Lagrange’s Theorem in Group Theory: Formalization and Proof with Coq

Machine-proof of mathematical theorems is a key component of the foundational theory of artificial intelligence. Lagrange’s Theorem in group theory, which reveals the crucial relationship between a finite group and its subgroups, plays a significant role in understanding group structures and properties. Utilizing the interactive theorem proving tool Coq, this paper formalizes essential concepts including mappings, surjections, injections, bijections, groups, subgroups, order, cosets, quotient sets, and coset operations, ultimately achieving the formal proof of Lagrange’s Theorem. This research establishes a rigorous framework for theorem proving in modern algebraic theory using Coq, enhancing our understanding of group theory’s fundamentals and demonstrating the effective integration of mathematical theory with advanced computational techniques.