Out Of The Blue (Blue Radio Edit)

⊢ Λ α . λ x α . x : ∀ α . α → α {\displaystyle \vdash \Lambda \alpha .\lambda x^{\alpha }.x:\forall \alpha .\alpha \to \alpha }

where α {\displaystyle \alpha } is a type variable. The upper-case Λ {\displaystyle \Lambda } is traditionally used to denote type-level functions, as opposed to the lower-case λ {\displaystyle \lambda } which is used for value-level functions. (The superscripted α {\displaystyle \alpha } means that the bound variable x is of type α {\displaystyle \alpha } ; the expression after the colon is the type of the lambda expression preceding it.) As a term rewriting system, System F is strongly normalizing. However, type inference in System F (without explicit type annotations) is undecidable. Under the Curry–Howard isomorphism, System F corresponds to second-order propositional intuitionistic logic. System F can be seen as part of the lambda cube, together with even more expressive typed lambda calculi, including those with dependent types. According to Girard, the "F" in System F was picked by chance.