```{-# OPTIONS -fglasgow-exts #-}
module Test.QuickCheck.Arbitrary
(
-- * Arbitrary and CoArbitrary classes.
Arbitrary(..)
, CoArbitrary(..)

-- ** Helper functions for implementing arbitrary
, arbitrarySizedIntegral   -- :: Num a => Gen a
, arbitrarySizedFractional -- :: Fractional a => Gen a
, arbitraryBoundedIntegral -- :: (Bounded a, Integral a) => Gen a
, arbitraryBoundedRandom   -- :: (Bounded a, Random a) => Gen a
-- ** Helper functions for implementing shrink
, shrinkNothing            -- :: 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

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

-- ** Type-level modifiers for changing generator behavior
, Blind(..)
, Fixed(..)
, OrderedList(..)
, NonEmptyList(..)
, Positive(..)
, NonZero(..)
, NonNegative(..)
, Smart(..)
, Shrinking(..)
, ShrinkState(..)
)
where

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

import Test.QuickCheck.Gen

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

import Data.Char
( chr
, ord
, isLower
)

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

import System.Random
( Random
)

import Data.List
( sort
, nub
)

( liftM
, liftM2
, liftM3
, liftM4
, liftM5
)

--------------------------------------------------------------------------
-- ** 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)

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 = removeChunks xs
++ shrinkOne xs
where
removeChunks xs = rem (length xs) xs
where
rem 0 _  = []
rem 1 _  = [[]]
rem n xs = xs1
: xs2
: ( [ xs1' ++ xs2 | xs1' <- rem n1 xs1, not (null xs1') ]
`ilv` [ xs1 ++ xs2' | xs2' <- rem n2 xs2, not (null xs2') ]
)
where
n1  = n `div` 2
xs1 = take n1 xs
n2  = n - n1
xs2 = drop n1 xs

[]     `ilv` ys     = ys
xs     `ilv` []     = xs
(x:xs) `ilv` (y:ys) = x : y : (xs `ilv` ys)

shrinkOne []     = []
shrinkOne (x:xs) = [ x':xs | x'  <- shrink x ]
++ [ x:xs' | xs' <- shrinkOne 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 (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 = arbitrarySizedIntegral
arbitrary = arbitrarySizedBoundedInt
shrink    = shrinkIntegral

instance Arbitrary Char where
arbitrary = chr `fmap` oneof [choose (0,127), choose (0,255)]
shrink c  = [ c' | c' <- ['a','b','c'], c' < c || not (isLower c) ]

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 from the entire
-- range of the type.
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 integral number from a bounded domain.
-- Inspired by demands from Phil Wadler.
arbitrarySizedBoundedInt :: Gen Int
arbitrarySizedBoundedInt =
sized \$ \s ->
do let mn = minBound
mx = maxBound `asTypeOf` mn
k  = 2^(s*2 `div` 5)
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' <- takeWhile (<< x) (0:[ x - i | i <- tail (iterate (`quot` 2) x) ])
]
where
x << y = abs x < abs y

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

--------------------------------------------------------------------------
-- ** 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 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 (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 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)

--------------------------------------------------------------------------
-- ** 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

--------------------------------------------------------------------------
-- ** arbitrary modifiers

-- These datatypes are mainly here to *pattern match* on in properties.
-- This is a stylistic alternative to using explicit quantification.
-- In other words, they should not be replaced by type synonyms, and their
-- constructors should be exported.

-- Examples:
{-
prop_TakeDropWhile (Blind p) (xs :: [A]) =           -- because functions cannot be shown
takeWhile p xs ++ dropWhile p xs == xs

prop_TakeDrop (NonNegative n) (xs :: [A]) =          -- (BTW, also works for negative n)
take n xs ++ drop n xs == xs

prop_Cycle (NonNegative n) (NonEmpty (xs :: [A])) =  -- cycle does not work for empty lists
take n (cycle xs) == take n (xs ++ cycle xs)

prop_Sort (Ordered (xs :: [OrdA])) =                 -- instead of "forAll orderedList"
sort xs == xs
-}

-- | @Blind x@: as x, but x does not have to be in the 'Show' class.
newtype Blind a = Blind a
deriving ( Eq, Ord, Num, Integral, Real, Enum )

instance Show (Blind a) where
show _ = "(*)"

instance Arbitrary a => Arbitrary (Blind a) where
arbitrary = Blind `fmap` arbitrary

shrink (Blind x) = [ Blind x' | x' <- shrink x ]

-- | @Fixed x@: as x, but will not be shrunk.
