Famously, Rosser introduced a provability predicate $\pi[A]$ that holds iff $\exists x(xP[A]\wedge\forall y(y\le x\to\lnot P[\lnot A])$.

Supposing $PA$ is consistent, what are the adequacy conditions for $\pi$ as compared with the Hilbert-Bernays-Löb derivability conditions for P?