Formalizing equivalence between real number completeness and intermediate value theorem

Abstract

This paper reports on the formalization of the completeness of intermediate value theorem. This theorem as a fundamental property of continuous function on a closed interval, can be used to solve the existence of roots of equations. The intermediate value theorem is usually regarded as a corollary of completeness theorem of real number. In fact, completeness theorem of real number also can be derived from the intermediate value theorem. This formalization is based on the formal system we developed strictly following Landau’s “Foundations of Analysis”. Dedekind fundament theorem has been proved in this system, and further we prove the supremum theorem. Then we formally verify the equivalence between supremum theorem and intermediate value theorem, which shows its completeness. All the proof has been checked by Coq, and the proof process is normalized, rigorous and reliable.