non-negative (empty) → 0.0.1
raw patch · 10 files changed
+1352/−0 lines, 10 filesdep +QuickCheckdep +basesetup-changed
Dependencies added: QuickCheck, base
Files
- LICENSE +674/−0
- Setup.lhs +3/−0
- Test/Main.hs +14/−0
- Test/Numeric/NonNegative/Chunky.hs +216/−0
- Test/Utility.hs +28/−0
- non-negative.cabal +34/−0
- src/Numeric/NonNegative/Chunky.hs +152/−0
- src/Numeric/NonNegative/Class.hs +35/−0
- src/Numeric/NonNegative/Utility.hs +11/−0
- src/Numeric/NonNegative/Wrapper.hs +185/−0
+ LICENSE view
@@ -0,0 +1,674 @@+ GNU GENERAL PUBLIC LICENSE+ Version 3, 29 June 2007++ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>+ Everyone is permitted to copy and distribute verbatim copies+ of this license document, but changing it is not allowed.++ Preamble++ The GNU General Public License is a free, copyleft license for+software and other kinds of works.++ The licenses for most software and other practical works are designed+to take away your freedom to share and change the works. By contrast,+the GNU General Public License is intended to guarantee your freedom to+share and change all versions of a program--to make sure it remains free+software for all its users. We, the Free Software Foundation, use the+GNU General Public License for most of our software; it applies also to+any other work released this way by its authors. You can apply it to+your programs, too.++ When we speak of free software, we are referring to freedom, not+price. 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+ Setup.lhs view
@@ -0,0 +1,3 @@+#! /usr/bin/env runhaskell+> import Distribution.Simple+> main = defaultMain
+ Test/Main.hs view
@@ -0,0 +1,14 @@+module Main where++import qualified Test.Numeric.NonNegative.Chunky as Chunky++prefix :: String -> [(String, IO ())] -> [(String, IO ())]+prefix msg =+ map (\(str,test) -> (msg ++ "." ++ str, test))++main :: IO ()+main =+ mapM_ (\(msg,io) -> putStr (msg++": ") >> io) $+ concat $+ prefix "NonNegative.Chunky" Chunky.tests :+ []
+ Test/Numeric/NonNegative/Chunky.hs view
@@ -0,0 +1,216 @@+{- |+Copyright : (c) Henning Thielemann 2007++Maintainer : haskell@henning-thielemann.de+Stability : stable+Portability : Haskell 98+-}+module Test.Numeric.NonNegative.Chunky (tests) where++-- import Test.Utility+import Test.QuickCheck (test, Property, (==>))++import qualified Numeric.NonNegative.Chunky as Chunky+import qualified Numeric.NonNegative.Class as NonNeg+import qualified Numeric.NonNegative.Wrapper as NonNegW+-- import Numeric.NonNegative.Class ((-|))++-- import Debug.Trace (trace)++import qualified Prelude as P+import Prelude hiding (min, max, abs, signum, compare)+++infixl 5 $~++($~) :: NonNeg.C a =>+ (a -> b) -> (Chunky.T a -> b)+($~) f = f . Chunky.toNumber++infixl 4 ==~++(==~) :: NonNeg.C a =>+ Chunky.T a -> a -> Bool+(==~) xs ys =+ Chunky.toNumber xs == ys++++add :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+add x y = x+y ==~ (+) $~ x $~ y++sub :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+sub x y =+ if x>y+ then x-y ==~ (-) $~ x $~ y+ else y-x ==~ (-) $~ y $~ x++mul :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+mul x y = x*y ==~ (*) $~ x $~ y++min :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+min x y = P.min x y ==~ P.min $~ x $~ y++max :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+max x y = P.max x y ==~ P.max $~ x $~ y++abs :: NonNeg.C a => Chunky.T a -> Bool+abs x = P.abs x ==~ P.abs $~ x++signum :: NonNeg.C a => Chunky.T a -> Bool+signum x = P.signum x ==~ P.signum $~ x++equal :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+equal x y = (x == y) == (==) $~ x $~ y++compare :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Bool+compare x y = P.compare x y == P.compare $~ x $~ y+++infinity :: NonNeg.C a => a -> Chunky.T a+infinity = Chunky.fromChunks . repeat . (1+)++checkInfinity :: NonNeg.C a => a -> Chunky.T a -> Bool+checkInfinity limit x =+ let y = Chunky.fromNumber limit+ in P.min x y == y++addInfiniteL :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+addInfiniteL limit x y =+ checkInfinity limit (infinity x + y)++addInfiniteR :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+addInfiniteR limit x y =+ checkInfinity limit (y + infinity x)+++mulInfiniteL :: NonNeg.C a => a -> a -> Chunky.T a -> Property+mulInfiniteL limit x y =+ Chunky.isPositive y ==>+ checkInfinity limit (infinity x * y)++{- |+Without normalization the test would fail for+@y = Chunky.fromChunks [0,1]@+-}+mulInfiniteR :: NonNeg.C a => a -> a -> Chunky.T a -> Property+mulInfiniteR limit x y =+ Chunky.isPositive y ==>+ checkInfinity limit (Chunky.normalize y * infinity x)++{-+mulInfiniteL :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+mulInfiniteL limit x y =+ if Chunky.isNull y+ then Chunky.isNull (infinity x * y)+ else checkInfinity limit (infinity x * y)++mulInfiniteR :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+mulInfiniteR limit x y =+ if Chunky.isNull y+ then Chunky.isNull (y * infinity x)+ else checkInfinity limit (y * infinity x)+-}++minRecursiveL :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Property+minRecursiveL x y =+ Chunky.isPositive x ==>+ let z = P.min (x+z) y+ in z == y++minRecursiveR :: NonNeg.C a => Chunky.T a -> Chunky.T a -> Property+minRecursiveR x y =+ Chunky.isPositive x ==>+ let z = P.min y (x+z)+ in z == y++minInfiniteL :: NonNeg.C a => a -> Chunky.T a -> Bool+minInfiniteL x y =+ P.min (infinity x) y == y++minInfiniteR :: NonNeg.C a => a -> Chunky.T a -> Bool+minInfiniteR x y =+ P.min y (infinity x) == y++maxInfiniteL :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+maxInfiniteL limit x y =+ checkInfinity limit (P.max (infinity x) y)++maxInfiniteR :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+maxInfiniteR limit x y =+ checkInfinity limit (P.max y (infinity x))++minInfiniteAssociative :: NonNeg.C a => a -> a -> Chunky.T a -> Bool+minInfiniteAssociative x y z =+ P.min (P.min (infinity x) (infinity y)) z == z++maxInfiniteAssociative :: NonNeg.C a => a -> a -> a -> Chunky.T a -> Bool+maxInfiniteAssociative limit x y z =+ checkInfinity limit+ (P.max (P.max (infinity x) (infinity y)) z)++compareInfiniteL :: NonNeg.C a => a -> Chunky.T a -> Bool+compareInfiniteL x y =+ P.compare (infinity x) y == GT++compareInfiniteR :: NonNeg.C a => a -> Chunky.T a -> Bool+compareInfiniteR x y =+ P.compare y (infinity x) == LT++++++type Card = NonNegW.T Int++tests :: [(String, IO ())]+tests =+ ("add",+ test (add :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("sub",+ test (sub :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("mul",+ test (mul :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("min",+ test (min :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("max",+ test (max :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("abs",+ test (abs :: Chunky.T Card -> Bool)) :+ ("signum",+ test (signum :: Chunky.T Card -> Bool)) :+ ("equal",+ test (equal :: Chunky.T Card -> Chunky.T Card -> Bool)) :+ ("compare",+ test (compare :: Chunky.T Card -> Chunky.T Card -> Bool)) :++ ("addInfiniteL",+ test (addInfiniteL :: Card -> Card -> Chunky.T Card -> Bool)) :+ ("addInfiniteR",+ test (addInfiniteR :: Card -> Card -> Chunky.T Card -> Bool)) :+ ("mulInfiniteL",+ test (mulInfiniteL :: Card -> Card -> Chunky.T Card -> Property)) :+ ("mulInfiniteR",+ test (mulInfiniteR :: Card -> Card -> Chunky.T Card -> Property)) :+ ("minRecursiveL",+ test (minRecursiveL :: Chunky.T Card -> Chunky.T Card -> Property)) :+ ("minRecursiveR",+ test (minRecursiveR :: Chunky.T Card -> Chunky.T Card -> Property)) :+ ("minInfiniteL",+ test (minInfiniteL :: Card -> Chunky.T Card -> Bool)) :+ ("minInfiniteR",+ test (minInfiniteR :: Card -> Chunky.T Card -> Bool)) :+ ("maxInfiniteL",+ test (maxInfiniteL :: Card -> Card -> Chunky.T Card -> Bool)) :+ ("maxInfiniteR",+ test (maxInfiniteR :: Card -> Card -> Chunky.T Card -> Bool)) :+ ("minInfiniteAssociative",+ test (minInfiniteAssociative :: Card -> Card -> Chunky.T Card -> Bool)) :+ ("maxInfiniteAssociative",+ test (maxInfiniteAssociative :: Card -> Card -> Card -> Chunky.T Card -> Bool)) :++ ("compareInfiniteL",+ test (compareInfiniteL :: Card -> Chunky.T Card -> Bool)) :+ ("compareInfiniteR",+ test (compareInfiniteR :: Card -> Chunky.T Card -> Bool)) :+ []
+ Test/Utility.hs view
@@ -0,0 +1,28 @@+module Test.Utility where++import Test.QuickCheck (Arbitrary(..))++import qualified Numeric.NonNegative.Wrapper as NonNeg+import Control.Monad (liftM)+import Data.Char (chr)+++type TimeDiff = NonNeg.Int++timeToDouble :: TimeDiff -> NonNeg.Double+timeToDouble = fromIntegral++makeFracTime :: (TimeDiff, TimeDiff) -> NonNeg.Double+makeFracTime (n,d) =+ timeToDouble n / (timeToDouble d + 1)+++instance Arbitrary Char where+ arbitrary = liftM (chr . (32+) . flip mod 96) arbitrary+ coarbitrary = undefined++instance Arbitrary a => Arbitrary (Maybe a) where+ arbitrary =+ arbitrary >>=+ \b -> if b then fmap Just arbitrary else return Nothing+ coarbitrary = undefined
+ non-negative.cabal view
@@ -0,0 +1,34 @@+Name: non-negative+Version: 0.0.1+License: GPL+License-File: LICENSE+Author: Henning Thielemann <haskell@henning-thielemann.de>+Maintainer: Henning Thielemann <haskell@henning-thielemann.de>+Homepage: http://darcs.haskell.org/non-negative/+Category: Math+Build-Type: Simple+Build-Depends: base>=1.0, QuickCheck+Synopsis: Non-negative numbers+Description:+ Provides a class for non-negative numbers,+ a wrapper which can turn any ordered numeric type into a member of that class,+ and a lazy number type for non-negative numbers+ (a generalization of Peano numbers).+ This library is used by the 'event-list' package.+GHC-Options: -Wall+Tested-With: GHC==6.4.1+Hs-Source-Dirs: src+Exposed-Modules:+ Numeric.NonNegative.Class+ Numeric.NonNegative.Wrapper+ Numeric.NonNegative.Chunky+Other-Modules:+ Numeric.NonNegative.Utility++Executable: test+Hs-source-dirs: src, .+GHC-Options: -Wall+Main-Is: Test/Main.hs+Other-Modules:+ Test.Utility+ Test.Numeric.NonNegative.Chunky
+ src/Numeric/NonNegative/Chunky.hs view
@@ -0,0 +1,152 @@+{- |+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 Numeric.NonNegative.Chunky+ (T, fromChunks, fromNumber, toNumber, normalize, isNull, isPositive) where++import qualified Numeric.NonNegative.Class as NonNeg+import Control.Monad (liftM, liftM2)++import Test.QuickCheck (Arbitrary(..))++{- |+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.+-}+newtype T a = Cons {decons :: [a]}+++fromChunks :: NonNeg.C a => [a] -> T a+fromChunks = Cons++fromNumber :: NonNeg.C a => a -> T a+fromNumber = fromChunks . (:[])++toNumber :: NonNeg.C a => T a -> a+toNumber = sum . decons+++instance (Show a) => Show (T a) where+ showsPrec p x =+ showParen (p>10)+ (showString "Chunky.fromChunks " . showsPrec 10 (decons x))+++lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)+lift2 f (Cons x) (Cons y) = Cons $ f x y++{- |+Remove zero chunks.