Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-Löf dependent type theory extended with a program for Spector double negation shift.

• 03B40 - Combinatory logic and lambda calculus
• 68N18 - Functional programming and lambda calculus
• 68N30 - Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
• 68Q55 - Semantics in the theory of computing