will help you This looks like a special case of a (jargon here but it can help with googling) paramorphism, a generalisation of

to all initial algebras.

code :

```
module ListCase where
paraList :: (a -> [a] -> b -> b) -- cons
-> b -- nil
-> [a] -> b -- resulting function on lists
paraList c n [] = n
paraList c n (x : xs) = c x xs $ paraList c n xs
```

```
listCase' :: (a -> [a] -> b) -> [a] -> [b]
listCase' c = paraList (\ x xs tl -> c x xs : tl) []
```

```
newtype Fix f = Fix { unFix :: f (Fix f) }
```

```
newtype ListF a as = ListF { unListF :: Maybe (a, as) }
type List a = Fix (ListF a)
nil :: List a
nil = Fix $ ListF $ Nothing
cons :: a -> List a -> List a
cons = curry $ Fix . ListF .Just
```

```
para :: Functor f => (f (Fix f, b) -> b) -> Fix f -> b
para alg = alg . fmap (\ rec -> (rec, para alg rec)) . unFix
```

```
instance Functor (ListF a) where fmap f = ListF . fmap (fmap f) . unListF
paraList' :: (a -> List a -> b -> b) -> b -> List a -> b
paraList' c n = para $ maybe n (\ (a, (as, b)) -> c a as b) . unListF
listCase'' :: (a -> List a -> b) -> List a -> List b
listCase'' c = paraList' (\ x xs tl -> cons (c x xs) tl) nil
```

```
toList :: [a] -> List a
toList = foldr cons nil
fromList :: List a -> [a]
fromList = paraList' (\ x _ tl -> x : tl) []
*ListCase> fmap fromList . fromList . listCase'' (\ _ as -> toList $ take 3 $ fromList as). toList $ [1..5]
[[2,3,4],[3,4,5],[4,5],[5],[]]
```