Data/Map/Base.hs

{-# LANGUAGE CPP #-}
#if __GLASGOW_HASKELL__
{-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-}
#endif
#if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
{-# LANGUAGE Trustworthy #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module : Data.Map.Base
-- Copyright : (c) Daan Leijen 2002
-- (c) Andriy Palamarchuk 2008
-- License : BSD-style
-- Maintainer : libraries@haskell.org
-- Stability : provisional
-- Portability : portable
--
-- An efficient implementation of maps from keys to values (dictionaries).
--
-- Since many function names (but not the type name) clash with
-- "Prelude" names, this module is usually imported @qualified@, e.g.
--
-- > import Data.Map (Map)
-- > import qualified Data.Map as Map
--
-- The implementation of 'Map' is based on /size balanced/ binary trees (or
-- trees of /bounded balance/) as described by:
--
-- * Stephen Adams, \"/Efficient sets: a balancing act/\",
-- Journal of Functional Programming 3(4):553-562, October 1993,
-- <http://www.swiss.ai.mit.edu/~adams/BB/>.
--
-- * J. Nievergelt and E.M. Reingold,
-- \"/Binary search trees of bounded balance/\",
-- SIAM journal of computing 2(1), March 1973.
--
-- Note that the implementation is /left-biased/ -- the elements of a
-- first argument are always preferred to the second, for example in
-- 'union' or 'insert'.
--
-- Operation comments contain the operation time complexity in
-- the Big-O notation <http://en.wikipedia.org/wiki/Big_O_notation>.
-----------------------------------------------------------------------------

-- [Note: Using INLINABLE]
-- ~~~~~~~~~~~~~~~~~~~~~~~
-- It is crucial to the performance that the functions specialize on the Ord
-- type when possible. GHC 7.0 and higher does this by itself when it sees th
-- unfolding of a function -- that is why all public functions are marked
-- INLINABLE (that exposes the unfolding).


-- [Note: Using INLINE]
-- ~~~~~~~~~~~~~~~~~~~~
-- For other compilers and GHC pre 7.0, we mark some of the functions INLINE.
-- We mark the functions that just navigate down the tree (lookup, insert,
-- delete and similar). That navigation code gets inlined and thus specialized
-- when possible. There is a price to pay -- code growth. The code INLINED is
-- therefore only the tree navigation, all the real work (rebalancing) is not
-- INLINED by using a NOINLINE.
--
-- All methods marked INLINE have to be nonrecursive -- a 'go' function doing
-- the real work is provided.


-- [Note: Type of local 'go' function]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- If the local 'go' function uses an Ord class, it sometimes heap-allocates
-- the Ord dictionary when the 'go' function does not have explicit type.
-- In that case we give 'go' explicit type. But this slightly decrease
-- performance, as the resulting 'go' function can float out to top level.


-- [Note: Local 'go' functions and capturing]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- As opposed to IntMap, when 'go' function captures an argument, increased
-- heap-allocation can occur: sometimes in a polymorphic function, the 'go'
-- floats out of its enclosing function and then it heap-allocates the
-- dictionary and the argument. Maybe it floats out too late and strictness
-- analyzer cannot see that these could be passed on stack.
--
-- For example, change 'member' so that its local 'go' function is not passing
-- argument k and then look at the resulting code for hedgeInt.


-- [Note: Order of constructors]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- The order of constructors of Map matters when considering performance.
-- Currently in GHC 7.0, when type has 2 constructors, a forward conditional
-- jump is made when successfully matching second constructor. Successful match
-- of first constructor results in the forward jump not taken.
-- On GHC 7.0, reordering constructors from Tip | Bin to Bin | Tip
-- improves the benchmark by up to 10% on x86.

module Data.Map.Base (
 -- * Map type
 Map(..) -- instance Eq,Show,Read

 -- * Operators
 , (!), (\\)

 -- * Query
 , null
 , size
 , member
 , notMember
 , lookup
 , findWithDefault
 , lookupLT
 , lookupGT
 , lookupLE
 , lookupGE

 -- * Construction
 , empty
 , singleton

 -- ** Insertion
 , insert
 , insertWith
 , insertWithKey
 , insertLookupWithKey

 -- ** Delete\/Update
 , delete
 , adjust
 , adjustWithKey
 , update
 , updateWithKey
 , updateLookupWithKey
 , alter

