How to map a function over multiple constructor arguments working with Monad class in Haskell?

Question

The problem I'm stumbled at has to do with >>= application to a sample type like that:

data ThreeArgs a = ThreeArgs a a a deriving (Show,Eq)

instance Functor ThreeArgs where
    fmap f (ThreeArgs a b c) = ThreeArgs (f a) (f b) (f c)

instance Applicative ThreeArgs where
    pure x = ThreeArgs x x x
    (ThreeArgs a b c) <*> (ThreeArgs p s q) = ThreeArgs (a p) (b s) (c q)

I'd declare a Monad instance as follows:

instance Monad ThreeArgs where
    return x = ThreeArgs x x x
    (ThreeArgs a b c) >>= f = f ... -- a code I need to complete

Yes, it looks as if f to be applied to all three ThreeArgs contructor arguments. If I complete last line

(ThreeArgs a b c) >>= f = f a

then compiler doesn't have any complaints, whereas result is:

*module1> let x = do { x <- ThreeArgs 1 2 3; y <- ThreeArgs 4 6 7; return $ x + y }
*module1> x
ThreeArgs 5 5 5 

it means that summation results into a context with same argument values, although correct output should be ThreeArgs 5 8 10. Once I edit to

(ThreeArgs a b c) >>= f = (f a) (f b) (f c)

compiler alerts:

 Couldn't match expected type `ThreeArgs b
                                -> ThreeArgs b -> ThreeArgs b -> ThreeArgs b'
              with actual type `ThreeArgs b'

So, I see a serious mistake guides my comprehension, but it's still rather hard to me to understand monadic class and another such things in Haskell. Presumably, do I want to use recursion here or what else?

Answer 1

ThreeArgs is isomorphic to ((->) Ordering). Witness:

to :: ThreeArgs a -> Ordering -> a
to (ThreeArgs x _ _) LT = x
to (ThreeArgs _ y _) EQ = y
to (ThreeArgs _ _ z) GT = z

from :: (Ordering -> a) -> ThreeArgs a
from f = ThreeArgs (f LT) (f EQ) (f GT)

Your Functor and Applicative instances match how the ones for ((->) r) work, so we can just make it match how its Monad one works too and we're done.

instance Monad ThreeArgs where
    ThreeArgs x y z >>= f = ThreeArgs x' y' z' where
        ThreeArgs x' _ _ = f x
        ThreeArgs _ y' _ = f y
        ThreeArgs _ _ z' = f z

By the way, the general term for data structures like ThreeArgs is "representable functor", if you want to look up more about this.