Call this type function M, and let P, Q, R, and S be variables over types.
Recall that a monad requires a singleton function 1:P-> MP, and a
-composition operator >=>: (P->MQ) -> (Q->MR) -> (R->MS) that obey the
-following laws:
+composition operator >=>: (P->MQ) -> (Q->MR) -> (P->MR) [the type for
+the composition operator given here corrects a "type"-o from the class handout]
+that obey the following laws:
1 >=> k = k
k >=> 1 = k
Then the obvious singleton for the Option monad is \p.Just p. Give
(or reconstruct) the composition operator >=> we discussed in class.
Show your composition operator obeys the monad laws.
+
+2. Do the same with crossy lists. That is, given an arbitrary type
+'a, let the boxed type be a list of objects of type 'a. The singleton
+is `\p.[p]`, and the composition operator is
+
+ >=> (first:P->[Q]) (second:Q->[R]) :(P->[R]) = fun p -> [r | q <- first p, r <- second q]
+
+Sanity check:
+
+ f p = [p, p+1]
+ s q = [q*q, q+q]
+ >=> f s 7 = [49, 14, 64, 16]
+
+3. Do the same for zippy lists. That is, you need to find a
+composition operator such that
+
+ f p = [p, p+1]
+ s q = [q*q, q+q]
+ >=> f s 7 = [49, 16]
+
+and then prove it obeys the monad laws.