Weak and Strong Beta Normalisations in Typed -calculi

We present a technique to study relations between weak and strong-normalisations in various typed-calculi. We rst introduce a translation which translates a-term into a I-term, and show that a-term is strongly-normalisable if and only if its translation is weakly-normalisable. We then prove that the translation preserves typability of-terms in various typed-calculi. This enables us to establish the equivalence between weak and strong-normalisations in these typed-calculi. This translation can deal with Curry typing as well as Church typing, strengthening some recent closely related results. This may bring some insights into answering whether weak and strong-normalisations in all pure type systems are equivalent.