+<!-- λ Λ ∀ ≡ α β ρ ω Ω -->
+
+
[[!toc levels=2]]
# System F: the polymorphic lambda calculus
Exercise: convince yourself that `zero` has type `N`.
+[By the way, in order to keep things as simple as possible here, the
+types used in this definition of the ancillary functions given here
+are not as general as they could be; see the discussion below of type
+inference and principle types in the OCaml type system.]
+
The key to the extra expressive power provided by System F is evident
in the typing imposed by the definition of `pred`. The variable `n`
is typed as a Church number, i.e., as <code>N ≡ ∀α.(α->α)->α->α</code>.
[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
Press, chapter 23.]
+
Typing ω
--------------
strongly normalizing, from which it follows that System F is not
Turing complete.
-
## Polymorphism in natural language
Is the simply-typed lambda calclus enough for analyzing natural
terminate?
+
+Type inference and principle types
+----------------------------------
+
+As we mentioned, the types given to some of the functions defined
+above in the System F definition of `pred` are not as general as they
+might be.
+
+ let pair = λx:N. λy:N. λz:N->N->N. z x y in
+ ...
+
+For instance, in the definition of `pair`, we assumed that the
+function `z` would return something of type `N`, i.e., a Church
+number. But we can give a more general treatment.
+
+ let general_pair = Λα. Λβ. λx:α. λy:β. Λρ. λz:α->β->ρ. z x y in
+ ...
+
+In this more general version, the pair function accepts any kind of
+objects as its first and second element. The resulting pair will
+expect any function that is ready to handle arguments of the same
+types the pair was built from, and there is no restriction on the type
+(ρ) of the result returned from the pair-manipulation function.
+
+The type we gave the `pair` function above is a specific instance of
+the more general type, with `α`, `β`, and `ρ` all set to `N`.
+Many practical type systems guarantee that under reasonably general
+conditions, there will be a ***principle type***: a type such that
+every other possible type for that expression is a more specific
+version of the principle type.
+
+As we have seen, it is often possible to infer constraints on the type
+of an expression based on its internal structure, as well as by the
+way in which it is used. When programming interpreters and compilers
+infer types, they often (but not always) aim for the principle type
+(if one is guaranteed to exist).
+
+ # let pair a b z = z a b;;
+ val pair : 'a -> 'b -> ('a -> 'b -> 'c) -> 'c = <fun>
+
+For instance, when we define the same `pair` function given above in
+the OCaml interpreter, it correctly infers the principle type we gave
+above (remember that OCaml doesn't bother giving the explicit
+universal type quantifiers required by System F).
+
+
Bottom type, divergence
-----------------------