Basic Theory of F-Bounded Quantification

System F-bounded is a second-order typed lambda calculus, where the basic features of object-oriented languages can be naturally modelled. F-bounded extends the better known system F⩽, in a way that provides an immediate solution for the treatment of the so-called “binary methods.” Although more powerful than F⩽ and also quite natural, system F-bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature. The aim of this paper is to give a solid foundation to F-bounded, by addressing and proving the key properties of the system. In particular, transitivity elimination, completeness of the type checking semi-algorithm, the subject reduction property for βη reduction, conservativity with respect to system F⩽, and antisymmetry of a “full” subsystem are considered, and various possible formulations for system F-bounded are compared. Finally, a semantic interpretation of system F-bounded is presented, based on partial equivalence relations.