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Yukihiro Oda, James Brotherston, Makoto Tatsuta, The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions, Journal of Logic and Computation, Volume 35, Issue 2, March 2025, exad068, https://doi.org/10.1093/logcom/exad068
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Abstract
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.
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