User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191031135205

Unfortunately, the new condition which you proposed does not work, because it rarely holds. For example, let us consider Wainer hierarchy. Let \(n\) be an arbitrary natural number. Then we have \begin{eqnarray*} \omega^{\omega} & > & \omega^n + \omega + \omega \\ \omega^{\omega}[0] = 1 & < & \omega^n + \omega = (\omega^n + \omega + \omega)[0] \\ \omega^{\omega}[1] = \omega & < & \omega^n + \omega + 1 = (\omega^n + \omega + \omega)[1] \\ \omega^{\omega}[2] = \omega^2 & < & \omega^n + \omega + 2 = (\omega^n + \omega + \omega)[2] \\ & \vdots & \\ \omega^{\omega}[n] = \omega^n & < & \omega^n + \omega + n = (\omega^n + \omega + \omega)[n], \end{eqnarray*} and hence even Wainer hierarchy does not satisfy the condition. We need more sensitive a condition.