X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=ae0b7e056f75c12f353d55a2a6b9ceca22759dea;hp=6b80c20a8c4adea2c75b44e6b7702ab368eec952;hb=b895ad012c22442151800347864a4b3f84f6de84;hpb=1cbc7b6a7a1ec26412c8fb695e65ee03c7e4fff0
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 6b80c20a..ae0b7e05 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -119,13 +119,13 @@ relevant evaluator is called "fullpoly"):
N = âα. (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 = Îα. 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.
-ω = lambda x:(â α. α->α) . x [â α . α->α] x
+ω = λlambda x:(â α. α->α) . x [â α . α->α] x
In order to see how this works, we'll apply ω to the identity
function.
ω id ==
- (lambda x:(â α . α->α) . x [â α . α->α] x) (lambda α . lambda x:α . x)
+ (λx:(âα. α->α) . x [âα.α->α] x) (Îα.λx:α. x)
-Since the type of the identity function is `(â α . α->α)`, it's the
+Since the type of the identity function is `âα.α->α`, it's the
right type to serve as the argument to ω. The definition of
ω instantiates the identity function by binding the type
-variable `α` to the universal type `â α . α->α`. Instantiating the
+variable `α` to the universal type `âα.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 = Îα.Îβ.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