How to define a Monad instance for types with multiple values?

Question

By many meanings, I mean something like this:

data Foo a = Bar a | Baz a a

I cannot provide a clear definition >>=for Baz:

instance Monad Foo where
    Bar x   >>= f = f x -- Great, that works perfectly!
    Baz x y >>= f = ??? -- What the heck do I even put here?

Answer 1

May be:

frst (Bar a) = a
frst (Baz a a') = a

scnd (Bar a) = a
scnd (Baz a a') = a'

instance Monad Foo where
    return = Bar
    Bar x >>= f = f x
    Baz x y >>= f = Baz (frst (f x)) (scnd (f y))

This definition is inspired by the definition (>>=)for (Bool ->). Ask me if it’s not clear how.

Let them check the laws. The laws of " returnis unit" are quite simple:

  return x >>= f
= Bar x >>= f
= f x

  m >>= return
= case m of
      Bar x -> return x
      Baz x y -> Baz (frst (return x)) (scnd (return y))
= case m of
      Bar x -> Bar x
      Baz x y -> Baz x y
= m

I believe that I have ascertained that it (>>=)is an “associative” law, but I am sure that this evidence is completely unreadable to anyone else ... I urge you to try to prove it yourself and turn to my calculations as a fraud if you stuck.

  m >>= (\v -> f v >>= g)
= case m of
      Bar x -> (\v -> f v >>= g) x
      Baz x y -> Baz (frst ((\v -> f v >>= g) x))
                     (scnd ((\v -> f v >>= g) y))
= case m of
      Bar x -> f x >>= g
      Baz x y -> Baz (frst (f x >>= g)) (scnd (f y >>= g))
= case m of
      Bar x -> case f x of
          Bar y -> g y
          Baz a b -> Baz (frst (g a)) (scnd (g b))
      Baz x y -> Baz (frst l) (scnd r) where
          l = case f x of
                  Bar a -> g a
                  Baz a b -> Baz (frst (g a)) (scnd (g b))
          r = case f y of
                  Bar a -> g a
                  Baz a b -> Baz (frst (g a)) (scnd (g b))
= case m of
      Bar x -> case f x of
          Bar y -> g y
          Baz a b -> Baz (frst (g a)) (scnd (g b))
      Baz x y -> Baz (frst (g (frst (f x))))
                     (scnd (g (scnd (f y))))
= case m of
      Bar a -> case f a of
          Bar x -> g x
          Baz x y -> Baz (frst (g x)) (scnd (g y))
      Baz a b -> case Baz (frst (f a)) (scnd (f b)) of
          Bar x -> g x
          Baz x y -> Baz (frst (g x)) (scnd (g y))
= case v of
      Bar x -> g x
      Baz x y -> Baz (frst (g x)) (scnd (g y))
  where v = case m of
                Bar a -> f a
                Baz a b -> Baz (frst (f a)) (scnd (f b))
= case m >>= f of
      Bar x -> g x
      Baz x y -> Baz (frst (g x)) (scnd (g y))
= (m >>= f) >>= g

edit , , (Bool ->), . , :

instance Monad (e ->) where
    m >>= f = \e -> f (m e) e

data Pair a = Pair a a

, Bool -> a Pair a :

to :: Pair a -> (Bool -> a)
to (Pair false true) = \bool -> case bool of
    False -> false
    True  -> true

from :: (Bool -> a) -> Pair a
from f = Pair (f False) (f True)

, from to . : Bool -> a " ". , (e ->) Monad Pair? , , , . , :

instance Monad Pair where
    return x = from (return x)
    m >>= f = from (to m >>= to . f)

" ":

  return x
= from (return x)
= from (\e -> x)
= Pair ((\e -> x) False) ((\e -> x) True)
= Pair x x

  m@(Pair false true) >>= f
= from (to m >>= to . f)
= from (\e -> (to . f) (to m e) e)
= from (\e -> to (f (to m e)) e)
= Pair (g False) (g True) where
      g = \e -> to (f (to m e)) e
= Pair (to (f (to m False)) False) (to (f (to m True)) True)
= Pair (case f (to m False) of Pair false true -> false)
       (case f (to m True ) of Pair false true -> true )
= Pair (case f false of Pair false true -> false)
       (case f true  of Pair false true -> true )

, , (Bool ->), :

frstPair (Pair false true) = false
scndPair (Pair false true) = true

instance Monad Pair where
    return x = Pair x x
    Pair false true >>= f = Pair (frstPair (f false)) (scndPair (f true))

, , (>>=), Foo.

edit 2 (!) ​​ . :

type Foo = Product Identity Maybe

Product. , :

instance Monad Foo where
    return x = Baz x x
    Bar x >>= f = Bar (frst (f x))
    Baz x y >>= f = case f y of
        Bar a -> Bar (frst (f x))
        Baz a b -> Baz (frst (f x)) b

, "" - Bar return Bar Baz bind - "" - Baz return Baz Bar bind. , ! , Monad Product (, ).