Haskell - The Writer type

Writer2.hs

{-
    learnyouahaskell.com/for-a-few-monads-more#writer
-}

newtype Writer2 w a = Writer2 { runWriter2 :: (a, w) } deriving (Eq, Show) 

instance Functor (Writer2 w) where
    fmap f m = let
        (a, w) = (runWriter2 m)
        in Writer2 (f a, w)

{-
    http://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Functor.html#t:Functor

    class Functor f where
        fmap :: (a -> b) -> f a -> f b

    Functor 至少要實作 fmap

    > fmap (+1) (Writer2 (0,"Hi"))
    Writer2 {runWriter2 = (1,"Hi")}
-}

instance (Monoid w) => Applicative (Writer2 w) where
    pure a  = Writer2 (a, mempty)
    f <*> v = let
        (a, w) = (runWriter2 f) 
        (b, w') = (runWriter2 v)
        in Writer2 (a b, w `mappend` w')

{-
    http://hackage.haskell.org/package/base-4.12.0.0/docs/Control-Applicative.html#t:Applicative

    class Functor f => Applicative f where
        pure :: a -> f a
        (<*>) :: f (a -> b) -> f a -> f b
        liftA2 :: (a -> b -> c) -> f a -> f b -> f c

    Applicative至少要實作 pure, ((<*>) 或是 liftA2)  
    根據 Applicative 的定義 還需要實作 Functor

    > (Writer2 ((+1),"plus one to ")) <*> (Writer2 (0,"zero"))
    Writer2 {runWriter2 = (1,"plus one to zero")}
-}

instance (Monoid w) => Monad (Writer2 w) where  
    return a = Writer2 (a, mempty)
    m >>= f = let 
        (a, w) = runWriter2 m
        (b, w') = runWriter2 (f a)
        in Writer2 (b, w `mappend` w')
        
{-
    http://hackage.haskell.org/package/base-4.12.0.0/docs/Control-Monad.html#t:Monad

    class Applicative m => Monad m where
        (>>=) :: forall a b. m a -> (a -> m b) -> m b
    
    Monad 至少要實作 (>>=)
    因此 若想實作 Monad (Writer2 w) 就必須同時實作其所倚賴的  Applicative

    > (Writer2 (0, "zero")) >>= (\ x-> Writer2 (x+1," ," ++ show x ++ "+1=" ++ show (x+1)))
    Writer2 {runWriter2 = (1,"zero ,0+1=1")}
-}