 -- * Combine

 -- ** Union
 , union
 , unionWith
 , unionWithKey
 , unions
 , unionsWith

 -- ** Difference
 , difference
 , differenceWith
 , differenceWithKey

 -- ** Intersection
 , intersection
 , intersectionWith
 , intersectionWithKey

 -- ** Universal combining function
 , mergeWithKey

 -- * Traversal
 -- ** Map
 , map
 , mapWithKey
 , traverseWithKey
 , mapAccum
 , mapAccumWithKey
 , mapAccumRWithKey
 , mapKeys
 , mapKeysWith
 , mapKeysMonotonic

 -- * Folds
 , foldr
 , foldl
 , foldrWithKey
 , foldlWithKey
 -- ** Strict folds
 , foldr'
 , foldl'
 , foldrWithKey'
 , foldlWithKey'

 -- * Conversion
 , elems
 , keys
 , assocs
 , keysSet
 , fromSet

 -- ** Lists
 , toList
 , fromList
 , fromListWith
 , fromListWithKey

 -- ** Ordered lists
 , toAscList
 , toDescList
 , fromAscList
 , fromAscListWith
 , fromAscListWithKey
 , fromDistinctAscList

 -- * Filter
 , filter
 , filterWithKey
 , partition
 , partitionWithKey

 , mapMaybe
 , mapMaybeWithKey
 , mapEither
 , mapEitherWithKey

 , split
 , splitLookup

 -- * Submap
 , isSubmapOf, isSubmapOfBy
 , isProperSubmapOf, isProperSubmapOfBy

 -- * Indexed
 , lookupIndex
 , findIndex
 , elemAt
 , updateAt
 , deleteAt

 -- * Min\/Max
 , findMin
 , findMax
 , deleteMin
 , deleteMax
 , deleteFindMin
 , deleteFindMax
 , updateMin
 , updateMax
 , updateMinWithKey
 , updateMaxWithKey
 , minView
 , maxView
 , minViewWithKey
 , maxViewWithKey

 -- * Debugging
 , showTree
 , showTreeWith
 , valid

 -- Used by the strict version
 , bin
 , balance
 , balanced
 , balanceL
 , balanceR
 , delta
 , join
 , merge
 , glue
 , trim
 , trimLookupLo
 , foldlStrict
 , MaybeS(..)
 , filterGt
 , filterLt
 ) where

import Prelude hiding (lookup,map,filter,foldr,foldl,null)
import qualified Data.Set.Base as Set
import Data.StrictPair
import Data.Monoid (Monoid(..))
import Control.Applicative (Applicative(..), (<$>))
import Data.Traversable (Traversable(traverse))
import qualified Data.Foldable as Foldable
import Data.Typeable
import Control.DeepSeq (NFData(rnf))

#if __GLASGOW_HASKELL__
import GHC.Exts ( build )
import Text.Read
import Data.Data
#endif

-- Use macros to define strictness of functions.
-- STRICT_x_OF_y denotes an y-ary function strict in the x-th parameter.
-- We do not use BangPatterns, because they are not in any standard and we
-- want the compilers to be compiled by as many compilers as possible.
#define STRICT_1_OF_2(fn) fn arg _ | arg `seq` False = undefined
#define STRICT_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined
#define STRICT_2_OF_3(fn) fn _ arg _ | arg `seq` False = undefined
#define STRICT_1_OF_4(fn) fn arg _ _ _ | arg `seq` False = undefined
#define STRICT_2_OF_4(fn) fn _ arg _ _ | arg `seq` False = undefined

{--------------------------------------------------------------------
 Operators
--------------------------------------------------------------------}
infixl 9 !,\\ --

-- | /O(log n)/. Find the value at a key.
-- Calls 'error' when the element can not be found.
--
-- > fromList [(5,'a'), (3,'b')] ! 1 Error: element not in the map
-- > fromList [(5,'a'), (3,'b')] ! 5 == 'a'

(!) :: Ord k => Map k a -> k -> a
m ! k = find k m
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE (!) #-}
#endif

-- | Same as 'difference'.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
m1 \\ m2 = difference m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE (\\) #-}
#endif

{--------------------------------------------------------------------
 Size balanced trees.
--------------------------------------------------------------------}
-- | A Map from keys @k@ to values @a@.

