The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions

A cyclic proof system is a proof system whose proof figure is a tree with cycles. The cut-elimination in a proof system is fundamental. It is conjectured that the cut-elimination in the cyclic proof system for first-order logic with inductive definitions does not hold. This paper shows that the conjecture is correct by giving a sequent not provable without the cut rule but provable in the cyclic proof system.

Automated summary

This paper investigates the cut-elimination in cyclic proof systems, focusing on the case of first-order logic with inductive definitions. By providing a specific example, it demonstrates that the use of cut rule is not possible in the cyclic proof system.