In the setting of classical first-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. In~\cite{Brotherston:2005qy,Brotherston:2006uy}, it has been shown how to transform any structural proof developed with the LKID system into a cyclic proof using the CLKID$^\omega$ system. However, the inverse transformation was only conjectured. We provide a simple procedure that performs the inverse transformation for an extension of LKID with explicit induction rules issued from the structural analysis of CLKID$^{\omega}$ proofs, then establish the equivalence of the two systems. This result is further refined for an extension of LKID with Noetherian induction rules. We show that Noetherian induction subsumes the two kinds of reasoning. This opens the perspective for building new effective induction proof methods and validation techniques supported by (higher-order) certification systems integrating the Noetherian induction principle, like Coq.
