Some thoughts on merging Church Free and State monad

ChurchState.hs

-- Option 1: State as an eDSL

data StateMethod s next
  = Put s (() -> next)
  | Get (s -> next)

instance Functor (StateMethod s) where
  fmap f (Put st next) = Put st (f . next)
  fmap f (Get next)    = Get (f . next)

type FreePoweredState st a = F (StateMethod st) a

put :: s -> FreePoweredState s ()
put st = liftF (Put st id)

get :: FreePoweredState s s
get = liftF (Get id)


-- Option 2: State as a part of Church Free (may be incorrect):



newtype StateMergedChurch s f a = MyChurch
  { runMyChurch
    :: forall r
     . (s -> a -> (s, r))    -- Pure case
    -> (s -> f r -> (s, r))  -- Continuation case
    -> s                     -- initial state
    -> (s, r)
  }