Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and effcient proof search. The 2011 paper by Brotherston and Simpson showed that the provability of CLKID-omega includes the provability of Martin-Lof's system of inductive definitions, called LKID, and conjectured the equivalence. Since then, the equivalence has been left an open question. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false.