From: chris <chris@web>
Date: Thu, 26 Feb 2015 02:18:04 +0000 (-0500)
Subject: (no commit message)
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=6c8edb34886abac6afe327d50ebfefeb19c85d4c

---

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index a80cc58e..f7c38eb1 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -117,13 +117,13 @@ 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 = All X . (X->X)->X->X;
+    N = ∀ α . (α->α)->α->α;
     Pair = (N -> N -> N) -> N;
-    let zero = lambda X . lambda s:X->X . lambda z:X. z in 
+    let zero = lambda α . lambda s:α->α . lambda z:α. z in 
     let fst = lambda x:N . lambda y:N . x in
     let snd = lambda x:N . lambda y:N . y in
     let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in
-    let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in
+    let suc = lambda n:N . lambda α . lambda s:α->α . lambda z:α . s (n [α] s z) in
     let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in
     let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
 
@@ -138,7 +138,7 @@ 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 `All X . (X->X)->X->X`.  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
@@ -151,8 +151,8 @@ that the ordered pairs we need here are pairs of numbers.  If we tried
 to replace the type for Church numbers with a concrete (simple) type,
 we would have to replace each `X` with the type for Pairs, `(N -> N ->
 N) -> N`.  But then we'd have to replace each of these `N`'s with the
-type for Church numbers, `(X -> X) -> X -> X`.  And then we'd have to
-replace each of these `X`'s with... ad infinitum.  If we had to choose
+type for Church numbers, `(α -> α) -> α -> α`.  And then we'd have to
+replace each of these `α`'s with... ad infinitum.  If we had to choose
 a concrete type built entirely from explicit base types, we'd be
 unable to proceed.
  
@@ -165,19 +165,19 @@ 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:(All X. X->X) . x [All X . X->X] x</code>
+<code>&omega; = lambda x:(∀ 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:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+    (lambda x:(∀ α . α->α) . x [∀ α . α->α] x) (lambda α . lambda x:α . x)
 
-Since the type of the identity function is `(All X . X->X)`, 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 `X` to the universal type `All X . X->X`.  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.
@@ -229,9 +229,9 @@ 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 'a . lambda 'b . 
-            lambda l:'a->'b . lambda r:'a->'b . 
-              lambda x:'a . and:'b (l x) (r x)
+    and = lambda α . lambda β . 
+            lambda l:α->β . lambda r:α->β . 
+              lambda x: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
@@ -258,7 +258,7 @@ argument of that type.  We would like to instantiate the recursive use
 of *and* in the definition by using the result type.  But fully
 instantiating the definition as given requires type application to a
 pair of types, not to just a single type.  We want to somehow
-guarantee that 'b will always itself be a complex type.
+guarantee that β will always itself be a complex type.
 
 So conjunction and disjunction provide a compelling motivation for
 polymorphism in natural language, but we don't yet have the ability to