In this paper we study induction in the context of the first-order µ-calculus with explicit approximations. We present and compare two Gentzen-style proof systems each using a different type of induction. The first is

We investigate a Gentzen-style proof system for the first-order µ-calculus based on cyclic proofs, produced by unfolding fixed point formulas and detecting repeated proof goals. Our system uses explicit ordinal variables and approximations to support a simple semantic induction discharge condition which ensures the well-foundedness of inductive reasoning. As the main result of this paper we propose a new syntactic discharge condition based on traces and establish its equivalence with the semantical condition. We give an automata-theoretic reformulation of this condition which is more suitable for practical proofs. For a detailed