-- See Note: Order of constructors
data Map k a = Bin {-# UNPACK #-} !Size !k a !(Map k a) !(Map k a)
 | Tip

type Size = Int

instance (Ord k) => Monoid (Map k v) where
 mempty = empty
 mappend = union
 mconcat = unions

#if __GLASGOW_HASKELL__

{--------------------------------------------------------------------
 A Data instance
--------------------------------------------------------------------}

-- This instance preserves data abstraction at the cost of inefficiency.
-- We omit reflection services for the sake of data abstraction.

instance (Data k, Data a, Ord k) => Data (Map k a) where
 gfoldl f z m = z fromList `f` toList m
 toConstr _ = error "toConstr"
 gunfold _ _ = error "gunfold"
 dataTypeOf _ = mkNoRepType "Data.Map.Map"
 dataCast2 f = gcast2 f

#endif

{--------------------------------------------------------------------
 Query
--------------------------------------------------------------------}
-- | /O(1)/. Is the map empty?
--
-- > Data.Map.null (empty) == True
-- > Data.Map.null (singleton 1 'a') == False

null :: Map k a -> Bool
null Tip = True
null (Bin {}) = False
{-# INLINE null #-}

-- | /O(1)/. The number of elements in the map.
--
-- > size empty == 0
-- > size (singleton 1 'a') == 1
-- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3

size :: Map k a -> Int
size Tip = 0
size (Bin sz _ _ _ _) = sz
{-# INLINE size #-}


-- | /O(log n)/. Lookup the value at a key in the map.
--
-- The function will return the corresponding value as @('Just' value)@,
-- or 'Nothing' if the key isn't in the map.
--
-- An example of using @lookup@:
--
-- > import Prelude hiding (lookup)
-- > import Data.Map
-- >
-- > employeeDept = fromList([("John","Sales"), ("Bob","IT")])
-- > deptCountry = fromList([("IT","USA"), ("Sales","France")])
-- > countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])
-- >
-- > employeeCurrency :: String -> Maybe String
-- > employeeCurrency name = do
-- > dept <- lookup name employeeDept
-- > country <- lookup dept deptCountry
-- > lookup country countryCurrency
-- >
-- > main = do
-- > putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))
-- > putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
--
-- The output of this program:
--
-- > John's currency: Just "Euro"
-- > Pete's currency: Nothing
lookup :: Ord k => k -> Map k a -> Maybe a
lookup = go
 where
 STRICT_1_OF_2(go)
 go _ Tip = Nothing
 go k (Bin _ kx x l r) = case compare k kx of
 LT -> go k l
 GT -> go k r
 EQ -> Just x
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookup #-}
#else
{-# INLINE lookup #-}
#endif

-- | /O(log n)/. Is the key a member of the map? See also 'notMember'.
--
-- > member 5 (fromList [(5,'a'), (3,'b')]) == True
-- > member 1 (fromList [(5,'a'), (3,'b')]) == False
member :: Ord k => k -> Map k a -> Bool
member = go
 where
 STRICT_1_OF_2(go)
 go _ Tip = False
 go k (Bin _ kx _ l r) = case compare k kx of
 LT -> go k l
 GT -> go k r
 EQ -> True
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE member #-}
#else
{-# INLINE member #-}
#endif

-- | /O(log n)/. Is the key not a member of the map? See also 'member'.
--
-- > notMember 5 (fromList [(5,'a'), (3,'b')]) == False
-- > notMember 1 (fromList [(5,'a'), (3,'b')]) == True

notMember :: Ord k => k -> Map k a -> Bool
notMember k m = not $ member k m
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE notMember #-}
#else
{-# INLINE notMember #-}
#endif

-- | /O(log n)/. Find the value at a key.
-- Calls 'error' when the element can not be found.
find :: Ord k => k -> Map k a -> a
find = go
 where
 STRICT_1_OF_2(go)
 go _ Tip = error "Map.!: given key is not an element in the map"
 go k (Bin _ kx x l r) = case compare k kx of
 LT -> go k l
 GT -> go k r
 EQ -> x
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE find #-}
#else
{-# INLINE find #-}
#endif

-- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns
-- the value at key @k@ or returns default value @def@
-- when the key is not in the map.
--
-- > findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
-- > findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
findWithDefault :: Ord k => a -> k -> Map k a -> a
findWithDefault = go
 where
 STRICT_2_OF_3(go)
 go def _ Tip = def
 go def k (Bin _ kx x l r) = case compare k kx of
 LT -> go def k l
 GT -> go def k r
 EQ -> x
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE findWithDefault #-}
#else
{-# INLINE findWithDefault #-}
#endif

-- | /O(log n)/. Find largest key smaller than the given one and return the
-- corresponding (key, value) pair.
--
-- > lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
-- > lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)
lookupLT = goNothing
 where
 STRICT_1_OF_2(goNothing)
 goNothing _ Tip = Nothing
 goNothing k (Bin _ kx x l r) | k <= kx = goNothing k l
 | otherwise = goJust k kx x r

