Formal Verification of Taylor’s Theorem in Coq

We have constructed a formalized system of unary differential calculus using the Coq tool. In this system, we formally defined the fundamental concepts of sequences, limits, functions, and derivatives in differential calculus. Furthermore, we carried out formal verification of various properties and theorems related to function limits, Local Preserving of Sign, Intermediate Value theorem, Uniform Continuity theorem, and the four Arithmetic Operations of Derivatives. Based on these efforts, we successfully completed the formal proof of Taylor’s theorem. The system’s code can be rigorously executed and validated on a computer, ensuring the entire proof process is rigorous and reliable. This work shows great potential in advancing the formalization of mathematics in the future, assisting educators in their teaching efforts, guiding learners in their studies, and making a positive contribution to the field of education and teaching.