.

Findings

Formal definition and verification of Laplace transform properties

Formalization of Lerch's theorem within HOL Light

Application to a nanotechnology crosstalk model

Abstract

The Laplace transform is an algebraic method that is widely used for analyzing physical systems by either solving the differential equations modeling their dynamics or by evaluating their transfer function. The dynamics of the given system are firstly modeled using differential equations and then Laplace transform is applied to convert these differential equations to their equivalent algebraic equations. These equations can further be simplified to either obtain the transfer function of the system or to find out the solution of the differential equations in frequency domain. Next, the uniqueness of the Laplace transform provides the solution of these differential equations in the time domain. The traditional Laplace transform based analysis techniques, i.e., paper-and-pencil proofs and computer simulation methods are error-prone due to their inherent limitations and thus are not suitable for the analysis of the systems. Higher-order-logic theorem proving can overcome these limitations of these techniques and can ascertain accurate analysis of the systems. 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. One of the main contributions of the paper is the formalization of Lerch's theorem, which describes the uniqueness of the Laplace transform and thus plays a vital role in solving linear differential equations in the frequency domain. For illustration, we present the formal analysis of a $4$-$\pi$ soft error crosstalk model, which is widely used in nanometer technologies, such as, Integrated Circuits (ICs).