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.

中文翻译：

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HFLN 循环证明系统

循环证明系统允许我们在没有明确归纳的情况下执行归纳推理。我们为 HFLN 提出了一个循环证明系统，这是一个具有自然数和交替定点的高阶谓词逻辑。据我们所知，我们是第一个用于高阶逻辑的循环证明系统。由于高阶谓词和交替不动点的存在，我们的循环证明系统需要比 Brotherston 和 Simpson 的原始系统更精细的循环证明全局条件。我们证明了检查该系统的全局条件和健全性的可判定性，并证明了我们循环证明系统的无限变体的标准完备性的受限形式。我们的循环证明系统的一个潜在应用是基于 Kobayashi 等人的高阶程序的半自动验证。