Cyclic proofs for the first-order µ-calculus - Afshari, Bahareh; Enqvist, Sebastian; Leigh, Graham E (Logic journal of the IGPL, 2024)
Abstract We introduce a path-based cyclic proof system for first-order $μ $-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 $μ $-calculus.