Simple field extensions in Haskell

FieldExtension.hs

module FieldExtension where

{-

This module lets you define the exact Fibonacci function using the mathematically
elegant definition in terms of phi = (1 + sqrt 5) / 2, without losing precision
due to calculations being carried out in floating point.

phi :: Q5
phi = 0.5 :+ 0.5

fib :: Integral a => a -> a
fib n = let a :+ b = phi^n in round $ 2 * b

-}

import Data.Ratio

class Nat n where
    nat :: f n -> Int

withType :: (a -> a) -> a
withType f = f undefined

withNat :: Nat n => (Int -> Ext n) -> Ext n
withNat foo = withType (foo . nat)

data Ext a = Rational :+ Rational deriving (Eq)

instance Nat a => Show (Ext a) where
    show z@(a :+ b) = show a ++ " + " ++ show b ++ " sqrt " ++ show (nat z)

instance Nat a => Num (Ext a) where

    (a :+ b) + (c :+ d) = withNat $ \_ -> (a + c) :+ (b + d)
    (a :+ b) * (c :+ d) = withNat $ \n -> (a * c + fromIntegral n * b * d) :+ (a * d + b * c) 
    (a :+ b) - (c :+ d) = withNat $ \_ -> (a - c) :+ (b - d)

    abs    (a :+ b) = abs a :+ abs b
    signum (a :+ b) = signum a :+ signum b
    fromInteger n   = fromInteger n :+ 0

data I2
data I3
data I5
data I6
data I7
data I8
data I10
data I11
data I12
data I13
data I14
data I15
data I17
data I18
data I19
data I20
data I21
data I22
data I23
data I24
data I26
data I27
data I28
data I29
data I30
data I31
data I32
data I33
data I34
data I35
data I37
data I38
data I39
data I40
data I41
data I42
data I43
data I44
data I45
data I46
data I47
data I48
data I50
data I51
data I52
data I53
data I54
data I55
data I56
data I57
data I58
data I59
data I60
data I61
data I62
data I63
data I65
data I66
data I67
data I68
data I69
data I70
data I71
data I72
data I73
data I74
data I75
data I76
data I77
data I78
data I79
data I80
data I82
data I83
data I84
data I85
data I86
data I87
data I88
data I89
data I90
data I91
data I92
data I93
data I94
data I95
data I96
data I97
data I98
data I99

instance Nat I2 where nat _ = 2
instance Nat I3 where nat _ = 3
instance Nat I5 where nat _ = 5
instance Nat I6 where nat _ = 6
instance Nat I7 where nat _ = 7
instance Nat I8 where nat _ = 8
instance Nat I10 where nat _ = 10
instance Nat I11 where nat _ = 11
instance Nat I12 where nat _ = 12
instance Nat I13 where nat _ = 13
instance Nat I14 where nat _ = 14
instance Nat I15 where nat _ = 15
instance Nat I17 where nat _ = 17
instance Nat I18 where nat _ = 18
instance Nat I19 where nat _ = 19
instance Nat I20 where nat _ = 20
instance Nat I21 where nat _ = 21
instance Nat I22 where nat _ = 22
instance Nat I23 where nat _ = 23
instance Nat I24 where nat _ = 24
instance Nat I26 where nat _ = 26
instance Nat I27 where nat _ = 27
instance Nat I28 where nat _ = 28
instance Nat I29 where nat _ = 29
instance Nat I30 where nat _ = 30
instance Nat I31 where nat _ = 31
instance Nat I32 where nat _ = 32
instance Nat I33 where nat _ = 33
instance Nat I34 where nat _ = 34
instance Nat I35 where nat _ = 35
instance Nat I37 where nat _ = 37
instance Nat I38 where nat _ = 38
instance Nat I39 where nat _ = 39
instance Nat I40 where nat _ = 40
instance Nat I41 where nat _ = 41
instance Nat I42 where nat _ = 42
instance Nat I43 where nat _ = 43
instance Nat I44 where nat _ = 44
instance Nat I45 where nat _ = 45
instance Nat I46 where nat _ = 46
instance Nat I47 where nat _ = 47
instance Nat I48 where nat _ = 48
instance Nat I50 where nat _ = 50
instance Nat I51 where nat _ = 51
instance Nat I52 where nat _ = 52
instance Nat I53 where nat _ = 53
instance Nat I54 where nat _ = 54
instance Nat I55 where nat _ = 55
instance Nat I56 where nat _ = 56
instance Nat I57 where nat _ = 57
instance Nat I58 where nat _ = 58
instance Nat I59 where nat _ = 59
instance Nat I60 where nat _ = 60
instance Nat I61 where nat _ = 61
instance Nat I62 where nat _ = 62
instance Nat I63 where nat _ = 63
instance Nat I65 where nat _ = 65
instance Nat I66 where nat _ = 66
instance Nat I67 where nat _ = 67
instance Nat I68 where nat _ = 68
instance Nat I69 where nat _ = 69
instance Nat I70 where nat _ = 70
instance Nat I71 where nat _ = 71
instance Nat I72 where nat _ = 72
instance Nat I73 where nat _ = 73
instance Nat I74 where nat _ = 74
instance Nat I75 where nat _ = 75
instance Nat I76 where nat _ = 76
instance Nat I77 where nat _ = 77
instance Nat I78 where nat _ = 78
instance Nat I79 where nat _ = 79
instance Nat I80 where nat _ = 80
instance Nat I82 where nat _ = 82
instance Nat I83 where nat _ = 83
instance Nat I84 where nat _ = 84
instance Nat I85 where nat _ = 85
instance Nat I86 where nat _ = 86
instance Nat I87 where nat _ = 87
instance Nat I88 where nat _ = 88
instance Nat I89 where nat _ = 89
instance Nat I90 where nat _ = 90
instance Nat I91 where nat _ = 91
instance Nat I92 where nat _ = 92
instance Nat I93 where nat _ = 93
instance Nat I94 where nat _ = 94
instance Nat I95 where nat _ = 95
instance Nat I96 where nat _ = 96
instance Nat I97 where nat _ = 97
instance Nat I98 where nat _ = 98
instance Nat I99 where nat _ = 99

