🤖 AI Summary
This work presents the first formalization of the multivariable Taylor’s theorem in synthetic differential geometry within a constructive mathematical framework, using the Lean proof assistant and its mathlib library. To meet the analytical demands of infinitesimal neighborhoods, the authors develop a novel proof technique compatible with constructive logic, successfully achieving a machine-verified formalization of the Taylor expansion. This contribution not only fills a notable gap in mathlib regarding synthetic differential geometry but also demonstrates the feasibility and potential of interactive theorem proving systems to support constructive infinitesimal analysis. The result underscores the capacity of modern proof assistants to handle sophisticated geometric and analytic concepts while adhering to the constraints of constructive mathematics.
📝 Abstract
This article is about the formalization of synthetic differential geometry with the Lean proof assistant and the mathematical library mathlib. The main result we prove and formalize is a Taylor theorem for functions of several variables, where the series expansion is around an infinitesimal neighborhood. Most of our proofs are in fact new. Our investigations highlight the possibility of using mathlib to do constructive mathematics.