Copyright	(c) Henning Thielemann 2007-2012
Maintainer	numericprelude@henning-thielemann.de
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell98

Number.Peano

Description

Lazy Peano numbers represent natural numbers inclusive infinity. Since they are lazily evaluated, they are optimally for use as number type of genericLength et.al.

Synopsis

data T
- = Zero
- | Succ T
infinity :: T
err :: String -> String -> a
add :: T -> T -> T
sub :: T -> T -> T
subNeg :: T -> T -> (Bool, T)
mul :: T -> T -> T
fromPosEnum :: (C a, Enum a) => a -> T
toPosEnum :: (C a, Enum a) => T -> a
ifLazy :: Bool -> T -> T -> T
argMinFull :: (T, a) -> (T, a) -> (T, a)
argMin :: (T, a) -> (T, a) -> a
argMinimum :: [(T, a)] -> a
argMaxFull :: (T, a) -> (T, a) -> (T, a)
argMax :: (T, a) -> (T, a) -> a
argMaximum :: [(T, a)] -> a
isAscendingFiniteList :: [T] -> Bool
isAscendingFiniteNumbers :: [T] -> Bool
toListMaybe :: a -> T -> [Maybe a]
glue :: T -> T -> (T, (Bool, T))
isAscending :: [T] -> Bool
data Valuable a = Valuable {
- costs :: T
- value :: a
}
increaseCosts :: T -> Valuable a -> Valuable a
(&&~) :: Valuable Bool -> Valuable Bool -> Valuable Bool
andW :: [Valuable Bool] -> Valuable Bool
leW :: T -> T -> Valuable Bool
isAscendingW :: [T] -> Valuable Bool
notImplemented :: String -> a

Documentation

data T Source #

Constructors

Zero
Succ T

Instances

Bounded T Source #
Instance details Defined in Number.Peano Methods minBound :: T # maxBound :: T #
Enum T Source #
Instance details Defined in Number.Peano Methods succ :: T -> T # pred :: T -> T # toEnum :: Int -> T # fromEnum :: T -> Int # enumFrom :: T -> [T] # enumFromThen :: T -> T -> [T] # enumFromTo :: T -> T -> [T] # enumFromThenTo :: T -> T -> T -> [T] #
Eq T Source #
Instance details Defined in Number.Peano Methods (==) :: T -> T -> Bool # (/=) :: T -> T -> Bool #
Integral T Source #
Instance details Defined in Number.Peano Methods quot :: T -> T -> T # rem :: T -> T -> T # div :: T -> T -> T # mod :: T -> T -> T # quotRem :: T -> T -> (T, T) # divMod :: T -> T -> (T, T) # toInteger :: T -> Integer #
Num T Source #
Instance details Defined in Number.Peano Methods (+) :: T -> T -> T # (-) :: T -> T -> T # (*) :: T -> T -> T # negate :: T -> T # abs :: T -> T # signum :: T -> T # fromInteger :: Integer -> T #
Ord T Source #
Instance details Defined in Number.Peano Methods compare :: T -> T -> Ordering # (<) :: T -> T -> Bool # (<=) :: T -> T -> Bool # (>) :: T -> T -> Bool # (>=) :: T -> T -> Bool # max :: T -> T -> T # min :: T -> T -> T #
Read T Source #
Instance details Defined in Number.Peano Methods readsPrec :: Int -> ReadS T # readList :: ReadS [T] # readPrec :: ReadPrec T # readListPrec :: ReadPrec [T] #
Real T Source #
Instance details Defined in Number.Peano Methods toRational :: T -> Rational #
Show T Source #
Instance details Defined in Number.Peano Methods showsPrec :: Int -> T -> ShowS # show :: T -> String # showList :: [T] -> ShowS #
Ix T Source #
Instance details Defined in Number.Peano Methods range :: (T, T) -> [T] # index :: (T, T) -> T -> Int # unsafeIndex :: (T, T) -> T -> Int inRange :: (T, T) -> T -> Bool # rangeSize :: (T, T) -> Int # unsafeRangeSize :: (T, T) -> Int
C T Source #
Instance details Defined in Number.Peano Methods compare :: T -> T -> Ordering Source #
C T Source #
Instance details Defined in Number.Peano Methods zero :: T Source # (+) :: T -> T -> T Source # (-) :: T -> T -> T Source # negate :: T -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods isZero :: T -> Bool Source #
C T Source #
Instance details Defined in Number.Peano Methods (*) :: T -> T -> T Source # one :: T Source # fromInteger :: Integer -> T Source # (^) :: T -> Integer -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods idt :: T Source # (<*>) :: T -> T -> T Source # cumulate :: [T] -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods split :: T -> T -> (T, (Bool, T)) Source #
C T Source #
Instance details Defined in Number.Peano Methods div :: T -> T -> T Source # mod :: T -> T -> T Source # divMod :: T -> T -> (T, T) Source #
C T Source #
Instance details Defined in Number.Peano Methods isUnit :: T -> Bool Source # stdAssociate :: T -> T Source # stdUnit :: T -> T Source # stdUnitInv :: T -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods extendedGCD :: T -> T -> (T, (T, T)) Source # gcd :: T -> T -> T Source # lcm :: T -> T -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods abs :: T -> T Source # signum :: T -> T Source #
C T Source #
Instance details Defined in Number.Peano Methods toRational :: T -> Rational Source #
C T Source #
Instance details Defined in Number.Peano Methods quot :: T -> T -> T Source # rem :: T -> T -> T Source # quotRem :: T -> T -> (T, T) Source #
C T Source #
Instance details Defined in Number.Peano Methods toInteger :: T -> Integer Source #

