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```
The Free On-line Dictionary of Computing (18 March 2015):polymorphic lambda-calculus
System F

(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].

(2005-03-07)
```