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[lambda.git] / topics / _week5_system_F.mdwn

diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 6b80c20..ae0b7e0 100644 (file)
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -119,13 +119,13 @@ 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
+    let zero = Î\9bα. 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
 
     pre (suc (suc (suc zero)));
 
@@ -165,19 +165,19 @@ Typing ω
 In fact, unlike in the simply-typed lambda calculus, 
 it is even possible to give a type for ω in System F. 
 
-<code>&omega; = lambda x:(∀ α. α->α) . x [∀ α . α->α] x</code>
+<code>&omega; = λlambda 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:(∀ 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 +228,8 @@ 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