 | Formalization of Lerch's theorem using HOL Light
Adnan Rashid and Osman Hasan
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. Higherorder-
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- soft
error crosstalk model, which is widely used in nanometer technologies, such as,
Integrated Circuits (ICs).
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