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**1 - 2**of**2**### Explicit substitutions for the lambda Delta-calculus

"... . The \Delta-calculus is a -calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce \Deltaexp, an explicit substitution calculus for \Delta, and study its properties. In particular, we show that \Deltaexp preserves ..."

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. The \Delta-calculus is a -calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce \Deltaexp, an explicit substitution calculus for \Delta, and study its properties. In particular, we show that \Deltaexp preserves strong normalisation, which provides us with the first example --moreover a very natural one indeed-- of explicit substitution calculus which is not structure-preserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of \Deltaexp is strongly normalising. In addition, we show that Plotkin's call-by-name continuation-passing style translation may be extended to \Deltaexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions. 1 Introduction Explicit substitutions were introduced by Abadi, Cardelli, Curien and L...

### Explicit Substitutions for the λΔ-calculus

"... The λΔ-calculus is a λ-calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisa ..."

Abstract
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The λΔ-calculus is a λ-calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisation, which provides us with the first example -- moreover a very natural one indeed -- of explicit substitution calculus which is not structure-preserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of λΔexp is strongly normalising. In addition, we show that Plotkin's call-by-name continuation-passing style translation may be extended to λΔexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions.