+<!-- 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 principal 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 principal 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`, repeated here, 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 a handler function (`z`) that is ready to handle arguments of
+the same types the pair was built from, but there is no restriction on
+the type (ρ) of the result returned by the handler function.
+
+The number specific type we gave the `pair` function above (i.e.,
+`N->N->(N->N->N)->N`) 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
+***principal type***: a type such that every other possible type for
+that expression is a more specific version of the principal type.
+
+As we have seen, it is often possible to infer a type for an
+expression based on its internal structure, as well as by the way in
+which it is used. In the simply-typed lambda calculus, types for a
+well-typed expression can always be inferred; this is what enables
+programs to be written "Curry-style", i.e., without explicit types.
+
+Unfortunately, reliably inferring types is not always possible. For
+unrestricted System F, type inference is undecidable, so a certain
+amount of explicit type specification is required. OCaml places
+restrictions on its type system that makes type inference feasible.
+
+When programming language interpreters and compilers infer types, they
+often (but not always) aim for the principal 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 principal type we gave
+above (remember that OCaml doesn't bother giving the explicit
+universal type quantifiers required by System F).
+
+Computing principal types involves unification algorithms. Inferring
+types is a subtle and complicated business, and all sorts of
+extensions to a programming language can interfere with it.
+
+
+
Bottom type, divergence
-----------------------