+-}+normalize :: NonNeg.C a => T a -> T a+normalize = Cons . filter (>0) . decons++isNullList :: NonNeg.C a => [a] -> Bool+isNullList = null . filter (>0)++isNull :: NonNeg.C a => T a -> Bool+isNull = isNullList . decons+ -- null . decons . normalize++isPositive :: NonNeg.C a => T a -> Bool+isPositive = 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+ (Cons x) -| (Cons w) =+ let sub _ [] = []+ sub z (y:ys) =+ if z<y then (y-z):ys else sub (z-y) ys+ in Cons (foldr sub x w)++instance (NonNeg.C a) => Num (T a) where+ (+) = lift2 (++)+ (Cons x) - (Cons y) =+ let (_,d,b) = glue x y+ d' = Cons d+ in check "-" (not b || isNull d') d'+ negate x = check "negate" (isNull x) x+ fromInteger = fromNumber . fromInteger+ (*) = lift2 (liftM2 (*))+ abs = id+ signum = fromNumber . (\b -> if b then 1 else 0) . isPositive+++instance (NonNeg.C a, Arbitrary a) => Arbitrary (T a) where+ arbitrary = liftM Cons arbitrary+ coarbitrary = undefined
+ src/Numeric/NonNegative/Class.hs view
@@ -0,0 +1,35 @@+{- |+Copyright : (c) Henning Thielemann 2007++Maintainer : haskell@henning-thielemann.de+Stability : stable+Portability : Haskell 98++A type class for non-negative numbers.+Prominent instances are 'Numeric.NonNegative.Wrapper.T' and peano numbers.+This class cannot do any checks,+but it let you show to the user what arguments your function expects.+In fact many standard functions ('take', '(!!)', ...)+should have this type class constraint.+Thus you must define class instances with care.+-}+module Numeric.NonNegative.Class (C(..)) where+++{- |+Instances of this class must ensure non-negative values.+We cannot enforce this by types, but the type class constraint @NonNegative.C@+avoids accidental usage of types which allow for negative numbers.+-}+class (Ord a, Num a) => C a where+ {- |+ @x -| y == max 0 (x-y)@++ The default implementation is not efficient,+ because it compares the values and then subtracts, again, if safe.+ @max 0 (x-y)@ is more elegant and efficient+ but not possible in the general case,+ since @x-y@ may already yield a negative number.+ -}+ (-|) :: a -> a -> a+ x -| y = if x >= y then x-y else 0
+ src/Numeric/NonNegative/Utility.hs view
@@ -0,0 +1,11 @@+module Numeric.NonNegative.Utility where+++mapPair :: (a -> c, b -> d) -> (a,b) -> (c,d)+mapPair ~(f,g) ~(x,y) = (f x, g y)++mapFst :: (a -> c) -> (a,b) -> (c,b)+mapFst f ~(x,y) = (f x, y)++mapSnd :: (b -> d) -> (a,b) -> (a,d)+mapSnd g ~(x,y) = (x, g y)
+ src/Numeric/NonNegative/Wrapper.hs view
@@ -0,0 +1,185 @@+{- |+Copyright : (c) Henning Thielemann 2007++Maintainer : haskell@henning-thielemann.de+Stability : stable+Portability : Haskell 98++A type for non-negative numbers.+It performs a run-time check at construction time (i.e. at run-time)+and is a member of the non-negative number type class+'Numeric.NonNegative.Class.C'.+-}+module Numeric.NonNegative.Wrapper+ (T, fromNumber, fromNumberMsg, fromNumberClip, fromNumberUnsafe, toNumber,+ Int, Integer, Float, Double, Ratio, Rational) where++import qualified Numeric.NonNegative.Class as NonNeg++import Test.QuickCheck (Arbitrary(..))+import Numeric.NonNegative.Utility(mapPair, mapSnd)+import Control.Monad (liftM)++import qualified Data.Ratio as R+import qualified Prelude as P+import Prelude hiding (Int, Integer, Float, Double, Rational)+++newtype T a = Cons {unwrap :: a}+ deriving (Eq, Ord)++instance Show a => Show (T a) where+ showsPrec p (Cons a) = showsPrec p a+++{- |+Convert a number to a non-negative number.+If a negative number is given, an error is raised.