 STRICT_1_OF_4(goJust)
 goJust _ kx' x' Tip = Just (kx', x')
 goJust k kx' x' (Bin _ kx x l r) | k <= kx = goJust k kx' x' l
 | otherwise = goJust k kx x r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookupLT #-}
#else
{-# INLINE lookupLT #-}
#endif

-- | /O(log n)/. Find smallest key greater than the given one and return the
-- corresponding (key, value) pair.
--
-- > lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
-- > lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)
lookupGT = goNothing
 where
 STRICT_1_OF_2(goNothing)
 goNothing _ Tip = Nothing
 goNothing k (Bin _ kx x l r) | k < kx = goJust k kx x l
 | otherwise = goNothing k r

 STRICT_1_OF_4(goJust)
 goJust _ kx' x' Tip = Just (kx', x')
 goJust k kx' x' (Bin _ kx x l r) | k < kx = goJust k kx x l
 | otherwise = goJust k kx' x' r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookupGT #-}
#else
{-# INLINE lookupGT #-}
#endif

-- | /O(log n)/. Find largest key smaller or equal to the given one and return
-- the corresponding (key, value) pair.
--
-- > lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
-- > lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
-- > lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)
lookupLE = goNothing
 where
 STRICT_1_OF_2(goNothing)
 goNothing _ Tip = Nothing
 goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goNothing k l
 EQ -> Just (kx, x)
 GT -> goJust k kx x r

 STRICT_1_OF_4(goJust)
 goJust _ kx' x' Tip = Just (kx', x')
 goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx' x' l
 EQ -> Just (kx, x)
 GT -> goJust k kx x r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookupLE #-}
#else
{-# INLINE lookupLE #-}
#endif

-- | /O(log n)/. Find smallest key greater or equal to the given one and return
-- the corresponding (key, value) pair.
--
-- > lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
-- > lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
-- > lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)
lookupGE = goNothing
 where
 STRICT_1_OF_2(goNothing)
 goNothing _ Tip = Nothing
 goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l
 EQ -> Just (kx, x)
 GT -> goNothing k r

 STRICT_1_OF_4(goJust)
 goJust _ kx' x' Tip = Just (kx', x')
 goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l
 EQ -> Just (kx, x)
 GT -> goJust k kx' x' r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookupGE #-}
#else
{-# INLINE lookupGE #-}
#endif

{--------------------------------------------------------------------
 Construction
--------------------------------------------------------------------}
-- | /O(1)/. The empty map.
--
-- > empty == fromList []
-- > size empty == 0

empty :: Map k a
empty = Tip
{-# INLINE empty #-}

-- | /O(1)/. A map with a single element.
--
-- > singleton 1 'a' == fromList [(1, 'a')]
-- > size (singleton 1 'a') == 1

singleton :: k -> a -> Map k a
singleton k x = Bin 1 k x Tip Tip
{-# INLINE singleton #-}

{--------------------------------------------------------------------
 Insertion
--------------------------------------------------------------------}
-- | /O(log n)/. Insert a new key and value in the map.
-- If the key is already present in the map, the associated value is
-- replaced with the supplied value. 'insert' is equivalent to
-- @'insertWith' 'const'@.
--
-- > insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
-- > insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
-- > insert 5 'x' empty == singleton 5 'x'

-- See Note: Type of local 'go' function
insert :: Ord k => k -> a -> Map k a -> Map k a
insert = go
 where
 go :: Ord k => k -> a -> Map k a -> Map k a
 STRICT_1_OF_3(go)
 go kx x Tip = singleton kx x
 go kx x (Bin sz ky y l r) =
 case compare kx ky of
 LT -> balanceL ky y (go kx x l) r
 GT -> balanceR ky y l (go kx x r)
 EQ -> Bin sz kx x l r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE insert #-}
#else
{-# INLINE insert #-}
#endif

-- Insert a new key and value in the map if it is not already present.
-- Used by `union`.

