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diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 2c37ae34..559135e0 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -25,49 +25,55 @@ continuations.)
System F enhances the simply-typed lambda calculus with abstraction
over types. In order to state System F, we'll need to adopt the
-notational convention that "x:α
" represents a
-expression whose type is α
.
+notational convention 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:
+ System F:
+ ---------
types Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï
expressions e ::= x | λx:Ï. e | e1 e2 | Î'a. e | e [Ï]
In the definition of the types, "`c`" is a type constant (e.g., `e` or
-`t`). "`'a`" is a type variable (the tick mark just indicates that
-the variable ranges over types rather than values). "`Ï1 -> Ï2`" is
-the type of a function from expressions of type `Ï1` to expressions of
-type `Ï2`. And "`â'a. Ï`" is called a universal type, since it
-universally quantifies over the type variable `'a`.
-
-In the definition of the expressions, we have variables "`x`".
+`t`, or in arithmetic contexts, `N` or `Int`). "`'a`" is a type
+variable (the tick mark just indicates that the variable ranges over
+types rather than over values). "`Ï1 -> Ï2`" is the type of a
+function from expressions of type `Ï1` to expressions of type `Ï2`.
+And "`â'a. Ï`" is called a universal type, since it universally
+quantifies over the type variable `'a`. (You can expect that in
+`â'a. Ï`, the type `Ï` 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:Ï. 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: "`Î'a. e`" is a type
-abstraction, and "`e [Ï]`" is a type application. The idea is that
-Λ
is a capital λ
. Just like
-the lower-case λ
, Λ
binds
-variables in its body; unlike λ
,
-Λ
binds type variables. So in the expression
+additional ways of forming expressions: "`Î'a. e`" is called a *type
+abstraction*, and "`e [Ï]`" 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. This expression is a polymorphic
-version of the identity function. It says that this one general
-identity function can be adapted for use with expressions of any
-type. In order to get it ready to apply to, say, a variable of type
-boolean, just do this:
+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 to, say, a variable of type boolean, just do
+this:
(Λ 'a (λ x:'a . x)) [t]
-The type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `α`, which is
-the type of the variable bound in the `λ` expression. Not
+This type application (where `t` is a type constant for Boolean truth
+values) specifies the value of the type variable α, which is
+the type of the variable bound in the λ expression. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
@@ -84,13 +90,16 @@ instantiated as a function from expresions of type `'a` to expressions
of type `'a`. In general, then, the type of the unapplied
(polymorphic) identity function is
-(Λ 'a (λ x:'a . x)): (\forall 'a . 'a -> 'a)
-
-
+(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.
+[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
+Press, pp. 350--353, for `tail` for lists in System F.]
Types in OCaml