{- | Copyright : (c) Henning Thielemann 2007 Maintainer : haskell@henning-thielemann.de Stability : stable Portability : Haskell 98 A lazy number type, which is a generalization of lazy Peano numbers. Comparisons can be made lazy and thus computations are possible which are impossible with strict number types, e.g. you can compute @let y = min (1+y) 2 in y@. You can even work with infinite values. However, depending on the granularity, the memory consumption is higher than that for strict number types. This number type is of interest for the merge operation of event lists, which allows for co-recursive merges. -} module Number.NonNegativeChunky (T, fromChunks, fromNumber, toNumber, fromChunky98, toChunky98, normalize, isNull, isPositive) where import qualified Numeric.NonNegative.ChunkyPrivate as Chunky98 import qualified Numeric.NonNegative.Class as NonNeg98 import qualified Algebra.Field as Field import qualified Algebra.Real as Real import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.NonNegative as NonNeg import qualified Algebra.ToInteger as ToInteger import qualified Algebra.ToRational as ToRational import qualified Algebra.IntegralDomain as Integral import qualified Algebra.RealIntegral as RealIntegral import qualified Algebra.ZeroTestable as ZeroTestable import Algebra.ZeroTestable (isZero, ) import Control.Monad (liftM, liftM2, ) import Test.QuickCheck (Arbitrary(..)) import NumericPrelude import Data.Tuple.HT (mapFst, mapPair, ) import PreludeBase import qualified Prelude as P98 {- | A chunky non-negative number is a list of non-negative numbers. It represents the sum of the list elements. It is possible to represent a finite number with infinitely many chunks by using an infinite number of zeros. Note the following problems: Addition is commutative only for finite representations. E.g. @let y = min (1+y) 2 in y@ is defined, @let y = min (y+1) 2 in y@ is not. The type is equivalent to 'Numeric.NonNegative.Chunky'. -} newtype T a = Cons {decons :: [a]} fromChunks :: NonNeg.C a => [a] -> T a fromChunks = Cons toChunks :: NonNeg.C a => T a -> [a] toChunks = decons fromChunky98 :: (NonNeg.C a, NonNeg98.C a) => Chunky98.T a -> T a fromChunky98 = fromChunks . Chunky98.toChunksUnsafe toChunky98 :: (NonNeg.C a, NonNeg98.C a) => T a -> Chunky98.T a toChunky98 = Chunky98.fromChunks . toChunks fromNumber :: NonNeg.C a => a -> T a fromNumber = fromChunks . (:[]) toNumber :: NonNeg.C a => T a -> a toNumber = sum . toChunks lift2 :: NonNeg.C a => ([a] -> [a] -> [a]) -> (T a -> T a -> T a) lift2 f x y = fromChunks $ f (toChunks x) (toChunks y) {- | Remove zero chunks. -} normalize :: NonNeg.C a => T a -> T a normalize = fromChunks . filter (>zero) . toChunks isNullList :: NonNeg.C a => [a] -> Bool isNullList = null . filter (>zero) isNull :: NonNeg.C a => T a -> Bool isNull = isNullList . toChunks -- null . toChunks . normalize isPositive :: NonNeg.C a => T a -> Bool isPositive = not . isNull {- normalizeZT :: ZeroTestable.C a => T a -> T a normalizeZT = fromChunks . filter (not . isZero) . toChunks -} isNullListZT :: ZeroTestable.C a => [a] -> Bool isNullListZT = null . filter (not . isZero) isNullZT :: ZeroTestable.C a => T a -> Bool isNullZT = isNullListZT . decons -- null . toChunks . normalize {- isPositiveZT :: ZeroTestable.C a => T a -> Bool isPositiveZT = not . isNull -} check :: String -> Bool -> a -> a check funcName b x = if b then x else error ("Numeric.NonNegative.Chunky."