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diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 55f49aed..4bde11e0 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -1,6 +1,13 @@
[[!toc levels=2]]
-# System F and recursive types
+# System F: the polymorphic lambda calculus
+
+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.
In the simply-typed lambda calculus, we write types like σ
-> τ
. This looks like logical implication. We'll take
@@ -118,13 +125,14 @@ relevant evaluator is called "fullpoly"):
N = âα.(α->α)->α->α;
Pair = (N->N->N)->N;
- 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.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
+
+ 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. Îα. 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)));
@@ -137,7 +145,7 @@ 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 `â α . (α->α)->α->α`. The type
+typed as a Church number, i.e., as `âα.(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
@@ -164,14 +172,14 @@ 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 = λlambda x:(â α. α->α) . x [â α . α->α] x
+ω = λx:(âα.α->α). x [âα.α->α] x
In order to see how this works, we'll apply ω to the identity
function.
ω id ==
- (λx:(âα. α->α) . x [âα.α->α] x) (Îα.λx:α. x)
+ (λx:(âα.α->α). x [âα.α->α] x) (Îα.λx:α.x)
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