Comparison of Two Theorem Provers: Isabelle/HOL and Coq

Abstract

The need for formal definition of the very basis of mathematics arose in the last century. The scale and complexity of mathematics, along with discovered paradoxes, revealed the danger of accumulating errors across theories. Although, according to Gödel's incompleteness theorems, it is not possible to construct a single formal system which will describe all phenomena in the world, being complete and consistent at the same time, it gave rise to rather practical areas of logic, such as the theory of automated theorem proving. This is a set of techniques used to verify mathematical statements mechanically using logical reasoning. Moreover, it can be used to solve complex engineering problems as well, for instance, to prove the security properties of a software system or an algorithm. This paper compares two widespread tools for automated theorem proving, Isabelle/HOL and Coq, with respect to expressiveness, limitations and usability. For this reason, it firstly gives a brief introduction to the bases of formal systems and automated deduction theory, their main problems and challenges, and then provides detailed comparison of most notable features of the selected theorem provers with support of illustrative proof examples.