type Q2 = Ext I2
type Q3 = Ext I3
type Q5 = Ext I5
type Q6 = Ext I6
type Q7 = Ext I7
type Q8 = Ext I8
type Q10 = Ext I10
type Q11 = Ext I11
type Q12 = Ext I12
type Q13 = Ext I13
type Q14 = Ext I14
type Q15 = Ext I15
type Q17 = Ext I17
type Q18 = Ext I18
type Q19 = Ext I19
type Q20 = Ext I20
type Q21 = Ext I21
type Q22 = Ext I22
type Q23 = Ext I23
type Q24 = Ext I24
type Q26 = Ext I26
type Q27 = Ext I27
type Q28 = Ext I28
type Q29 = Ext I29
type Q30 = Ext I30
type Q31 = Ext I31
type Q32 = Ext I32
type Q33 = Ext I33
type Q34 = Ext I34
type Q35 = Ext I35
type Q37 = Ext I37
type Q38 = Ext I38
type Q39 = Ext I39
type Q40 = Ext I40
type Q41 = Ext I41
type Q42 = Ext I42
type Q43 = Ext I43
type Q44 = Ext I44
type Q45 = Ext I45
type Q46 = Ext I46
type Q47 = Ext I47
type Q48 = Ext I48
type Q50 = Ext I50
type Q51 = Ext I51
type Q52 = Ext I52
type Q53 = Ext I53
type Q54 = Ext I54
type Q55 = Ext I55
type Q56 = Ext I56
type Q57 = Ext I57
type Q58 = Ext I58
type Q59 = Ext I59
type Q60 = Ext I60
type Q61 = Ext I61
type Q62 = Ext I62
type Q63 = Ext I63
type Q65 = Ext I65
type Q66 = Ext I66
type Q67 = Ext I67
type Q68 = Ext I68
type Q69 = Ext I69
type Q70 = Ext I70
type Q71 = Ext I71
type Q72 = Ext I72
type Q73 = Ext I73
type Q74 = Ext I74
type Q75 = Ext I75
type Q76 = Ext I76
type Q77 = Ext I77
type Q78 = Ext I78
type Q79 = Ext I79
type Q80 = Ext I80
type Q82 = Ext I82
type Q83 = Ext I83
type Q84 = Ext I84
type Q85 = Ext I85
type Q86 = Ext I86
type Q87 = Ext I87
type Q88 = Ext I88
type Q89 = Ext I89
type Q90 = Ext I90
type Q91 = Ext I91
type Q92 = Ext I92
type Q93 = Ext I93
type Q94 = Ext I94
type Q95 = Ext I95
type Q96 = Ext I96
type Q97 = Ext I97
type Q98 = Ext I98
type Q99 = Ext I99