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We present a novel compiled approach to Normalisation by Evaluation (NBE) for ML-like languages. It supports efficient normalisation of open λ-terms with respect to β-reduction and rewrite rules. We have implemented NBE and show both a detailed formal model of our implementation and its verification in Isabelle. Finally we discuss how NBE is turned into a proof rule in Isabelle.
A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way. that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae increases the strength by one inductive definition.
A purely syntactic and untyped variant of Normalisation by Evaluation for the $\lambda$-calculus is presented in the framework of a two-level $\lambda$-calculus with rewrite rules to model the inverse of the evaluation functional. Among its operational properties there is a standardisation theorem that formally establishes the adequacy of implementation in functional programming languages. An example implementation in Haskell is provided. The relation to the usual type-directed Normalisation by Evaluation is highlighted, using a short analysis of $\eta$-expansion that leads to a perspicuous strong normalisation and confluence proof for $\beta\eta\!\up$-reduction as a byproduct.
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