A common and very convenient tool for proving termination of a reduction system consists in defining an interpretation function I from the set of terms in question into the set of natural numbers

Let T be Gödel’s system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term A reduces to a term B and if a natural number I(A) is assigned to A then a natural number I(B) can be assigned to B such that I(A) is greater than I(B). The construction of I is based on Howard’s 1970 ordinal assignment for T and Weiermann’s 1998 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation lengths classification for the lambda formulation of T and its fragments. Compared with Weiermann’s 1998 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems.