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A Formalization of the Laplace Transform and Its Inversion in Lean 4

This paper presents a Lean 4 formalization of the Laplace transform and its inversion via a Bromwich-type theorem, demonstrating its application to the harmonic oscillator while addressing key analytic and formalization challenges.

Original authors: Daniel Goldberg, Antoine Vinciguerra

Published 2026-08-10

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel Goldberg, Antoine Vinciguerra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the messy, chaotic motion of things changing over time—like a swinging pendulum, a vibrating guitar string, or a signal traveling through a wire—could be instantly translated into a clean, static algebra problem. This is the magic of the Laplace transform, a mathematical tool used by engineers and scientists for over a century. Think of it as a universal translator that converts a story written in the language of "time" (where things move, speed up, and slow down) into a story written in the language of "complex numbers" (where those same movements become simple multiplication and division).

Why does this matter? Because solving equations about how things change is often incredibly hard, like trying to untangle a knot while it's being pulled. But if you can translate that knot into a different language where it just looks like a straight line, you can solve it easily and then translate the answer back. This paper is about building a "proof-checker" for this translator. The authors didn't just write down the rules; they used a computer program called Lean 4 to mathematically prove, step-by-step, that the translator works exactly as promised, even for the trickiest parts of the process. They wanted to make sure that when we use these powerful tools to design bridges, circuits, or control systems, the underlying math is rock-solid and free from hidden errors.

The Digital Proof-Checker

Imagine you have a very strict, very literal robot friend who loves math but hates guessing. You tell it, "Here is a formula that turns a wiggly line into a smooth curve," and it asks, "Are you sure? What if the line wiggles too much? What if it goes on forever?" This paper is the result of two researchers, Daniel and Antoine, teaching that robot friend everything it needs to know about the Laplace transform.

They didn't just write a textbook; they built a complete, computer-verified library in Lean 4. This is a programming language designed specifically for writing mathematical proofs that a computer can check for errors. Their goal was to take the Laplace transform—a method that turns functions of time into functions of complex numbers—and prove that every single rule works, from the basic definitions to the complex "inversion" process that turns the answer back into time.

The Translator and the Magic Mirror

The Laplace transform is like a magical mirror. You put a function $f(t)$ (which describes something happening over time) into the mirror, and it reflects back a new function $(Lf)(s)$ (which describes the same thing in a "frequency" world).

The Forward Trip: The paper proves that if you have a function that behaves nicely (it doesn't explode to infinity too fast), the mirror works. They proved the rules for how to translate simple things like constants, powers of time, and even sine waves. For example, they showed that the mirror turns the derivative (the rate of change) into a simple multiplication by a number $s$ , minus a starting value. This is the "secret sauce" that makes solving differential equations so easy.
The Return Trip (Inversion): The real challenge is getting the answer back. How do you look at the reflection and know exactly what the original object was? This is called the inverse Laplace transform. The paper proves a specific method for doing this, known as the Bromwich formula.

Why They Didn't Take the "Easy" Route

Usually, mathematicians prove the inversion formula using a technique called complex contour integration. Imagine drawing a loop around a shape on a map and using a special theorem (the Residue Theorem) to count the "treasures" inside. It's a powerful tool, but the authors found that the computer's library didn't have enough of these "map-drawing" tools built in yet.

So, they took a different, more ground-level path. Instead of drawing loops in the complex plane, they treated the problem as a real-world integration over a straight line. They broke the problem down into smaller, manageable pieces:

Truncation: They pretended the infinite line was just a short, finite segment from $-T$ to $T$ .
The Sinc Function: As they made this segment longer and longer, a specific pattern emerged involving a function called sinc (which looks like a wave that gets smaller and smaller).
The Dirichlet Integral: They relied on a famous, pre-proven fact about the area under this sinc wave (the Dirichlet integral) to show that as the segment gets infinitely long, the result perfectly reconstructs the original function.

This approach was harder to set up but safer for the computer to verify because it relied on real-number calculus, which the computer was already very good at.

The Swinging Pendulum Test

To prove their system actually works, they didn't just check abstract math; they solved a classic physics problem: the harmonic oscillator. This is the math behind a swinging pendulum or a spring bouncing up and down.

The Setup: They defined a spring that starts at rest but is given a quick push, described by the equation $y''(t) + \omega^2 y(t) = 0$ .
The Translation: They fed this equation into their computer-verified Laplace translator.
The Result: The computer successfully converted the messy differential equation into a simple algebraic equation: $(s^2 + \omega^2)Y(s) = \omega$ .
The Solution: Solving for $Y(s)$ gave them $\frac{\omega}{s^2 + \omega^2}$ .
The Verification: The computer then checked its own library and confirmed that this specific result is exactly the Laplace transform of $\sin(\omega t)$ .

This was a huge success. It meant the computer didn't just calculate the answer; it proved that the answer is indeed a sine wave, matching what human physicists have known for centuries, but with a level of certainty that leaves no room for human error.

The Strict Rules of the Game

The paper is also a lesson in how careful you have to be when you stop guessing and start proving. The authors highlight several "gotchas" that often get swept under the rug in textbooks:

Infinity is Tricky: You can't just assume an integral goes to infinity. The proof had to explicitly state that the function must decay fast enough so that the "tail" of the integral vanishes.
The Edge Cases: When doing the math, there are specific points (like time $t=0$ ) where the rules change. The computer forced them to be precise about exactly where the function is defined and continuous.
Swapping Order: In the inversion proof, they had to swap the order of two integrals. In casual math, you might just do this. In their formal proof, they had to rigorously prove that the "area" under the combined surface was finite before they were allowed to swap the order.

