import Prelude hiding (const, id, (.))
import Control.Category

data St a b = St (a -> (b, St a b))

ap :: St a b -> [a] -> [b]
ap _ [] = []
ap (St f) (a:as) = let (b, nSt) = f a in b : ap nSt as

const :: a -> St b a
const x = St $ \_ -> (x, const x)

integral :: Num a => St a a
integral = from 0 where
  from n = St $ \x -> st x where
    st x = let s = x + n in (s, from s)

instance Category St where
  --  id :: St a a
  id = St $ (\x -> (x, id))
  --  (.) :: St b c -> St a b -> St a c
  (.) (St f) (St g) = St $ \x -> compose x where
    compose x = (x2, f1 . g1) where
      (x1, g1) = g x
      (x2, f1) = f x1

main = do
  print $ ap (integral . id) [0,1,2] == ap (id . integral) [0,1,2]
  print $ ap (integral . (const 42 . integral)) [0,1,2] == ap ((integral . const 42) . integral) [0,1,2]
  print $ ap (const 42) [0,1,2]  == [42,42,42]
  print $ ap integral [0,1,2,3] == [0,1,3,6]