[Haskell-cafe] How do I simulate dependent types using phantom types?

[DavidA]

Hi,

I am trying to implement quadratic fields Q(sqrt d). These are numbers of the 
form a + b sqrt d, where a and b are rationals, and d is an integer.

In an earlier attempt, I tried
data QF = QF Integer Rational Rational
(see http://www.polyomino.f2s.com/david/haskell/hs/QuadraticField.hs.txt)
The problem with this approach is that it's not really type-safe:
I can attempt to add a + b sqrt 2 to c + d sqrt 3, whereas this should be a 
type error because 2 /= 3.

So I thought I'd have a go at doing it with phantom types. In effect I'd be 
using phantom types to simulate dependent types. Here's the code:

{-# OPTIONS_GHC -fglasgow-exts #-}

import Data.Ratio

class IntegerType a where
    value :: Integer

data Two
instance IntegerType Two where value = 2

data Three
instance IntegerType Three where value = 3

data QF d = QF Rational Rational deriving (Eq)

instance IntegerType d => Show (QF d) where
    show (QF a b) = show a ++ " + " ++ show b ++ " sqrt " ++ show value

instance IntegerType d => Num (QF d) where
    QF a b + QF a' b' = QF (a+a') (b+b')
    negate (QF a b) = QF (-a) (-b)
    QF a b * QF c d = QF (a*c + b*d*value) (a*d + b*c)
    fromInteger n = QF (fromInteger n) 0

The problem is, this doesn't work. GHC complains:
    The class method `value'
    mentions none of the type variables of the class IntegerType a
    When checking the class method: value :: Integer
    In the class declaration for `IntegerType'

Is what I'm trying to do reasonable? If no, what should I be doing instead? If 
yes, why doesn't GHC like it?

Thanks, David