fix System F booleans

diff --git a/exercises/assignment5_answers.mdwn b/exercises/assignment5_answers.mdwn
index 0575276..de91618 100644 (file)
--- a/exercises/assignment5_answers.mdwn
+++ b/exercises/assignment5_answers.mdwn
@@ -496,8 +496,15 @@ type `Bool`.
         true ≡ Λα. λy:α. λn:α. y
         false ≡ Λα. λy:α. λn:α. n
         not ≡ λp:Bool. p [Bool] false true
         true ≡ Λα. λy:α. λn:α. y
         false ≡ Λα. λy:α. λn:α. n
         not ≡ λp:Bool. p [Bool] false true

-        and ≡ λp:Bool. λq:Bool. p [Bool] (q [Bool]) false
-        or ≡ λp:Bool. λq:Bool. p [Bool] true (q [Bool])
+        and ≡ λp:Bool. λq:Bool. p [Bool] q false
+        or ≡ λp:Bool. λq:Bool. p [Bool] true q
+
+    When I first wrote up these answers, I had put `(q [Bool])` where I now have just `q` in the body of `and` and `or`. On reflection,
+    this isn't necessary, because `q` is already of that type. But as I learned by checking these answers with Pierce's evaluator, it's
+    also a mistake. What we want is a result whose type is `Bool`, that is, `∀α. α -> α -> α`. `(q [Bool])` doesn't have that type, but
+    rather the type `Bool -> Bool -> Bool`. The first, desired, type has an outermost `∀`. The second, wrong type doesn't; it only has `∀`s
+    inside the antecedents and consequents of the various arrows. The last one of those could be promoted to be an outermost `∀`, since
+    `P -> ∀α. Q ≡ ∀α. P -> Q` when `α` is not free in `P`. But that couldn't be done with the others.

 
 
 The type `Nat` (for "natural number") may be encoded as follows:
 
 
 The type `Nat` (for "natural number") may be encoded as follows: