We show that reducing any simply-typed $\lambda\sigma$-term by applying the rules in $\sigma$ eagerly always terminates, by a translation to the simply-typed $\lambda$-calculus, and similarly for $\lambda\sigma_\lift$-terms with $\sigma_\lift$-eager rewrites. This holds even with term and substitution meta-variables. In fact, every reduction terminates provided that $(\beta)$-redexes are only contracted under so-called safe contexts. The previous results follow because in $\sigma$, resp. $\sigma_{\lift}$-normal forms, all contexts around terms of sort $T$ are safe.