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Formalization of Transfinite Induction in Coq*

Transfinite induction is a proof method of mathematics and has important applications in computer science. In this paper, a formal proof of transfinite induction is proposed based on Morse-Kelley set theory. It is the first Coq formalization about transfinite induction to our knowledge. Furthermore, according to the minimum principle, we formally prove mathematical induction which is a special form on non-negative integers of transfinite induction. The whole proofs in this paper are verified in Coq proof assistant. The Morse-Kelley set theory is also formalized in Coq. The work shows that the formal proof of mathematical theorems is credible and religious. The formal work in this paper has important applications in set theory, algebra, and topology. In addition, according to transfinite induction, we can formally prove the general well-founded structures in computer science.