  A cyclic proof system allows us to perform inductive reasoning without
explicit inductions. We propose a cyclic proof system for HFLN, which is a
higher-order predicate logic with natural numbers and alternating fixed-points.
Ours is the first cyclic proof system for a higher-order logic, to our
knowledge. Due to the presence of higher-order predicates and alternating
fixed-points, our cyclic proof system requires a more delicate global condition
on cyclic proofs than the original system of Brotherston and Simpson. We prove
the decidability of checking the global condition and soundness of this system,
and also prove a restricted form of standard completeness for an infinitary
variant of our cyclic proof system. A potential application of our cyclic proof
system is semi-automated verification of higher-order programs, based on
Kobayashi et al.'s recent work on reductions from program verification to HFLN
validity checking.
