Haskell difference lists and point free function

Question

I was researching difference lists and found the DList type

newtype DList a = DL { unDL :: [a] -> [a] }

and the function

dlToList :: DList a -> [a]
dlToList = ($[]) . unDL

I am wondering what is the non point free version of the function and what does ($[]) do?

Answer 1

The first step in seeing into a point-free definition of a function is to revert the η-reduction:

dlToList = ($[]) . unDL
dlToList dl = (($[]) . unDL) dl

Then you start applying to the composition-chain, right-to-left:

dlToList dl = ($[]) (unDL dl)

You could then unpack the operator section^†

dlToList dl = unDL dl $ []

However, keeping the ($[]) as it is actually makes sense, because this is the essential converter between difference lists and ordinary lists: it takes a [a]->[a]-prepender-function and applies it to the terminator [], resulting in a concrete list.

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^† We could simplify that further:

dlToList dl = unDL dl []

which, incidentally, could be made point-free again in a shorter manner:

dlToList = (`unDL`[])