++funcName++": negative number") {- | In @glue x y == (z,r,b)@ @z@ represents @min x y@, @r@ represents @max x y - min x y@, and @x<y ==> b@ or @x>y ==> not b@, for @x==y@ the value of b is arbitrary. -} glue :: (NonNeg.C a) => [a] -> [a] -> ([a], [a], Bool) glue [] ys = ([], ys, True) glue xs [] = ([], xs, False) glue (x:xs) (y:ys) = let (z,(zs,rs,b)) = case compare x y of LT -> (x, glue xs ((y-x):ys)) GT -> (y, glue ((x-y):xs) ys) EQ -> (x, glue xs ys) in (z:zs,rs,b) equalList :: (NonNeg.C a) => [a] -> [a] -> Bool equalList x y = let (_,r,_) = glue x y in isNullList r compareList :: (NonNeg.C a) => [a] -> [a] -> Ordering compareList x y = let (_,r,b) = glue x y in if isNullList r then EQ else if b then LT else GT minList :: (NonNeg.C a) => [a] -> [a] -> [a] minList x y = let (z,_,_) = glue x y in z maxList :: (NonNeg.C a) => [a] -> [a] -> [a] maxList x y = let (z,r,_) = glue x y in z++r instance (NonNeg.C a) => Eq (T a) where (Cons x) == (Cons y) = equalList x y instance (NonNeg.C a) => Ord (T a) where compare (Cons x) (Cons y) = compareList x y min = lift2 minList max = lift2 maxList instance (NonNeg.C a) => NonNeg.C (T a) where (-|) = lift2 (\x w -> let sub _ [] = [] sub z (y:ys) = if z<y then (y-z):ys else sub (z-y) ys in foldr sub x w) instance (ZeroTestable.C a) => ZeroTestable.C (T a) where isZero = isNullZT instance (NonNeg.C a) => Additive.C (T a) where zero = Cons [] (+) = lift2 (++) x - y = let (_,d,b) = glue (toChunks x) (toChunks y) d' = fromChunks d in check "-" (not b || isNull d') d' negate x = check "negate" (isNull x) x {- x0 - y0 = let d' = lift2 (\x y -> let (_,d,b) = glue x y in d) x0 y0 in check "-" (not b || isNull d') d' -} instance (Ring.C a, NonNeg.C a) => Ring.C (T a) where one = fromNumber one (*) = lift2 (liftM2 (*)) fromInteger = fromNumber . fromInteger instance (Ring.C a, ZeroTestable.C a, NonNeg.C a) => Real.C (T a) where abs = id signum = fromNumber . (\b -> if b then one else zero) . isPositive instance (ToInteger.C a, NonNeg.C a) => ToInteger.C (T a) where toInteger = sum . map toInteger . toChunks instance (ToRational.C a, NonNeg.C a) => ToRational.C (T a) where toRational = sum . map toRational . toChunks instance (RealIntegral.C a, NonNeg.C a) => RealIntegral.C (T a) where quot = div rem = mod quotRem = divMod instance (Ord a, Integral.C a, NonNeg.C a) => Integral.C (T a) where divMod x0 y0 = let y = toChunks y0 recurse x = let (r,d,b) = glue y x in if not b then ([], r) else mapFst (one:) (recurse d) in mapPair (fromChunks, fromChunks) (recurse (toChunks x0)) instance (Show a) => Show (T a) where showsPrec p x = showParen (p>10) (showString "Chunky.fromChunks " . showsPrec 10 (decons x)) instance (NonNeg.C a, Arbitrary a) => Arbitrary (T a) where arbitrary = liftM Cons arbitrary coarbitrary = undefined {- * legacy instances -} legacyInstance :: a legacyInstance = error "legacy Ring.C instance for simple input of numeric literals" instance (Ring.C a, Eq a, Show a, NonNeg.C a) => P98.Num (T a) where fromInteger = fromNumber . fromInteger negate = Additive.negate -- for unary minus (+) = legacyInstance (*) = legacyInstance abs = legacyInstance signum = legacyInstance instance (Field.C a, Eq a, Show a, NonNeg.C a) => P98.Fractional (T a) where fromRational = fromNumber . fromRational (/) = legacyInstance