The Bottom Line

This paper is a milestone in formal verification. It doesn't discover a new law of physics or invent a new type of wave. Instead, it builds a fortress of certainty around a tool that is already widely used. By translating the Laplace transform and its inversion into a language a computer can check, the authors have created a reference standard.

They proved that the "magic mirror" works, provided you follow the strict rules about how the function behaves at infinity and at the start. They showed that the path to the answer involves careful, step-by-step logic rather than shortcuts. For anyone building the next generation of software that relies on these mathematical tools, this work ensures that the foundation is not just strong, but unbreakable. The harmonic oscillator example serves as the final seal of approval: the computer agrees with the human, and for the first time, the computer has signed off on the proof.

Technical Summary: A Formalization of the Laplace Transform and Its Inversion in Lean 4

Problem Statement
The paper addresses the formalization of the Laplace transform and its inversion within the Lean 4 theorem prover, specifically leveraging the mathlib library. While the Laplace transform is a standard tool in applied mathematics for converting linear differential equations into algebraic ones, its rigorous formalization presents significant challenges. These include handling complex-valued functions, managing improper integrals (limits of truncated integrals versus Lebesgue integrals), and proving the inversion theorem without relying on complex contour integration and residue theorems, which are not yet sufficiently developed in the current Lean 4 library.

Methodology
The authors adopt a measure-theoretic approach grounded in the Bochner integral framework available in mathlib. The development proceeds through several key methodological choices:

Abstract Kernel Construction: The transform is first defined in a general setting over a complete normed algebra $E$ using a kernel $e^{-L(e,p)}$ . This abstract layer isolates the algebraic properties of the exponential kernel from specific measure-theoretic choices.
Specialization to Real-Valued Functions: The construction is specialized to complex-valued functions on the real line. The authors utilize two representations:
- A mapped-measure approach, pushing forward the Lebesgue measure on $[0, \infty)$ to the real axis within $\mathbb{C}$ , used primarily for the inversion theorem.
- A direct set-integral approach over $(0, \infty)$ , used for operational rules and the harmonic oscillator example.
Handling Improper Integrals: The formalization explicitly treats the Laplace transform as a limit of truncated integrals (over $(0, T]$ ) rather than a direct improper integral notation. This distinction is crucial for applying the Dominated Convergence Theorem correctly within the Lean framework.
Real-Variable Inversion: Instead of using the standard complex-analytic proof involving contour deformation and the Residue Theorem, the authors prove the inversion theorem using real-variable integration. By parameterizing the Bromwich contour as $s = \gamma + ir$ $s = γ + i r$ , the vertical line integral is converted into an integral over $r \in \mathbb{R}$ $r \in R$ . The proof then relies on:
- Fubini's Theorem to exchange the order of integration.
- The Dirichlet integral (formalized in a separate module [9]) to handle the oscillatory kernel $\frac{\sin(T(t-a))}{t-a}$ .
- Integration by parts and properties of the Dirichlet integral's primitive to show pointwise convergence to the original function.

Key Contributions
The paper presents four principal contributions to the formal mathematics community:

Generalized Transform Formalization: A definition of the Laplace kernel and transform on a complete normed algebra, including proofs of scalar and additive laws.
Operational Rules and Convergence: A formalization of the transform as a Bochner integral over the positive half-line. The authors prove convergence of truncated transforms and establish closed-form formulas for constants, powers, exponentials, derivatives (including iterated derivatives), trigonometric functions (sine and cosine), and scaling. These proofs are conducted under explicit integrability and decay assumptions.
Bromwich-Type Inversion Theorem: A formal proof that the inverse Laplace transform recovers a differentiable function $f(t)$ at every point in a prescribed set $S \subset (0, \infty)$ . This proof utilizes the formalized Dirichlet integral and avoids complex contour integration, relying instead on real parameter integrals and dominated convergence.
Harmonic Oscillator Application: A complete formal derivation of the Laplace-domain solution for the harmonic oscillator equation $y''(t) + \omega^2 y(t) = 0$ with initial conditions $y(0)=0, y'(0)=\omega$ . The theorem identifies the transform of the solution with $\frac{\omega}{s^2 + \omega^2}$ , explicitly verifying all analytic hypotheses (regularity, integrability, decay) usually suppressed in textbook derivations.

Results
The formal development consists of approximately 2,900 lines of Lean code across four source files. Key results include:

Theorem RealLaplaceTransformIs: Unfolds the abstract construction into the familiar integral form.
Theorem IsInverseLaplace: Establishes the pointwise convergence of the inverse transform to the original function $f(t)$ under conditions of continuity, differentiability, and integrability of $f(t)e^{-\gamma t}$ and its derivative.
Theorem harmonic_oscillator_laplace_eq: Proves that the Laplace transform of a function satisfying the harmonic oscillator equation equals the transform of $\sin(\omega t)$ .

Significance and Scope
The authors position this work as a foundational step in formalizing applied analysis. The significance lies in:

Bridging the Gap: Providing a working inversion theorem in Lean 4 without waiting for the development of a full complex residue theory.
Explicit Hypotheses: The formalization exposes analytic conditions (such as boundary decay and integrability of weighted derivatives) that are often implicit in classical treatments, ensuring rigorous correctness.
Modular Architecture: The separation of the abstract kernel, the real-axis specialization, and the application examples creates a reusable structure for future developments.

The paper concludes by noting that while convolution and other operational rules are natural next steps, they require careful treatment and are not covered in this manuscript. The work is presented as a specific formalization of the forward transform, the inversion theorem, and a canonical application, rather than a comprehensive library of all Laplace transform properties.