-- See Note: Type of local 'go' function
insertR :: Ord k => k -> a -> Map k a -> Map k a
insertR = go
 where
 go :: Ord k => k -> a -> Map k a -> Map k a
 STRICT_1_OF_3(go)
 go kx x Tip = singleton kx x
 go kx x t@(Bin _ ky y l r) =
 case compare kx ky of
 LT -> balanceL ky y (go kx x l) r
 GT -> balanceR ky y l (go kx x r)
 EQ -> t
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE insertR #-}
#else
{-# INLINE insertR #-}
#endif

-- | /O(log n)/. Insert with a function, combining new value and old value.
-- @'insertWith' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
-- insert the pair @(key, f new_value old_value)@.
--
-- > insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
-- > insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
-- > insertWith (++) 5 "xxx" empty == singleton 5 "xxx"

insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWith f = insertWithKey (\_ x' y' -> f x' y')
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE insertWith #-}
#else
{-# INLINE insertWith #-}
#endif

-- | /O(log n)/. Insert with a function, combining key, new value and old value.
-- @'insertWithKey' f key value mp@
-- will insert the pair (key, value) into @mp@ if key does
-- not exist in the map. If the key does exist, the function will
-- insert the pair @(key,f key new_value old_value)@.
-- Note that the key passed to f is the same key passed to 'insertWithKey'.
--
-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
-- > insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
-- > insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
-- > insertWithKey f 5 "xxx" empty == singleton 5 "xxx"

-- See Note: Type of local 'go' function
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
insertWithKey = go
 where
 go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
 STRICT_2_OF_4(go)
 go _ kx x Tip = singleton kx x
 go f kx x (Bin sy ky y l r) =
 case compare kx ky of
 LT -> balanceL ky y (go f kx x l) r
 GT -> balanceR ky y l (go f kx x r)
 EQ -> Bin sy kx (f kx x y) l r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE insertWithKey #-}
#else
{-# INLINE insertWithKey #-}
#endif

-- | /O(log n)/. Combines insert operation with old value retrieval.
-- The expression (@'insertLookupWithKey' f k x map@)
-- is a pair where the first element is equal to (@'lookup' k map@)
-- and the second element equal to (@'insertWithKey' f k x map@).
--
-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
-- > insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
-- > insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")])
-- > insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")
--
-- This is how to define @insertLookup@ using @insertLookupWithKey@:
--
-- > let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t
-- > insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
-- > insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")])

-- See Note: Type of local 'go' function
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a
 -> (Maybe a, Map k a)
insertLookupWithKey = go
 where
 go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)
 STRICT_2_OF_4(go)
 go _ kx x Tip = (Nothing, singleton kx x)
 go f kx x (Bin sy ky y l r) =
 case compare kx ky of
 LT -> let (found, l') = go f kx x l
 in (found, balanceL ky y l' r)
 GT -> let (found, r') = go f kx x r
 in (found, balanceR ky y l r')
 EQ -> (Just y, Bin sy kx (f kx x y) l r)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE insertLookupWithKey #-}
#else
{-# INLINE insertLookupWithKey #-}
#endif

{--------------------------------------------------------------------
 Deletion
--------------------------------------------------------------------}
-- | /O(log n)/. Delete a key and its value from the map. When the key is not
-- a member of the map, the original map is returned.
--
-- > delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
-- > delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > delete 5 empty == empty

-- See Note: Type of local 'go' function
delete :: Ord k => k -> Map k a -> Map k a
delete = go
 where
 go :: Ord k => k -> Map k a -> Map k a
 STRICT_1_OF_2(go)
 go _ Tip = Tip
 go k (Bin _ kx x l r) =
 case compare k kx of
 LT -> balanceR kx x (go k l) r
 GT -> balanceL kx x l (go k r)
 EQ -> glue l r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE delete #-}
#else
{-# INLINE delete #-}
#endif

-- | /O(log n)/. Update a value at a specific key with the result of the provided function.
-- When the key is not
-- a member of the map, the original map is returned.
--
-- > adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
-- > adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > adjust ("new " ++) 7 empty == empty

adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a
adjust f = adjustWithKey (\_ x -> f x)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE adjust #-}
#else
{-# INLINE adjust #-}
#endif

-- | /O(log n)/. Adjust a value at a specific key. When the key is not
-- a member of the map, the original map is returned.
--
-- > let f key x = (show key) ++ ":new " ++ x
-- > adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
-- > adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > adjustWithKey f 7 empty == empty

adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
adjustWithKey f = updateWithKey (\k' x' -> Just (f k' x'))
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE adjustWithKey #-}
#else
{-# INLINE adjustWithKey #-}
#endif

-- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@
-- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is
-- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.
--
-- > let f x = if x == "a" then Just "new a" else Nothing
-- > update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
-- > update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a
update f = updateWithKey (\_ x -> f x)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE update #-}
#else
{-# INLINE update #-}
#endif

-- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the
-- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',
-- the element is deleted. If it is (@'Just' y@), the key @k@ is bound
-- to the new value @y@.
--
-- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
-- > updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
-- > updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

-- See Note: Type of local 'go' function
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
updateWithKey = go
 where
 go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a
 STRICT_2_OF_3(go)
 go _ _ Tip = Tip
 go f k(Bin sx kx x l r) =
 case compare k kx of
 LT -> balanceR kx x (go f k l) r
 GT -> balanceL kx x l (go f k r)
 EQ -> case f kx x of
 Just x' -> Bin sx kx x' l r
 Nothing -> glue l r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE updateWithKey #-}
#else
{-# INLINE updateWithKey #-}
#endif

-- | /O(log n)/. Lookup and update. See also 'updateWithKey'.
-- The function returns changed value, if it is updated.
-- Returns the original key value if the map entry is deleted.
--
-- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
-- > updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
-- > updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")])
-- > updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")

-- See Note: Type of local 'go' function
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
updateLookupWithKey = go
 where
 go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
 STRICT_2_OF_3(go)
 go _ _ Tip = (Nothing,Tip)
 go f k (Bin sx kx x l r) =
 case compare k kx of
 LT -> let (found,l') = go f k l in (found,balanceR kx x l' r)
 GT -> let (found,r') = go f k r in (found,balanceL kx x l r')
 EQ -> case f kx x of
 Just x' -> (Just x',Bin sx kx x' l r)
 Nothing -> (Just x,glue l r)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE updateLookupWithKey #-}
#else
{-# INLINE updateLookupWithKey #-}
#endif

-- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.
-- 'alter' can be used to insert, delete, or update a value in a 'Map'.
-- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@.
--
-- > let f _ = Nothing
-- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
-- > alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
-- >
-- > let f _ = Just "c"
-- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
-- > alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]

-- See Note: Type of local 'go' function
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
alter = go
 where
 go :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a
 STRICT_2_OF_3(go)
 go f k Tip = case f Nothing of
 Nothing -> Tip
 Just x -> singleton k x

 go f k (Bin sx kx x l r) = case compare k kx of
 LT -> balance kx x (go f k l) r
 GT -> balance kx x l (go f k r)
 EQ -> case f (Just x) of
 Just x' -> Bin sx kx x' l r
 Nothing -> glue l r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE alter #-}
#else
{-# INLINE alter #-}
#endif

{--------------------------------------------------------------------
 Indexing
--------------------------------------------------------------------}
-- | /O(log n)/. Return the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map. Calls 'error' when
-- the key is not a 'member' of the map.
--
-- > findIndex 2 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map
-- > findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
-- > findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
-- > findIndex 6 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map

-- See Note: Type of local 'go' function
findIndex :: Ord k => k -> Map k a -> Int
findIndex = go 0
 where
 go :: Ord k => Int -> k -> Map k a -> Int
 STRICT_1_OF_3(go)
 STRICT_2_OF_3(go)
 go _ _ Tip = error "Map.findIndex: element is not in the map"
 go idx k (Bin _ kx _ l r) = case compare k kx of
 LT -> go idx k l
 GT -> go (idx + size l + 1) k r
 EQ -> idx + size l
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE findIndex #-}
#endif

-- | /O(log n)/. Lookup the /index/ of a key. The index is a number from
-- /0/ up to, but not including, the 'size' of the map.
--
-- > isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")])) == False
-- > fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
-- > fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
-- > isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")])) == False

-- See Note: Type of local 'go' function
lookupIndex :: Ord k => k -> Map k a -> Maybe Int
lookupIndex = go 0
 where
 go :: Ord k => Int -> k -> Map k a -> Maybe Int
 STRICT_1_OF_3(go)
 STRICT_2_OF_3(go)
 go _ _ Tip = Nothing
 go idx k (Bin _ kx _ l r) = case compare k kx of
 LT -> go idx k l
 GT -> go (idx + size l + 1) k r
 EQ -> Just $! idx + size l
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE lookupIndex #-}
#endif

-- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an
-- invalid index is used.
--
-- > elemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
-- > elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
-- > elemAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range

elemAt :: Int -> Map k a -> (k,a)
STRICT_1_OF_2(elemAt)
elemAt _ Tip = error "Map.elemAt: index out of range"
elemAt i (Bin _ kx x l r)
 = case compare i sizeL of
 LT -> elemAt i l
 GT -> elemAt (i-sizeL-1) r
 EQ -> (kx,x)
 where
 sizeL = size l

