Let $\rho$ be an ordinal different from $0$.
Show that the two classes of ordinals are equivalent:
Ordinals $\alpha < \omega^{\rho}$ such that $\alpha = \omega^{\xi_0} + \ldots +\omega^{\xi_{p-1}}$ where $\rho > \xi_0 > \ldots > \xi_{p-1}$.
Ordinal $\alpha < \omega^{\rho}$ such that $\alpha = 0$ or there do not exist $\beta,\gamma,\delta < \omega^{\rho}$ such that $\alpha = \beta + \gamma + \gamma + \delta$ and $\gamma+\delta \neq \delta$.
I'm not very familiar with ordinal arithmetic. What kind of information could I use to prove this characterization?