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                <term xml:lang="ro">LAMBDA-SIGMA-CALCULUS</term>
                <term xml:lang="ro">EXPLICIT SUBSTITUTIONS</term>
                <term xml:lang="ro">TERMINATION</term>
                <term xml:lang="ro">LAMBDA-CALCULUS</term>
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              <p>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.</p>
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