-- | /O(log n)/. Update the element at /index/. Calls 'error' when an
-- invalid index is used.
--
-- > updateAt (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
-- > updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
-- > updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range
-- > updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range
-- > updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
-- > updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
-- > updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range
-- > updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range

updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a
updateAt f i t = i `seq`
 case t of
 Tip -> error "Map.updateAt: index out of range"
 Bin sx kx x l r -> case compare i sizeL of
 LT -> balanceR kx x (updateAt f i l) r
 GT -> balanceL kx x l (updateAt f (i-sizeL-1) r)
 EQ -> case f kx x of
 Just x' -> Bin sx kx x' l r
 Nothing -> glue l r
 where
 sizeL = size l

-- | /O(log n)/. Delete the element at /index/.
-- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).
--
-- > deleteAt 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
-- > deleteAt 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
-- > deleteAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range
-- > deleteAt (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range

deleteAt :: Int -> Map k a -> Map k a
deleteAt i t = i `seq`
 case t of
 Tip -> error "Map.deleteAt: index out of range"
 Bin _ kx x l r -> case compare i sizeL of
 LT -> balanceR kx x (deleteAt i l) r
 GT -> balanceL kx x l (deleteAt (i-sizeL-1) r)
 EQ -> glue l r
 where
 sizeL = size l


{--------------------------------------------------------------------
 Minimal, Maximal
--------------------------------------------------------------------}
-- | /O(log n)/. The minimal key of the map. Calls 'error' if the map is empty.
--
-- > findMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
-- > findMin empty Error: empty map has no minimal element

findMin :: Map k a -> (k,a)
findMin (Bin _ kx x Tip _) = (kx,x)
findMin (Bin _ _ _ l _) = findMin l
findMin Tip = error "Map.findMin: empty map has no minimal element"

-- | /O(log n)/. The maximal key of the map. Calls 'error' if the map is empty.
--
-- > findMax (fromList [(5,"a"), (3,"b")]) == (5,"a")
-- > findMax empty Error: empty map has no maximal element

findMax :: Map k a -> (k,a)
findMax (Bin _ kx x _ Tip) = (kx,x)
findMax (Bin _ _ _ _ r) = findMax r
findMax Tip = error "Map.findMax: empty map has no maximal element"

-- | /O(log n)/. Delete the minimal key. Returns an empty map if the map is empty.
--
-- > deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
-- > deleteMin empty == empty

deleteMin :: Map k a -> Map k a
deleteMin (Bin _ _ _ Tip r) = r
deleteMin (Bin _ kx x l r) = balanceR kx x (deleteMin l) r
deleteMin Tip = Tip

-- | /O(log n)/. Delete the maximal key. Returns an empty map if the map is empty.
--
-- > deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
-- > deleteMax empty == empty

deleteMax :: Map k a -> Map k a
deleteMax (Bin _ _ _ l Tip) = l
deleteMax (Bin _ kx x l r) = balanceL kx x l (deleteMax r)
deleteMax Tip = Tip

-- | /O(log n)/. Update the value at the minimal key.
--
-- > updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
-- > updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

updateMin :: (a -> Maybe a) -> Map k a -> Map k a
updateMin f m
 = updateMinWithKey (\_ x -> f x) m

-- | /O(log n)/. Update the value at the maximal key.
--
-- > updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
-- > updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

updateMax :: (a -> Maybe a) -> Map k a -> Map k a
updateMax f m
 = updateMaxWithKey (\_ x -> f x) m


-- | /O(log n)/. Update the value at the minimal key.
--
-- > updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
-- > updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMinWithKey _ Tip = Tip
updateMinWithKey f (Bin sx kx x Tip r) = case f kx x of
 Nothing -> r
 Just x' -> Bin sx kx x' Tip r
updateMinWithKey f (Bin _ kx x l r) = balanceR kx x (updateMinWithKey f l) r

-- | /O(log n)/. Update the value at the maximal key.
--
-- > updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
-- > updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"

updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a
updateMaxWithKey _ Tip = Tip
updateMaxWithKey f (Bin sx kx x l Tip) = case f kx x of
 Nothing -> l
 Just x' -> Bin sx kx x' l Tip
updateMaxWithKey f (Bin _ kx x l r) = balanceL kx x l (updateMaxWithKey f r)

