We introduce a path-based cyclic proof system for first-order $\mu $-calculus, the extension of first-order logic by second-order quantifiers for least and greatest fixed points of definable monotone functions. We prove soundness of the system and demonstrate it to be as expressive as the known trace-based cyclic systems of Dam and Sprenger. Furthermore, we establish cut-free completeness of our system for the fragment corresponding to the modal $\mu $-calculus.

中文翻译：

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一阶 µ 演算的循环证明

我们为一阶 $\mu $-演算引入了一个基于路径的循环证明系统，它是通过二阶量词对可定义单调函数的最小和最大不动点的一阶逻辑的扩展。我们证明了该系统的稳健性，并证明它与已知的基于轨迹的 Dam 和 Sprenger 循环系统一样具有表现力。此外，我们为与模态 $\mu $-calculus 对应的片段建立了系统的无切割完整性。