+-}+fromNumber :: (Ord a, Num a) =>+ a+ -> T a+fromNumber = fromNumberMsg "fromNumber"++fromNumberMsg :: (Ord a, Num a) =>+ String {- ^ name of the calling function to be used in the error message -}+ -> a+ -> T a+fromNumberMsg funcName x =+ if x>=0+ then Cons x+ else error (funcName++": negative number")++fromNumberWrap :: (Ord a, Num a) =>+ String+ -> a+ -> T a+fromNumberWrap funcName =+ fromNumberMsg ("NonNegative.Wrapper."++funcName)++{- |+Convert a number to a non-negative number.+A negative number will be replaced by zero.+Use this function with care since it may hide bugs.+-}+fromNumberClip :: (Ord a, Num a) =>+ a+ -> T a+fromNumberClip = Cons . max 0++{- |+Wrap a number into a non-negative number without doing checks.+This routine exists entirely for efficiency reasons+and must be used only in cases where you are absolutely sure,+that the input number is non-negative.+-}+fromNumberUnsafe ::+ a+ -> T a+fromNumberUnsafe = Cons++{-+export only this in order to disable direct access to the record field+by record update syntax+-}+toNumber :: T a -> a+toNumber = unwrap+++{- |+Results are not checked for positivity.+-}+lift :: (a -> a) -> (T a -> T a)+lift f = Cons . f . toNumber++liftWrap :: (Ord a, Num a) => String -> (a -> a) -> (T a -> T a)+liftWrap msg f = fromNumberWrap msg . f . toNumber+++{- |+Results are not checked for positivity.+-}+lift2 :: (a -> a -> a) -> (T a -> T a -> T a)+lift2 f (Cons x) (Cons y) = Cons $ f x y++++instance (Ord a, Num a) => NonNeg.C (T a) where+ (Cons x) -| (Cons y) = fromNumberClip (x-y)++instance (Ord a, Num a) => Num (T a) where+ (+) = lift2 (+)+ (Cons x) - (Cons y) = fromNumberWrap "-" (x-y)+ negate = liftWrap "negate" negate+ fromInteger x = fromNumberWrap "fromInteger" (fromInteger x)+ (*) = lift2 (*)+ abs = lift abs+ signum = lift signum++instance Real a => Real (T a) where+ toRational = toRational . toNumber++{- required for Integral instance -}+instance (Ord a, Num a, Enum a) => Enum (T a) where+ toEnum = fromNumberWrap "toEnum" . toEnum+ fromEnum = fromEnum . toNumber++instance (Ord a, Num a, Bounded a) => Bounded (T a) where+ minBound = fromNumberClip minBound+ maxBound = fromNumberWrap "maxBound" maxBound++instance Integral a => Integral (T a) where+ toInteger = toInteger . toNumber+ quot = lift2 quot+ rem = lift2 rem+ quotRem (Cons x) (Cons y) =+ mapPair (Cons, Cons) (quotRem x y)+ div = lift2 div+ mod = lift2 mod+ divMod (Cons x) (Cons y) =+ mapPair (Cons, Cons) (divMod x y)++instance (Ord a, Fractional a) => Fractional (T a) where+ fromRational = fromNumberWrap "fromRational" . fromRational+ (/) = lift2 (/)+++instance (RealFrac a) => RealFrac (T a) where+ properFraction = mapSnd fromNumberUnsafe . properFraction . toNumber+ truncate = truncate . toNumber+ round = round . toNumber+ ceiling = ceiling . toNumber+ floor = floor . toNumber++instance (Ord a, Floating a) => Floating (T a) where+ pi = fromNumber pi+ exp = lift exp+ sqrt = lift sqrt+ log = liftWrap "log" log+ (**) = lift2 (**)+ logBase (Cons x) = liftWrap "logBase" (logBase x)+ sin = liftWrap "sin" sin+ tan = liftWrap "tan" tan+ cos = liftWrap "cos" cos+ asin = liftWrap "asin" asin+ atan = liftWrap "atan" atan+ acos = liftWrap "acos" acos+ sinh = liftWrap "sinh" sinh+ tanh = liftWrap "tanh" tanh+ cosh = liftWrap "cosh" cosh+ asinh = liftWrap "asinh" asinh+ atanh = liftWrap "atanh" atanh+ acosh = liftWrap "acosh" acosh+++instance (Num a, Arbitrary a) => Arbitrary (T a) where+ arbitrary = liftM (Cons . abs) arbitrary+ coarbitrary = undefined+++type Int = T P.Int+type Integer = T P.Integer+type Ratio a = T (R.Ratio a)+type Rational = T P.Rational+type Float = T P.Float+type Double = T P.Double