X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=0dcb0957803ffa0855c51e792a0c0979a4590683;hp=ae0b7e056f75c12f353d55a2a6b9ceca22759dea;hb=4d9fe4645917a4966694c7c4bf8edde8b243745f;hpb=b895ad012c22442151800347864a4b3f84f6de84

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index ae0b7e05..0dcb0957 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -1,23 +1,13 @@
 [[!toc levels=2]]
 
-# System F and recursive types
+# System F: the polymorphic lambda calculus
 
-In the simply-typed lambda calculus, we write types like <code>&sigma;
--> &tau;</code>.  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: 
-
-<pre>
-Expression    Type      Implication
------------------------------------
-fn            &alpha; -> &beta;    &alpha; &sup; &beta;
-arg           &alpha;         &alpha;
-------        ------    --------
-(fn arg)      &beta;         &beta;
-</pre>
-
-The implication in the right-hand column is modus ponens, of course.
+The simply-typed lambda calculus is beautifully simple, but it can't
+even express the predecessor function, let alone full recursion.  And
+we'll see shortly that there is good reason to be unsatisfied with the
+simply-typed lambda calculus as a way of expressing natural language
+meaning.  So we will need to get more sophisticated about types.  The
+next step in that journey will be to consider System F.
 
 System F was discovered by Girard (the same guy who invented Linear
 Logic), but it was independently proposed around the same time by
@@ -34,8 +24,7 @@ notational convention (which will last throughout the rest of the
 course) that "<code>x:&alpha;</code>" represents an expression `x`
 whose type is <code>&alpha;</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:
 	---------
@@ -47,7 +36,7 @@ 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`.  "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
 (&alpha;, &beta;, etc.), or by using capital roman letters (X, Y,
 etc.).  "`τ1 -> τ2`" is the type of a function from expressions of
@@ -57,7 +46,7 @@ universal type, since it universally quantifies over the type variable
 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.
 
@@ -117,13 +106,14 @@ however.  Here is one way, coded in
 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
 
@@ -138,7 +128,7 @@ 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 `∀ 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
@@ -165,14 +155,14 @@ Typing &omega;
 In fact, unlike in the simply-typed lambda calculus, 
 it is even possible to give a type for &omega; in System F. 
 
-<code>&omega; = λlambda x:(∀ α. α->α) . x [∀ α . α->α] x</code>
+<code>&omega; = λx:(∀α.α->α). x [∀α.α->α] x</code>
 
 In order to see how this works, we'll apply &omega; to the identity
 function.  
 
 <code>&omega; 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 &omega;.  The definition of