Generalized beta-Reduction and Explicit Substitution

Extending the λ-calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus λsg and shows that it is a desirable extension of the λ-calculus. In particular, we show that λsg preserves strong normalisation, is sound and it simulates classical β-reduction. Furthermore, we study the simply typed λ-calculus extended with both generalised reduction and explicit substitution and show that well-typed terms are strongly normalising and that other properties such as subtyping and subject reduction hold.

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