Adequacy for untyped translations of typed lambda -calculi

PCF is a simply typed lambda -calculus with ground types iota (natural numbers) and omicron (Booleans); there are no type variables and implies is the only type constructor. There is a natural way to translate any PCF term t into an untyped lambda -expression Lambda (t), such that if t is a program, i.e. a closed term of ground type (say integer type) and t implies /sub N/ n then Lambda (t) implies /sub beta / c/sub n/, where implies /sub N/ denotes call-by-name evaluation and c/sub n/ denotes the nth Church numeral. This paper contains a proof of the converse: if Lambda (t) implies /sub beta / c/sub n/ then t implies /sub N/ n; this tells us that the translation is adequate. The proof is semantic, and uses synthetic domain theory to reduce the question to the original Plotkin/Sazonov adequacy theorem for standard domain models of call-by-name PCF. This argument generalises easily to extensions of PCF which can be translated into the untyped lambda -calculus: we illustrate this by proving an analogous result for a ´second-order´ PCF with type quantification. We also discuss how to extend the result to versions of PCF with recursive types and subtyping.