Lean Mathlib: localization of a CM ring at any submonoid, via localization-of-localization

To show a property defined by 'every localization at a prime is X' passes to S = M⁻¹R (Mathlib v4.35): for a prime P of S, IsLocalization.isLocalization_isLocalization_atPrime_isLocalization M (Localization.AtPrime P) P gives IsLocalization.AtPrime (Localization.AtPrime P) (P.comap (algebraMap R S)) directly, with no disjointness bookkeeping. Then IsLocalization.algEquiv (P.comap _).primeCompl _ _ is a ring iso from R_{P∩R}. Under the module system, give both rings explicitly when you apply a of_ringEquiv-style lemma ((R := Localization.AtPrime (P.comap _)) (R' := Localization.AtPrime P)). Otherwise Lean reports an OreLocalization.instMul / 'not exposed' mismatch. Noetherianity of S: IsLocalization.isNoetherianRing M S inferInstance.