```module Test.QuickCheck.Arbitrary
(
-- * Arbitrary and CoArbitrary classes
Arbitrary(..)
, CoArbitrary(..)

-- ** Helper functions for implementing arbitrary
, arbitrarySizedIntegral        -- :: Num a => Gen a
, arbitraryBoundedIntegral      -- :: (Bounded a, Integral a) => Gen a
, arbitrarySizedBoundedIntegral -- :: (Bounded a, Integral a) => Gen a
, arbitrarySizedFractional      -- :: Fractional a => Gen a
, arbitraryBoundedRandom        -- :: (Bounded a, Random a) => Gen a
, arbitraryBoundedEnum          -- :: (Bounded a, Enum a) => Gen a
-- ** Helper functions for implementing shrink
, shrinkNothing            -- :: a -> [a]
, shrinkList               -- :: (a -> [a]) -> [a] -> [[a]]
, shrinkIntegral           -- :: Integral a => a -> [a]
, shrinkRealFrac           -- :: RealFrac a => a -> [a]
-- ** Helper functions for implementing coarbitrary
, (><)
, coarbitraryIntegral      -- :: Integral a => a -> Gen b -> Gen b
, coarbitraryReal          -- :: Real a => a -> Gen b -> Gen b
, coarbitraryShow          -- :: Show a => a -> Gen b -> Gen b
, coarbitraryEnum          -- :: Enum a => a -> Gen b -> Gen b

-- ** Generators which use arbitrary
, vector      -- :: Arbitrary a => Int -> Gen [a]
, orderedList -- :: (Ord a, Arbitrary a) => Gen [a]
)
where

--------------------------------------------------------------------------
-- imports

import Test.QuickCheck.Gen

{-
import Data.Generics
( (:*:)(..)
, (:+:)(..)
, Unit(..)
)
-}

import Data.Char
( chr
, ord
, isLower
, isUpper
, toLower
, isDigit
, isSpace
)

import Data.Fixed
( Fixed
, HasResolution
)

import Data.Ratio
( Ratio
, (%)
, numerator
, denominator
)

import Data.Complex
( Complex((:+)) )

import System.Random
( Random
)

import Data.List
( sort
, nub
)

( liftM
, liftM2
, liftM3
, liftM4
, liftM5
)

import Data.Int(Int8, Int16, Int32, Int64)
import Data.Word(Word, Word8, Word16, Word32, Word64)

--------------------------------------------------------------------------
-- ** class Arbitrary

-- | Random generation and shrinking of values.
class Arbitrary a where
-- | A generator for values of the given type.
arbitrary :: Gen a
arbitrary = error "no default generator"

-- | Produces a (possibly) empty list of all the possible
-- immediate shrinks of the given value.
shrink :: a -> [a]
shrink _ = []

-- instances

instance (CoArbitrary a, Arbitrary b) => Arbitrary (a -> b) where
arbitrary = promote (`coarbitrary` arbitrary)

instance Arbitrary () where
arbitrary = return ()

instance Arbitrary Bool where
arbitrary = choose (False,True)
shrink True = [False]
shrink False = []

instance Arbitrary Ordering where
arbitrary = arbitraryBoundedEnum
shrink GT = [EQ, LT]
shrink LT = [EQ]
shrink EQ = []

instance Arbitrary a => Arbitrary (Maybe a) where
arbitrary = frequency [(1, return Nothing), (3, liftM Just arbitrary)]

shrink (Just x) = Nothing : [ Just x' | x' <- shrink x ]
shrink _        = []

instance (Arbitrary a, Arbitrary b) => Arbitrary (Either a b) where
arbitrary = oneof [liftM Left arbitrary, liftM Right arbitrary]

shrink (Left x)  = [ Left  x' | x' <- shrink x ]
shrink (Right y) = [ Right y' | y' <- shrink y ]

instance Arbitrary a => Arbitrary [a] where
arbitrary = sized \$ \n ->
do k <- choose (0,n)
sequence [ arbitrary | _ <- [1..k] ]

shrink xs = shrinkList shrink xs

shrinkList :: (a -> [a]) -> [a] -> [[a]]
shrinkList shr xs = concat [ removes k n xs | k <- takeWhile (>0) (iterate (`div`2) n) ]
++ shrinkOne xs
where
n = length xs

shrinkOne []     = []
