previous type constructor roughly corresponds to the following definition of a tree with labeled leaves in OCaml:type 'a tree = | Leaf of 'a | Node of 'a tree * 'a tree This type constructor can be applied to other types to obtain new types. E.g., to obtain type of trees of integers:type int_tree = int tree System F is generally not used on its own, but is useful to isolate the independent feature of type constructors. (λP) Lambda-P In the λP system, also named ΛΠ, and closely related to the LF Logical Framework, one has so called dependent types. These are types that are allowed to depend on terms. The crucial introduction rule of the system is where represents valid types. The new type constructor corresponds via the Curry-Howard isomorphism to a universal quantifier, and the system λP as a whole corresponds to first-order logic with implication as only connective. An example of these dependent types in concrete programming is the type of vectors on a certain length: the length is a term, on which the type depends. (Fω) System Fω System Fω combines both the constructor of System F and the type constructors from System F. Thus System Fω provides both terms that depend on types and types that depend on types. (λC) Calculus of constructions In the calculus of constructions, denoted as λC in the cube or as λPω, these four features cohabit, so that both types and terms can depend on types and terms. The clear border that exists in λ→ between terms and types is somewhat abolished, as all types except the universal are themselves terms with a type. Formal definition As for all systems based upon the simply typed lambda calculus, all systems in the cube are given in two steps: first, raw terms, together with a notion of β-reduction, and then typing rules that allow to type those terms. The set of sorts is defined as , sorts are represented with the letter . There is also a set of variables, represented by the letters . The raw terms of the eight systems of the cube are given by the following syntax: and denoting when does not occur free in . The environment, as is usual in typed systems, are given by The notion of β-reduction is common to all systems in the cube. It is written and given by the rulesIts reflexive,