monadcomplex.hs

-- In an alternate universe of Haskell where this worked...
data (Num a) => Complex a = a :+ a

instance Functor Complex where
    fmap f (x :+ y) = f x :+ f y

instance Monad Complex where
    return x = x :+ 0
    join ((a :+ b) :+ (c :+ d)) = (a - d) :+ (b + c)

-- Associativity
   join . fmap join $ ((a :+ b) :+ (c :+ d)) :+ ((e :+ f) :+ (g :+ h))
 = join             $ join ((a :+ b) :+ (c :+ d)) :+ join ((e :+ f) :+ (g :+ h))
 = join             $ ((a - d) :+ (b + c)) :+ ((e - h) :+ (f + g))
 =                    (a - d - f - g) :+ (b + c + e - h)
 =                    (a - g - d - f) :+ (b - h + c + e)
 = join             $ ((a - g) :+ (b - h)) :+ ((c + e) :+ (d + f))
 = join             $ ((a :+ b) - (g :+ h)) :+ ((c :+ d) + (e :+ f))
 = join . join      $ ((a :+ b) :+ (c :+ d)) :+ ((e :+ f) :+ (g :+ h))

-- Right unit
   join . fmap return $ a :+ b
 = join               $ return a :+ return b
 = join               $ (a :+ 0) :+ (b :+ 0)
 =                      (a - 0) :+ (0 + b)
 = id                 $ a :+ b
 
-- Left unit
   join . return $ a :+ b
 = join          $ (a :+ b) :+ (0 :+ 0)
 =                 (a - 0) :+ (b + 0)
 = id            $ a :+ b