Dear Haskell,

Most of the time we get along well.  But, I'm growing weary of the arguments, fights, and nitpicking when I try to implement new mathematical types and overload your operators.  I don't know how to cooperate with your type systems.  At moments like this, I think about getting back together with C++.

I love you.  But, I also love implementing complex numbers, vectors, matrices, and quaternions, and Galois fields.  C++ is not nearly as elegant and beautiful as you.  But, C++ doesn't complain when I try to do this.  Isn't there some way we can work things out so I can implement these types with you?

Seriously, I'm trying to implement a vector.  I'm starting with vector addition:

{-
   This code is works with Glasgow, ghci, with these options:
  -fglasgow-exts
  -fallow-undecidable-instances
  -fno-monomorphism-restriction
  -fallow-incoherent-instances
-}

data Vector a = Vector [a] deriving Show

class Add a b c | a b -> c where
   (.+) :: a -> b -> c

instance Add Int Int Int where
   (.+) x y = x + y

instance Add Int Double Double where
   (.+) x y = (fromIntegral x) + y

instance Add Double Int Double where
   (.+) x y = x + (fromIntegral y)

instance Add Double Double Double where
   (.+) x y = x + y


instance (Add a b c) => Add (Vector a) (Vector b) (Vector c) where
   (.+) (Vector x) (Vector y) = Vector (zipWith (.+) x y)

vi1 = Vector [(1::Int)..3]
vi2 = Vector [(10::Int),15,2]
vd1 = Vector [(1::Double)..3]
vd2 = Vector [(10::Double),15,2]
test1 = vi1 .+ vi2
test2 = vi1 .+ vd2
test3 = vd1 .+ vi2
test4 = vd1 .+ vd2

v1 = Vector [1,2,3]
v2 = Vector [10,15,2]


However, it is necessary to explicitly nail down the type of the Vector.  v1 and v2 are more general.

*Main> :t v1
v1 :: forall a. (Num a) => Vector a
*Main> :t v2
v2 :: forall a. (Num a) => Vector a
*Main> test2

I'd like for .+ to work with v1 and v2.  So, I can use things like Vector [1,2,3] in expressions, instead of Vector[(1::Int),2,3].  However, v1 and v2 do not work with .+ in the code I produced above.

Does anyone have any ideas how to make this work?  I hoped defining .+ more generally for instances of Num would make my vector addition code work with v1 and v2.  My failed attempt involved making the following changes . . .

-- I added this
instance (Num d) => Add d d d where
   (.+) x y = x + y

-- instance Add Int Int Int where
--    (.+) x y = x + y

instance Add Int Double Double where
   (.+) x y = (fromIntegral x) + y

instance Add Double Int Double where
   (.+) x y = x + (fromIntegral y)

-- instance Add Double Double Double where
--    (.+) x y = x + y

When I make these changes and compile, I get the following error messages on the declaration of test1 and test4. . .

Vector2.hs:38:12:
    Overlapping instances for Add (Vector Int) (Vector Int) (Vector Int)
      arising from use of `.+' at Vector2.hs:38:12-13
    Matching instances:
      Vector2.hs:31:0: instance (Add a b c) => Add (Vector a) (Vector b) (Vector c)
      Vector2.hs:15:0: instance (Num d) => Add d d d
    In the definition of `test1': test1 = vi1 .+ vi2

Vector2.hs:41:12:
    Overlapping instances for Add (Vector Double) (Vector Double) (Vector Double)
      arising from use of `.+' at Vector2.hs:41:12-13
    Matching instances:
      Vector2.hs:31:0: instance (Add a b c) => Add (Vector a) (Vector b) (Vector c)
      Vector2.hs:15:0: instance (Num d) => Add d d d
    In the definition of `test4': test4 = vd1 .+ vd2

I interpret this as saying that the compiler doesn't know if the .+ in "test1 = vi1 .+ vi2" should match the Vector instance or the Num instance.  I could understand this if Vector was an instance of class Num.  However, this is not the case.  I figure either Glasgow has a bug or I don't really understand the error message.

I'd be grateful for any suggestions or pointers to information on how to implement vectors (or other mathematical types) so they seamlessly and intuitively work with types, classes and operators already built into Haskell.  Or, if someone could point to a more intermediate level book on working with the Haskell type system, that would be great.