X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=f75370908dfe78fb0f081af4b6cbb074402e27b4;hp=6b80c20a8c4adea2c75b44e6b7702ab368eec952;hb=9873c23f4967dc8a1dc03bd57a20bf73db1ac721;hpb=1cbc7b6a7a1ec26412c8fb695e65ee03c7e4fff0

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 6b80c20a..f7537090 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -34,8 +34,7 @@ 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 +46,7 @@ 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 +56,7 @@ 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.
 
@@ -117,15 +116,15 @@ 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 = α . 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
-    let shift = λp:Pair . pair (suc (p fst)) (p fst) in
-    let pre = λn:N . n [Pair] shift (pair zero zero) snd 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
 
     pre (suc (suc (suc zero)));
 
@@ -138,7 +137,7 @@ 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
@@ -165,19 +164,19 @@ 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; = lambda 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>
 
-    (lambda x:(∀ α . α->α) . x [∀ α . α->α] x) (lambda α . lambda x: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
+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
 &omega; instantiates the identity function by binding the type
-variable `α` to the universal type `∀ α . α->α`.  Instantiating the
+variable `α` to the universal type `∀α.α->α`.  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.
@@ -228,10 +227,8 @@ uses.  Can we capture this using polymorphic types?
 
 With these basic types, we want to say something like this:
 
-    and:t->t->t = lambda l:t . lambda r:t . l r false
-    and = lambda α . lambda β . 
-            lambda l:α->β . lambda r:α->β . 
-              lambda x:α . and:β (l x) (r x)
+    and:t->t->t = λl:t. λr:t. l r false
+    and = Λα.Λβ.λl:α->β.λr:α->β.λx:α. and [β] (l x) (r x)
 
 The idea is that the basic *and* conjoins expressions of type `t`, and
 when *and* conjoins functional types, it builds a function that