Bounded Quanti cation with Bottom

While numerous extensions of Cardelli and Wegner's calculus of polymorphism and subtyping, Kernel Fun, have been studied during the past decade, one quite simple one has received little attention: the addition of a minimal type Bot, dual to the familiar maximal type Top. We develop basic meta-theory for this extension. Although most of the usual properties of Kernel Fun (existence of meets and joins, decidability of subtyping and typing, subject reduction, etc.) also hold for the extended system, the presence of Bot introduces some surprising intricacies. In particular, a type variable bounded by Bot is actually a synonym for Bot; such \bottom variables" must be treated specially at several points.