Strong Normalisation for Applied Lambda Calculi

We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting ⊥ is strongly normalising provided all its ‘stratified approximations’ are. From this we derive a general normalisation theorem for applied typed λ-calculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of Gödel’s system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown.