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Let T be Goedel's system of primitive recursive functionals of finite type inthe lambda formulation. We define by constructive means using recursion onnested multisets a multivalued function I from the set of terms of T into theset of natural numbers such that if a term a reduces to a term b and if anatural number I(a) is assigned to a then a natural number I(b) can be assignedto b such that I(a) is greater than I(b). The construction of I is based onHoward's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T inthe combinatory logic version. As a corollary we obtain an optimal derivationlength classification for the lambda formulation of T and its fragments.Compared with Weiermann's 1996 exposition this article yields solutions toseveral non-trivial problems arising from dealing with lambda terms instead ofcombinatory logic terms. It is expected that the methods developed here can beapplied to other higher order rewrite systems resulting in new powerfultermination orderings since T is a paradigm for such systems.

Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment