  Due to the undecidability of most type-related properties of System F like
type inhabitation or type checking, restricted polymorphic systems have been
widely investigated (the most well-known being ML-polymorphism). In this paper
we investigate System Fat, or atomic System F, a very weak predicative fragment
of System F whose typable terms coincide with the simply typable ones. We show
that the type-checking problem for Fat is decidable and we propose an algorithm
which sheds some new light on the source of undecidability in full System F.
Moreover, we investigate free theorems and contextual equivalence in this
fragment, and we show that the latter, unlike in the simply typed
lambda-calculus, is undecidable.
