Cyclic and Inductive Calculi are equivalent

Brotherston and Simpson [citation] have formalized and investigated cyclic reasoning, reaching the important conclusion that it is at least as powerful as inductive reasoning (specifically, they showed that each inductive proof can be translated into a cyclic proof). We add to their investigation by proving the converse of this result, namely that each inductive proof can be translated into an inductive one. This, in effect, establishes the equivalence between first order cyclic and inductive calculi.