Characterisation of Normalisation Properties for λμ using Strict Negated Intersection Types

Abstract

We show characterisation results for normalisation, head-normalisation, and strong normalisation for λ μ using intersection types. We reach these results for a strict notion of type assignment for λ μ that is the natural restriction of the domain-based system of van Bakel et al. (2011) for λ μ by limiting the type inclusion relation to just intersection elimination. We show that this system respects β μ-equality, by showing both soundness and completeness results. We then define a notion of reduction on derivations that corresponds to cut-elimination, and show that this is strongly normalisable. We use this strong normalisation result to show an approximation result, and through that a characterisation of head-normalisation. Using the approximation result, we show that there is a very strong relation between the system of van Bakel et al. (2011) and ours. We then introduce a notion of type assignment that eliminates ω as an assignable type, and show, using the strong normalisation result for derivation reduction, that all terms typeable in this system are strongly normalisable as well, and show that all strongly normalisable terms are typeable. We conclude by adding type variables to our system, and show that system essentially is that of van Bakel (2010b).

Adding Negation to Lambda Mu

“…We will write Γ ⊢ λµ M : A | ∆ for statements derivable in this system. (5) We extend Barendregt's convention on free and bound variables and names to judgements (for all the notions of type assignment we define here), so in Γ, x:A ⊢ λµ M : B | α:C, ∆, both x and α cannot appear bound in M.…”