shrinkOne (x:xs) = [ x':xs | x'  <- shr x ]
++ [ x:xs' | xs' <- shrinkOne xs ]

removes k n xs
| k > n     = []
| null xs2  = [[]]
| otherwise = xs2 : map (xs1 ++) (removes k (n-k) xs2)
where
xs1 = take k xs
xs2 = drop k xs

{-
-- "standard" definition for lists:
shrink []     = []
shrink (x:xs) = [ xs ]
++ [ x:xs' | xs' <- shrink xs ]
++ [ x':xs | x'  <- shrink x ]
-}

instance (Integral a, Arbitrary a) => Arbitrary (Ratio a) where
arbitrary = arbitrarySizedFractional
shrink    = shrinkRealFrac

instance (RealFloat a, Arbitrary a) => Arbitrary (Complex a) where
arbitrary = liftM2 (:+) arbitrary arbitrary
shrink (x :+ y) = [ x' :+ y | x' <- shrink x ] ++
[ x :+ y' | y' <- shrink y ]

instance HasResolution a => Arbitrary (Fixed a) where
arbitrary = arbitrarySizedFractional
shrink    = shrinkRealFrac

instance (Arbitrary a, Arbitrary b)
=> Arbitrary (a,b)
where
arbitrary = liftM2 (,) arbitrary arbitrary

shrink (x,y) = [ (x',y) | x' <- shrink x ]
++ [ (x,y') | y' <- shrink y ]

instance (Arbitrary a, Arbitrary b, Arbitrary c)
=> Arbitrary (a,b,c)
where
arbitrary = liftM3 (,,) arbitrary arbitrary arbitrary

shrink (x,y,z) = [ (x',y,z) | x' <- shrink x ]
++ [ (x,y',z) | y' <- shrink y ]
++ [ (x,y,z') | z' <- shrink z ]

instance (Arbitrary a, Arbitrary b, Arbitrary c, Arbitrary d)
=> Arbitrary (a,b,c,d)
where
arbitrary = liftM4 (,,,) arbitrary arbitrary arbitrary arbitrary

shrink (w,x,y,z) = [ (w',x,y,z) | w' <- shrink w ]
++ [ (w,x',y,z) | x' <- shrink x ]
++ [ (w,x,y',z) | y' <- shrink y ]
++ [ (w,x,y,z') | z' <- shrink z ]

instance (Arbitrary a, Arbitrary b, Arbitrary c, Arbitrary d, Arbitrary e)
=> Arbitrary (a,b,c,d,e)
where
arbitrary = liftM5 (,,,,) arbitrary arbitrary arbitrary arbitrary arbitrary

shrink (v,w,x,y,z) = [ (v',w,x,y,z) | v' <- shrink v ]
++ [ (v,w',x,y,z) | w' <- shrink w ]
++ [ (v,w,x',y,z) | x' <- shrink x ]
++ [ (v,w,x,y',z) | y' <- shrink y ]
++ [ (v,w,x,y,z') | z' <- shrink z ]

-- typical instance for primitive (numerical) types

instance Arbitrary Integer where
arbitrary = arbitrarySizedIntegral
shrink    = shrinkIntegral

instance Arbitrary Int where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Int8 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Int16 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Int32 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Int64 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Word where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Word8 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Word16 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Word32 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Word64 where
arbitrary = arbitrarySizedBoundedIntegral
shrink    = shrinkIntegral

instance Arbitrary Char where
arbitrary = chr `fmap` oneof [choose (0,127), choose (0,255)]
shrink c  = filter (<. c) \$ nub
\$ ['a','b','c']
++ [ toLower c | isUpper c ]
++ ['A','B','C']
++ ['1','2','3']
++ [' ','\n']
where
a <. b  = stamp a < stamp b
stamp a = ( (not (isLower a)
, not (isUpper a)
, not (isDigit a))
, (not (a==' ')
, not (isSpace a)
, a)
)

instance Arbitrary Float where
arbitrary = arbitrarySizedFractional
shrink    = shrinkRealFrac

instance Arbitrary Double where
arbitrary = arbitrarySizedFractional
shrink    = shrinkRealFrac

-- ** Helper functions for implementing arbitrary

-- | Generates an integral number. The number can be positive or negative
-- and its maximum absolute value depends on the size parameter.
