🤖 AI Summary
This work addresses the absence of a comprehensive machine-verified foundation for the extended complex plane, Möbius transformations, and their fundamental invariant—the cross-ratio—in existing formal mathematics. Building upon the Mathlib library in Lean 4, the authors construct the extended complex plane using the Option type and present the first formalization of the group structure of Möbius transformations, the uniqueness of such transformations determined by three points, and the invariance of the cross-ratio. The development, grounded in dependent type theory, comprises approximately 6,000 lines of code, including 40 definitions and 150 theorems, thereby establishing the first formally verified basis for conformal geometry, hyperbolic models, and mathematical physics.
📝 Abstract
The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.