Use value :: a -> Integer On 8/18/07, DavidA wrote:

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

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DavidA wrote:

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.

...

class IntegerType a where value :: Integer

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?

You are on the right track. The problem with the class method is that it doesn't use type 'a' anywhere, consider

f :: Integer f = value What class instance should be used here?

The solution is to use a dummy parameter:

class IntegerType a where value :: a -> Integer And call it like: f = value (undefined :: Two)

So for instance:

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

The problem is that this doesn't work, because d is not in scope, you need the scoped type variables extension:

valueOfQF :: forall a. IntegerType a => QF a -> Integer valueOfQF qf = value (undefined :: a)

or maybe better, change the class:

class IntegerType a where value :: QF a -> Integer

Now you can simply use

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

Twan