Comparing cubes of typed and type assignment systems

We study the cube of type assignment systems, as introduced in Giannini et al. (Fund. Inform. 19 (1993) 87-126), and confront it with Barendregt's typed lambda-cube (Barendregt, in: Handbook of Logic in Computer Science, Vol. 2, Clarenden Press, Oxford, 1992). The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for systems without polymorphism. The type assignment systems we consider satisfy the properties 'subject reduction' and 'strong normalization'. Moreover, we define a new type assignment cube that is isomorphic to the typed one.