arbitrarySizedIntegral :: Num a => Gen a
arbitrarySizedIntegral =
sized \$ \n ->
let n' = toInteger n in
fmap fromInteger (choose (-n', n'))

-- | Generates a fractional number. The number can be positive or negative
-- and its maximum absolute value depends on the size parameter.
arbitrarySizedFractional :: Fractional a => Gen a
arbitrarySizedFractional =
sized \$ \n ->
let n' = toInteger n in
do a <- choose ((-n') * precision, n' * precision)
b <- choose (1, precision)
return (fromRational (a % b))
where
precision = 9999999999999 :: Integer

-- | Generates an integral number. The number is chosen uniformly from
-- the entire range of the type. You may want to use
arbitraryBoundedIntegral :: (Bounded a, Integral a) => Gen a
arbitraryBoundedIntegral =
do let mn = minBound
mx = maxBound `asTypeOf` mn
n <- choose (toInteger mn, toInteger mx)
return (fromInteger n `asTypeOf` mn)

-- | Generates an element of a bounded type. The element is
-- chosen from the entire range of the type.
arbitraryBoundedRandom :: (Bounded a, Random a) => Gen a
arbitraryBoundedRandom = choose (minBound,maxBound)

-- | Generates an element of a bounded enumeration.
arbitraryBoundedEnum :: (Bounded a, Enum a) => Gen a
arbitraryBoundedEnum =
do let mn = minBound
mx = maxBound `asTypeOf` mn
return (toEnum n `asTypeOf` mn)

-- | Generates an integral number from a bounded domain. The number is
-- chosen from the entire range of the type, but small numbers are
-- generated more often than big numbers. Inspired by demands from
arbitrarySizedBoundedIntegral :: (Bounded a, Integral a) => Gen a
arbitrarySizedBoundedIntegral =
sized \$ \s ->
do let mn = minBound
mx = maxBound `asTypeOf` mn
bits n | n `quot` 2 == 0 = 0
| otherwise = 1 + bits (n `quot` 2)
k  = 2^(s*(bits mn `max` bits mx `max` 40) `div` 100)
n <- choose (toInteger mn `max` (-k), toInteger mx `min` k)
return (fromInteger n `asTypeOf` mn)

-- ** Helper functions for implementing shrink

-- | Returns no shrinking alternatives.
shrinkNothing :: a -> [a]
shrinkNothing _ = []

-- | Shrink an integral number.
shrinkIntegral :: Integral a => a -> [a]
shrinkIntegral x =
nub \$
[ -x
| x < 0, -x > x
] ++
[ x'
| x' <- takeWhile (<< x) (0:[ x - i | i <- tail (iterate (`quot` 2) x) ])
]
where
-- a << b is "morally" abs a < abs b, but taking care of overflow.
a << b = case (a >= 0, b >= 0) of
(True,  True)  -> a < b
(False, False) -> a > b
(True,  False) -> a + b < 0
(False, True)  -> a + b > 0

-- | Shrink a fraction.
shrinkRealFrac :: RealFrac a => a -> [a]
shrinkRealFrac x =
nub \$
[ -x
| x < 0
] ++
[ x'
| x' <- [fromInteger (truncate x)]
, x' << x
]
where
a << b = abs a < abs b

--------------------------------------------------------------------------
-- ** CoArbitrary

-- | Used for random generation of functions.
class CoArbitrary a where
-- | Used to generate a function of type @a -> c@. The implementation
-- should use the first argument to perturb the random generator
-- given as the second argument. the returned generator
-- is then used to generate the function result.
-- You can often use 'variant' and '><' to implement
-- 'coarbitrary'.
coarbitrary :: a -> Gen c -> Gen c

{-
-- GHC definition:
coarbitrary{| Unit |}    Unit      = id
coarbitrary{| a :*: b |} (x :*: y) = coarbitrary x >< coarbitrary y
coarbitrary{| a :+: b |} (Inl x)   = variant 0    . coarbitrary x
coarbitrary{| a :+: b |} (Inr y)   = variant (-1) . coarbitrary y
-}

-- | Combine two generator perturbing functions, for example the
-- results of calls to 'variant' or 'coarbitrary'.
(><) :: (Gen a -> Gen a) -> (Gen a -> Gen a) -> (Gen a -> Gen a)
(><) f g gen =
do n <- arbitrary
(g . variant (n :: Int) . f) gen

-- for the sake of non-GHC compilers, I have added definitions
-- for coarbitrary here.

