The f-cube introduces an extension to the λ-cube by incorporating finite-set declarations (FSDs) to represent intersection types. This extension allows for the encoding of intersection types as Π-types, providing a new way to handle type intersections without the complexities of traditional intersection-introduction rules. The f-cube can type the same pure untyped λ-terms as the higher-order polymorphic implicitly typed λ-calculus Fω, but it also addresses the typability of terms that are not typable in Fω or the λ-cube.
The theoretical limits of the f-cube lie in its ability to handle intersection types and represent them as Π-types using FSDs. While this approach simplifies the encoding of intersection types and avoids the challenges posed by traditional intersection-introduction rules, it may have limitations in handling more complex type systems or constructs that go beyond simple intersection types. The f-cube may excel in scenarios where intersection types are a crucial feature, but it may not be as versatile in handling a wide range of type-theoretic constructs compared to more comprehensive type systems like Fω.