Bas van Dijk v.dijk.bas at gmail.com
Wed Jul 2 06:53:14 EDT 2008

```Dear list,

I have I question about the following code I was playing with:
(you can past the following right into your editor)

----------------------------------------------------------------------

import Data.Foldable       (Foldable, foldMap)
import Data.Monoid         (mempty, mappend)
import Data.Traversable    (Traversable, traverse)
import Control.Applicative (pure, (<\$>), (<*>))

-- I was playing with the following tree-like datastructure (my plan
-- is to make some kind of kd-tree but that's not important now):

data T a = L | N C2 a (T a) (T a)
(T a) (T a)

type C2 = (Float, Float)

-- A fold always comes in handy:

foldT :: b -> (C2 -> a -> b -> b
-> b -> b -> b) -> T a -> b
foldT e _ L = e
foldT e n (N c x tl tr
bl br) = n c x (foldT e n tl) (foldT e n tr)
(foldT e n bl) (foldT e n br)

instance Functor T where
fmap f = foldT L (\p -> N p . f)

-- Now I defined the following instances:

instance Foldable T where
foldMap f = foldT mempty \$ \_ x tl tr
bl br -> f x `mappend` tl `mappend` tr
`mappend` bl `mappend` br

instance Traversable T where
traverse f = foldT (pure L) \$ \p x tl tr
bl br -> N p <\$> f x <*> tl <*> tr
<*> bl <*> br

----------------------------------------------------------------------

-- If you look at the previous two functions you see a similar pattern:
-- they both combine an initial value:  'f x'  and  'N p <\$> f x'  respectively
-- with the childs using a combining function:  'mappend'  and  '<*>'
respectively.

-- My question is: can I abstract from that?

-- It looks like I can using a function like:

combineWith :: b -> (b -> a -> b) -> a -> a
-> a -> a -> b
n `combineWith` f = \tl tr
bl br -> n `f` tl `f` tr
`f` bl `f` br

-- Now 'foldMap' becomes:

instance Foldable T where
foldMap f = foldT mempty \$ \_ x -> f x `combineWith` mappend

-- But 'traverse' won't typecheck:

instance Traversable T where
traverse f = foldT (pure L) \$ \p x -> (N p <\$> f x) `combineWith` (<*>)

-- Is it possible to make 'combineWith' more general so that the
-- previous typechecks (maybe using arbitrary-rank polymorphism but I
-- don't see how)?

----------------------------------------------------------------------

Thanks,

Bas van Dijk
```