A completeness result for a realisability semantics for an intersection type system. (English summary) Ann. Pure Appl. Logic 146 (2007), no. 2-3, 180–198. This paper refines Hindley’s completeness theorem [J. R. Hindley, Theoret. Comput. Sci. 22 (1983), no. 1-2, 1–17; MR0693047 (85e:03030a)] in various ways. One considers a type system with intersection and a universal type ω. For Hindley a type is interpreted as a set of λ-terms closed under β-conversion. Here one semantics involves weak-head reduction and another normal β-reduction. The authors define then a notion of positive types and show that if a term inhabits a positive type then it is β-normalisable and reduces to a closed term. Like Hindley, they prove also the soundness and completeness of their semantics. For positive types, there is equivalence between being a term of this type and inhabiting this type. They show also that this notion of positive type is in some sense maximal for this property
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