instance (Arbitrary a, CoArbitrary b) => CoArbitrary (a -> b) where
coarbitrary f gen =
do xs <- arbitrary
coarbitrary (map f xs) gen

instance CoArbitrary () where
coarbitrary _ = id

instance CoArbitrary Bool where
coarbitrary False = variant 0
coarbitrary True  = variant (-1)

instance CoArbitrary Ordering where
coarbitrary GT = variant 1
coarbitrary EQ = variant 0
coarbitrary LT = variant (-1)

instance CoArbitrary a => CoArbitrary (Maybe a) where
coarbitrary Nothing  = variant 0
coarbitrary (Just x) = variant (-1) . coarbitrary x

instance (CoArbitrary a, CoArbitrary b) => CoArbitrary (Either a b) where
coarbitrary (Left x)  = variant 0    . coarbitrary x
coarbitrary (Right y) = variant (-1) . coarbitrary y

instance CoArbitrary a => CoArbitrary [a] where
coarbitrary []     = variant 0
coarbitrary (x:xs) = variant (-1) . coarbitrary (x,xs)

instance (Integral a, CoArbitrary a) => CoArbitrary (Ratio a) where
coarbitrary r = coarbitrary (numerator r,denominator r)

instance HasResolution a => CoArbitrary (Fixed a) where
coarbitrary = coarbitraryReal

instance (RealFloat a, CoArbitrary a) => CoArbitrary (Complex a) where
coarbitrary (x :+ y) = coarbitrary x >< coarbitrary y

instance (CoArbitrary a, CoArbitrary b)
=> CoArbitrary (a,b)
where
coarbitrary (x,y) = coarbitrary x
>< coarbitrary y

instance (CoArbitrary a, CoArbitrary b, CoArbitrary c)
=> CoArbitrary (a,b,c)
where
coarbitrary (x,y,z) = coarbitrary x
>< coarbitrary y
>< coarbitrary z

instance (CoArbitrary a, CoArbitrary b, CoArbitrary c, CoArbitrary d)
=> CoArbitrary (a,b,c,d)
where
coarbitrary (x,y,z,v) = coarbitrary x
>< coarbitrary y
>< coarbitrary z
>< coarbitrary v

instance (CoArbitrary a, CoArbitrary b, CoArbitrary c, CoArbitrary d, CoArbitrary e)
=> CoArbitrary (a,b,c,d,e)
where
coarbitrary (x,y,z,v,w) = coarbitrary x
>< coarbitrary y
>< coarbitrary z
>< coarbitrary v
>< coarbitrary w

-- typical instance for primitive (numerical) types

instance CoArbitrary Integer where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Int where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Int8 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Int16 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Int32 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Int64 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Word where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Word8 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Word16 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Word32 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Word64 where
coarbitrary = coarbitraryIntegral

instance CoArbitrary Char where
coarbitrary = coarbitrary . ord

instance CoArbitrary Float where
coarbitrary = coarbitraryReal

instance CoArbitrary Double where
coarbitrary = coarbitraryReal

-- ** Helpers for implementing coarbitrary

-- | A 'coarbitrary' implementation for integral numbers.
coarbitraryIntegral :: Integral a => a -> Gen b -> Gen b
coarbitraryIntegral = variant

-- | A 'coarbitrary' implementation for real numbers.
coarbitraryReal :: Real a => a -> Gen b -> Gen b
coarbitraryReal x = coarbitrary (toRational x)

-- | 'coarbitrary' helper for lazy people :-).
coarbitraryShow :: Show a => a -> Gen b -> Gen b
coarbitraryShow x = coarbitrary (show x)

-- | A 'coarbitrary' implementation for enums.
coarbitraryEnum :: Enum a => a -> Gen b -> Gen b

--------------------------------------------------------------------------
-- ** arbitrary generators

-- these are here and not in Gen because of the Arbitrary class constraint

-- | Generates a list of a given length.
vector :: Arbitrary a => Int -> Gen [a]
vector k = vectorOf k arbitrary

-- | Generates an ordered list of a given length.
orderedList :: (Ord a, Arbitrary a) => Gen [a]
orderedList = sort `fmap` arbitrary

--------------------------------------------------------------------------
-- the end.
```