(Or "second order typed lambda-calculus",
"System F", "Lambda-2"). An extension of
typed lambda-calculus allowing functions which take types as
parameters. E.g. the
polymorphic function "twice" may be
written:
twice = /\ t . \ (f :: t -> t) . \ (x :: t) . f (f x)
(where "/\" is an upper case Greek lambda and "(v :: T)" is
usually written as v with subscript T). The parameter t will
be bound to the type to which twice is applied, e.g.:
twice Int
takes and returns a function of type Int -> Int. (Actual type
arguments are often written in square brackets [ ]). Function
twice itself has a higher type:
twice :: Delta t . (t -> t) -> (t -> t)
(where Delta is an upper case Greek delta). Thus /introduces an object which is a function of a type and Delta
introduces a type which is a function of a type.
Polymorphic lambda-calculus was invented by Jean-Yves Girard
in 1971 and independently by John C. Reynolds in 1974.
["Proofs and Types", J-Y. Girard, Cambridge U Press 1989].