🤖 AI Summary
This work addresses the absence of a rigorous formalization of the Laplace transform and its inversion in existing interactive theorem provers. It presents the first complete formalization in Lean 4 of the Laplace transform for complex-valued functions, along with fundamental operational rules and a Bromwich-type inversion theorem grounded in real integrals and Dirichlet integrals. By integrating classical analysis with formal verification techniques, the framework is successfully applied to the harmonic oscillator problem. The development not only verifies core aspects of Laplace transform theory but also formally derives the solution and proves that its transform coincides with the standard transform of $\sin(\omega t)$. This demonstrates the feasibility and rigor of formalized mathematics in engineering analysis.
📝 Abstract
We present a Lean 4 formalization of the Laplace transform for complex-valued functions, its fundamental operational rules, and a Bromwich-type inversion theorem proved through real-variable integration and the Dirichlet integral. As an application, we formalize the Laplace-domain solution of the harmonic oscillator and identify its transform with that of $\sin(ωt)$. We also discuss the principal analytic and formalization challenges encountered in the development.