course) that "<code>x:α</code>" represents an expression `x`
whose type is <code>α</code>.
-Then System F can be specified as follows (choosing notation that will
-match up with usage in O'Caml, whose type system is based on System F):
+Then System F can be specified as follows:
System F:
---------
simply-typed lambda calculus. So in a lingusitics context, type
constants might include `e` and `t`. "α" is a type variable. The
tick mark just indicates that the variable ranges over types rather
-than over values; in various discussion below and later, type variable
+than over values; in various discussion below and later, type variables
can be distinguished by using letters from the greek alphabet
(α, β, etc.), or by using capital roman letters (X, Y,
etc.). "`τ1 -> τ2`" is the type of a function from expressions of
have at least one free occurrence of `α` somewhere inside of it.
In the definition of the expressions, we have variables "`x`" as usual.
-Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda
+Abstracts "`λx:τ.e`" are similar to abstracts in the simply-typed lambda
calculus, except that they have their shrug variable annotated with a
type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
relevant evaluator is called "fullpoly"):
- N = ∀α. (α->α)->α->α;
- Pair = (N->N->N) -> N;
- let zero = Λα. λs:α->α . λz:α. z in
- let fst = λx:N. λy:N . x in
- let snd = λx:N. λy:N . y in
- let pair = λx:N. λy:N . λz:N->N->N . z x y in
- let suc = λn:N. λα . λlambda s:α->α . λz:α. s (n [α] s z) in
+ N = ∀α.(α->α)->α->α;
+ Pair = (N->N->N)->N;
+
+ let zero = Λα. λs:α->α. λz:α. z in
+ let fst = λx:N. λy:N. x in
+ let snd = λx:N. λy:N. y in
+ let pair = λx:N. λy:N. λz:N->N->N. z x y in
+ let suc = λn:N. Λα. λs:α->α. λz:α. s (n [α] s z) in
let shift = λp:Pair. pair (suc (p fst)) (p fst) in
let pre = λn:N. n [Pair] shift (pair zero zero) snd in
The key to the extra expressive power provided by System F is evident
in the typing imposed by the definition of `pre`. The variable `n` is
-typed as a Church number, i.e., as `∀ α . (α->α)->α->α`. The type
+typed as a Church number, i.e., as `∀α.(α->α)->α->α`. The type
application `n [Pair]` instantiates `n` in a way that allows it to
manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`. In
other words, the instantiation turns a Church number into a
In fact, unlike in the simply-typed lambda calculus,
it is even possible to give a type for ω in System F.
-<code>ω = λlambda x:(∀ α. α->α) . x [∀ α . α->α] x</code>
+<code>ω = λx:(∀α.α->α). x [∀α.α->α] x</code>
In order to see how this works, we'll apply ω to the identity
function.
<code>ω id ==</code>
- (λx:(∀α. α->α) . x [∀α.α->α] x) (Λα.λx:α. x)
+ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)
Since the type of the identity function is `∀α.α->α`, it's the
right type to serve as the argument to ω. The definition of