newtype Fixed a = Fixed a
deriving ( Eq, Ord, Num, Integral, Real, Enum, Show, Read )

instance Arbitrary a => Arbitrary (Fixed a) where
arbitrary = Fixed `fmap` arbitrary

-- no shrink function

-- | @Ordered xs@: guarantees that xs is ordered.
newtype OrderedList a = Ordered [a]
deriving ( Eq, Ord, Show, Read )

instance (Ord a, Arbitrary a) => Arbitrary (OrderedList a) where
arbitrary = Ordered `fmap` orderedList

shrink (Ordered xs) =
[ Ordered xs'
| xs' <- shrink xs
, sort xs' == xs'
]

-- | @NonEmpty xs@: guarantees that xs is non-empty.
newtype NonEmptyList a = NonEmpty [a]
deriving ( Eq, Ord, Show, Read )

instance Arbitrary a => Arbitrary (NonEmptyList a) where
arbitrary = NonEmpty `fmap` (arbitrary `suchThat` (not . null))

shrink (NonEmpty xs) =
[ NonEmpty xs'
| xs' <- shrink xs
, not (null xs')
]

-- | @Positive x@: guarantees that @x \> 0@.
newtype Positive a = Positive a
deriving ( Eq, Ord, Num, Integral, Real, Enum, Show, Read )

instance (Num a, Ord a, Arbitrary a) => Arbitrary (Positive a) where
arbitrary =
(Positive . abs) `fmap` (arbitrary `suchThat` (/= 0))

shrink (Positive x) =
[ Positive x'
| x' <- shrink x
, x' > 0
]

-- | @NonZero x@: guarantees that @x \/= 0@.
newtype NonZero a = NonZero a
deriving ( Eq, Ord, Num, Integral, Real, Enum, Show, Read )

instance (Num a, Ord a, Arbitrary a) => Arbitrary (NonZero a) where
arbitrary = fmap NonZero \$ arbitrary `suchThat` (/= 0)

shrink (NonZero x) = [ NonZero x' | x' <- shrink x, x' /= 0 ]

-- | @NonNegative x@: guarantees that @x \>= 0@.
newtype NonNegative a = NonNegative a
deriving ( Eq, Ord, Num, Integral, Real, Enum, Show, Read )

instance (Num a, Ord a, Arbitrary a) => Arbitrary (NonNegative a) where
arbitrary =
frequency
-- why is this distrbution like this?
[ (5, (NonNegative . abs) `fmap` arbitrary)
, (1, return 0)
]

shrink (NonNegative x) =
[ NonNegative x'
| x' <- shrink x
, x' >= 0
]

-- | @Smart _ x@: tries a different order when shrinking.
data Smart a =
Smart Int a

instance Show a => Show (Smart a) where
showsPrec n (Smart _ x) = showsPrec n x

instance Arbitrary a => Arbitrary (Smart a) where
arbitrary =
do x <- arbitrary
return (Smart 0 x)

shrink (Smart i x) = take i' ys `ilv` drop i' ys
where
ys = [ Smart i y | (i,y) <- [0..] `zip` shrink x ]
i' = 0 `max` (i-2)

[]     `ilv` bs     = bs
as     `ilv` []     = as
(a:as) `ilv` (b:bs) = a : b : (as `ilv` bs)

{-
shrink (Smart i x) = part0 ++ part2 ++ part1
where
ys = [ Smart i y | (i,y) <- [0..] `zip` shrink x ]
i' = 0 `max` (i-2)
k  = i `div` 10

part0 = take k ys
part1 = take (i'-k) (drop k ys)
part2 = drop i' ys
-}

-- drop a (drop b xs) == drop (a+b) xs           | a,b >= 0
-- take a (take b xs) == take (a `min` b) xs
-- take a xs ++ drop a xs == xs

--    take k ys ++ take (i'-k) (drop k ys) ++ drop i' ys
-- == take k ys ++ take (i'-k) (drop k ys) ++ drop (i'-k) (drop k ys)
-- == take k ys ++ take (i'-k) (drop k ys) ++ drop (i'-k) (drop k ys)
-- == take k ys ++ drop k ys
-- == ys

-- | @Shrinking _ x@: allows for maintaining a state during shrinking.
data Shrinking s a =
Shrinking s a

class ShrinkState s a where
shrinkInit  :: a -> s
shrinkState :: a -> s -> [(a,s)]

instance Show a => Show (Shrinking s a) where
showsPrec n (Shrinking _ x) = showsPrec n x

instance (Arbitrary a, ShrinkState s a) => Arbitrary (Shrinking s a) where
arbitrary =
do x <- arbitrary
return (Shrinking (shrinkInit x) x)

shrink (Shrinking s x) =
[ Shrinking s' x'
| (x',s') <- shrinkState x s
]

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