infinity :: T Source #

err :: String -> String -> a Source #

add :: T -> T -> T Source #

sub :: T -> T -> T Source #

subNeg :: T -> T -> (Bool, T) Source #

mul :: T -> T -> T Source #

fromPosEnum :: (C a, Enum a) => a -> T Source #

toPosEnum :: (C a, Enum a) => T -> a Source #

ifLazy :: Bool -> T -> T -> T Source #

If all values are completely defined, then it holds

if b then x else y == ifLazy b x y

However if b is undefined, then it is at least known that the result is larger than min x y.

argMinFull :: (T, a) -> (T, a) -> (T, a) Source #

cf. To how to find the shortest list in a list of lists efficiently, this means, also in the presence of infinite lists. http://www.haskell.org/pipermail/haskell-cafe/2006-October/018753.html

argMin :: (T, a) -> (T, a) -> a Source #

On equality the first operand is returned.

argMinimum :: [(T, a)] -> a Source #

argMaxFull :: (T, a) -> (T, a) -> (T, a) Source #

argMax :: (T, a) -> (T, a) -> a Source #

On equality the first operand is returned.

argMaximum :: [(T, a)] -> a Source #

isAscendingFiniteList :: [T] -> Bool Source #

x0 <= x1 && x1 <= x2 ... for possibly infinite numbers in finite lists.

isAscendingFiniteNumbers :: [T] -> Bool Source #

toListMaybe :: a -> T -> [Maybe a] Source #

glue :: T -> T -> (T, (Bool, T)) Source #

In glue x y == (z,(b,r)) z represents min x y, r represents max x y - min x y, and x<=y == b.

Cf. Numeric.NonNegative.Chunky

isAscending :: [T] -> Bool Source #

data Valuable a Source #

Constructors

Valuable
Fields costs :: T value :: a

Instances

Eq a => Eq (Valuable a) Source #
Instance details Defined in Number.Peano Methods (==) :: Valuable a -> Valuable a -> Bool # (/=) :: Valuable a -> Valuable a -> Bool #
Ord a => Ord (Valuable a) Source #
Instance details Defined in Number.Peano Methods compare :: Valuable a -> Valuable a -> Ordering # (<) :: Valuable a -> Valuable a -> Bool # (<=) :: Valuable a -> Valuable a -> Bool # (>) :: Valuable a -> Valuable a -> Bool # (>=) :: Valuable a -> Valuable a -> Bool # max :: Valuable a -> Valuable a -> Valuable a # min :: Valuable a -> Valuable a -> Valuable a #
Show a => Show (Valuable a) Source #
Instance details Defined in Number.Peano Methods showsPrec :: Int -> Valuable a -> ShowS # show :: Valuable a -> String # showList :: [Valuable a] -> ShowS #

increaseCosts :: T -> Valuable a -> Valuable a Source #

(&&~) :: Valuable Bool -> Valuable Bool -> Valuable Bool infixr 3 Source #

Compute '(&&)' with minimal costs.

andW :: [Valuable Bool] -> Valuable Bool Source #

leW :: T -> T -> Valuable Bool Source #

isAscendingW :: [T] -> Valuable Bool Source #

notImplemented :: String -> a Source #