module Data.Number.Nat1
(
Nat1(..),
cmpNat1LT, invOrd, minusNat1, fromNat1, toNat1
) where
import Prelude hiding ( Int )
import Data.Ratio ( (%) )
data Nat1
= IHi
| O Nat1
| I Nat1
deriving Eq
instance Show Nat1 where
show = show . fromEnum
instance Read Nat1 where
readsPrec n = map (\(x,str) -> (toEnum x,str)) . readsPrec n
instance Ord Nat1 where
compare IHi IHi = EQ
compare IHi (O _) = LT
compare IHi (I _) = LT
compare (O _) IHi = GT
compare (I _) IHi = GT
compare (O x) (O y) = compare x y
compare (I x) (I y) = compare x y
compare (O x) (I y) = cmpNat1LT x y
compare (I x) (O y) = invOrd (cmpNat1LT y x)
x < y = cmpNat1LT y x == GT
x > y = cmpNat1LT x y == GT
x <= y = cmpNat1LT x y == LT
x >= y = cmpNat1LT y x == LT
cmpNat1LT :: Nat1 -> Nat1 -> Ordering
cmpNat1LT IHi _ = LT
cmpNat1LT (O _) IHi = GT
cmpNat1LT (I _) IHi = GT
cmpNat1LT (O x) (O y) = cmpNat1LT x y
cmpNat1LT (I x) (I y) = cmpNat1LT x y
cmpNat1LT (O x) (I y) = cmpNat1LT x y
cmpNat1LT (I x) (O y) = invOrd (cmpNat1LT y x)
invOrd :: Ordering -> Ordering
invOrd EQ = EQ
invOrd LT = GT
invOrd GT = LT
instance Enum Nat1 where
succ (O bs) = I bs
succ (I bs) = O (succ bs)
succ IHi = O IHi
pred IHi = error "predecessor of 1"
pred (O IHi) = IHi
pred (O x@(O _)) = I (pred x)
pred (O (I x)) = I (O x)
pred (I x) = O x
fromEnum = fromNat1
toEnum = toNat1
instance Num Nat1 where
O x + O y = O (x + y)
O x + I y = I (x + y)
O x + IHi = I x
I x + O y = I (x + y)
I x + I y = O (succ x + y)
I x + IHi = O (succ x)
IHi + y = succ y
x y =
case minusNat1 x y of
IHi -> error "result zero in (-)"
n -> pred n
IHi * y = y
I x * y = O (y * x) + y
O x * y = O (x * y)
negate = error "no non-positive numbers in Nat1"
abs = id
signum = const IHi
fromInteger = toNat1
minusNat1 :: Nat1 -> Nat1 -> Nat1
minusNat1 x IHi = x
minusNat1 IHi (O _) = error "negative result in (-)"
minusNat1 IHi (I _) = error "negative result in (-)"
minusNat1 (O x) (O y) = pred (O $! minusNat1 x y)
minusNat1 (O x) (I y) = O $! pred (minusNat1 x y)
minusNat1 (I x) (O y) = O $! minusNat1 x y
minusNat1 (I x) (I y) = pred (O $! minusNat1 x y)
instance Real Nat1 where
toRational n = fromNat1 n % 1
fromNat1 :: Num n => Nat1 -> n
fromNat1 IHi = 1
fromNat1 (O n) = 2 * fromNat1 n
fromNat1 (I n) = 2 * fromNat1 n + 1
toNat1 :: (Integral n,Num n) => n -> Nat1
toNat1 n
| n<0 = error "fromInteger/toEnum of negative number"
| n==0 = error "fromInteger/toEnum of zero"
| n==1 = IHi
| even n = O (toNat1 (n `div` 2))
| otherwise = I (toNat1 (n `div` 2))