```
σ
-> τ
```

. This looks like logical implication. We'll take
that resemblance seriously when we discuss the Curry-Howard
correspondence. In the meantime, note that types respect modus
ponens:
Expression Type Implication ----------------------------------- fn α -> β α ⊃ β arg α α ------ ------ -------- (fn arg) β βThe implication in the right-hand column is modus ponens, of course. System F was discovered by Girard (the same guy who invented Linear Logic), but it was independently proposed around the same time by Reynolds, who called his version the *polymorphic lambda calculus*. (Reynolds was also an early player in the development of continuations.) System F enhances the simply-typed lambda calculus with abstraction over types. Normal lambda abstraction abstracts (binds) an expression (a term); type abstraction abstracts (binds) a type. In order to state System F, we'll need to adopt the notational convention (which will last throughout the rest of the course) that "

`x:α`

" represents an expression `x`
whose type is `α`

.
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):
System F:
---------
types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
In the definition of the types, "`c`" is a type constant. Type
constants play the role in System F that base types play in the
simply-typed lambda calculus. So in a lingusitics context, type
constants might include `e` and `t`. "`'a`" 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
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
type `τ1` to expressions of type `τ2`. And "`∀'a. τ`" is called a
universal type, since it universally quantifies over the type variable
`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually
have at least one free occurrence of `'a` 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
calculus, except that they have their shrug variable annotated with a
type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
In addition to variables, abstracts, and applications, we have two
additional ways of forming expressions: "`Λ'a. e`" is called a *type
abstraction*, and "`e [τ]`" is called a *type application*. The idea
is that `Λ`

is a capital `λ`

: just
like the lower-case `λ`

, `Λ`

binds
variables in its body, except that unlike `λ`

,
`Λ`

binds type variables instead of expression
variables. So in the expression
`Λ 'a (λ x:'a . x)`

the `Λ`

binds the type variable `'a` that occurs in
the `λ`

abstract. Of course, as long as type
variables are carefully distinguished from expression variables (by
tick marks, Grecification, or capitalization), there is no need to
distinguish expression abstraction from type abstraction by also
changing the shape of the lambda.
The expression immediately below is a polymorphic version of the
identity function. It defines one general identity function that can
be adapted for use with expressions of any type. In order to get it
ready to apply this identity function to, say, a variable of type
boolean, just do this:
`(Λ 'a (λ x:'a . x)) [t]`

This type application (where `t` is a type constant for Boolean truth
values) specifies the value of the type variable `'a`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
`((Λ 'a (λ x:'a . x)) [t]): (b -> b)`

Likewise, if we had instantiated the type variable as an entity (base
type `e`), the resulting identity function would have been a function
of type `e -> e`:
`((Λ 'a (λ x:'a . x)) [e]): (e -> e)`

Clearly, for any choice of a type `'a`, the identity function can be
instantiated as a function from expresions of type `'a` to expressions
of type `'a`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
`(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)`

Pred in System F
----------------
We saw that the predecessor function couldn't be expressed in the
simply-typed lambda calculus. It *can* be expressed in System F,
however. Here is one way, coded in
[[Benjamin Pierce's type-checker and evaluator for
System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
relevant evaluator is called "fullpoly"):
N = All X . (X->X)->X->X;
Pair = (N -> N -> N) -> N;
let zero = lambda X . lambda s:X->X . lambda z:X. z in
let fst = lambda x:N . lambda y:N . x in
let snd = lambda x:N . lambda y:N . y in
let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in
let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in
let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in
let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
pre (suc (suc (suc zero)));
We've truncated the names of "suc(c)" and "pre(d)", since those are
reserved words in Pierce's system. Note that in this code, there is
no typographic distinction between ordinary lambda and type-level
lambda, though the difference is encoded in whether the variables are
lower case (for ordinary lambda) or upper case (for type-level
lambda).
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 `All X . (X->X)->X->X`. 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
pair-manipulating function, which is the heart of the strategy for
this version of predecessor.
Could we try to build a system for doing Church arithmetic in which
the type for numbers always manipulated ordered pairs? The problem is
that the ordered pairs we need here are pairs of numbers. If we tried
to replace the type for Church numbers with a concrete (simple) type,
we would have to replace each `X` with the type for Pairs, `(N -> N ->
N) -> N`. But then we'd have to replace each of these `N`'s with the
type for Church numbers, `(X -> X) -> X -> X`. And then we'd have to
replace each of these `X`'s with... ad infinitum. If we had to choose
a concrete type built entirely from explicit base types, we'd be
unable to proceed.
[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
Press, chapter 23.]
Typing ω
--------------
In fact, unlike in the simply-typed lambda calculus,
it is even possible to give a type for ω in System F.
`ω = lambda x:(All X. X->X) . x [All X . X->X] x`

In order to see how this works, we'll apply ω to the identity
function.
`ω id ==`

(lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
Since the type of the identity function is `(All X . X->X)`, it's the
right type to serve as the argument to ω. The definition of
ω instantiates the identity function by binding the type
variable `X` to the universal type `All X . X->X`. Instantiating the
identity function in this way results in an identity function whose
type is (in some sense, only accidentally) the same as the original
fully polymorphic identity function.
So in System F, unlike in the simply-typed lambda calculus, it *is*
possible for a function to apply to itself!
Does this mean that we can implement recursion in System F? Not at
all. In fact, despite its differences with the simply-typed lambda
calculus, one important property that System F shares with the
simply-typed lambda calculus is that they are both strongly
normalizing: *every* expression in either system reduces to a normal
form in a finite number of steps.
Not only does a fixed-point combinator remain out of reach, we can't
even construct an infinite loop. This means that although we found a
type for ω, there is no general type for Ω ≡ ω
ω. Furthermore, it turns out that no Turing complete system can
be strongly normalizing, from which it follows that System F is not
Turing complete.
Types in OCaml
--------------
OCaml has type inference: the system can often infer what the type of
an expression must be, based on the type of other known expressions.
For instance, if we type
# let f x = x + 3;;
The system replies with
val f : int -> int =