  A cyclic proof system, called CLKID-omega, gives us another way of
representing inductive definitions and efficient proof search. The 2005 paper
by Brotherston showed that the provability of CLKID-omega includes the
provability of LKID, first order classical logic with inductive definitions in
Martin-L\"of's style, and conjectured the equivalence. The equivalence has been
left an open question since 2011. 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.