-- | /O(log n)/. Retrieves the minimal (key,value) pair of the map, and
-- the map stripped of that element, or 'Nothing' if passed an empty map.
--
-- > minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
-- > minViewWithKey empty == Nothing

minViewWithKey :: Map k a -> Maybe ((k,a), Map k a)
minViewWithKey Tip = Nothing
minViewWithKey x = Just (deleteFindMin x)

-- | /O(log n)/. Retrieves the maximal (key,value) pair of the map, and
-- the map stripped of that element, or 'Nothing' if passed an empty map.
--
-- > maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
-- > maxViewWithKey empty == Nothing

maxViewWithKey :: Map k a -> Maybe ((k,a), Map k a)
maxViewWithKey Tip = Nothing
maxViewWithKey x = Just (deleteFindMax x)

-- | /O(log n)/. Retrieves the value associated with minimal key of the
-- map, and the map stripped of that element, or 'Nothing' if passed an
-- empty map.
--
-- > minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
-- > minView empty == Nothing

minView :: Map k a -> Maybe (a, Map k a)
minView Tip = Nothing
minView x = Just (first snd $ deleteFindMin x)

-- | /O(log n)/. Retrieves the value associated with maximal key of the
-- map, and the map stripped of that element, or 'Nothing' if passed an
--
-- > maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
-- > maxView empty == Nothing

maxView :: Map k a -> Maybe (a, Map k a)
maxView Tip = Nothing
maxView x = Just (first snd $ deleteFindMax x)

-- Update the 1st component of a tuple (special case of Control.Arrow.first)
first :: (a -> b) -> (a,c) -> (b,c)
first f (x,y) = (f x, y)

{--------------------------------------------------------------------
 Union.
--------------------------------------------------------------------}
-- | The union of a list of maps:
-- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).
--
-- > unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
-- > == fromList [(3, "b"), (5, "a"), (7, "C")]
-- > unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
-- > == fromList [(3, "B3"), (5, "A3"), (7, "C")]

unions :: Ord k => [Map k a] -> Map k a
unions ts
 = foldlStrict union empty ts
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE unions #-}
#endif

-- | The union of a list of maps, with a combining operation:
-- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).
--
-- > unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
-- > == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]

unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a
unionsWith f ts
 = foldlStrict (unionWith f) empty ts
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE unionsWith #-}
#endif

-- | /O(n+m)/.
-- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.
-- It prefers @t1@ when duplicate keys are encountered,
-- i.e. (@'union' == 'unionWith' 'const'@).
-- The implementation uses the efficient /hedge-union/ algorithm.
-- Hedge-union is more efficient on (bigset \``union`\` smallset).
--
-- > union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]

union :: Ord k => Map k a -> Map k a -> Map k a
union Tip t2 = t2
union t1 Tip = t1
union t1 t2 = hedgeUnion NothingS NothingS t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE union #-}
#endif

-- left-biased hedge union
hedgeUnion :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a b -> Map a b
hedgeUnion _ _ t1 Tip = t1
hedgeUnion blo bhi Tip (Bin _ kx x l r) = join kx x (filterGt blo l) (filterLt bhi r)
hedgeUnion _ _ t1 (Bin _ kx x Tip Tip) = insertR kx x t1 -- According to benchmarks, this special case increases
 -- performance up to 30%. It does not help in difference or intersection.
hedgeUnion blo bhi (Bin _ kx x l r) t2 = join kx x (hedgeUnion blo bmi l (trim blo bmi t2))
 (hedgeUnion bmi bhi r (trim bmi bhi t2))
 where bmi = JustS kx
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE hedgeUnion #-}
#endif

{--------------------------------------------------------------------
 Union with a combining function
--------------------------------------------------------------------}
-- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
--
-- > unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]

unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWith f m1 m2
 = unionWithKey (\_ x y -> f x y) m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE unionWith #-}
#endif

-- | /O(n+m)/.
-- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.
-- Hedge-union is more efficient on (bigset \``union`\` smallset).
--
-- > let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
-- > unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]

unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a
unionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) id id t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE unionWithKey #-}
#endif

{--------------------------------------------------------------------
 Difference
--------------------------------------------------------------------}
-- | /O(n+m)/. Difference of two maps.
-- Return elements of the first map not existing in the second map.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
--
-- > difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"

difference :: Ord k => Map k a -> Map k b -> Map k a
difference Tip _ = Tip
difference t1 Tip = t1
difference t1 t2 = hedgeDiff NothingS NothingS t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE difference #-}
#endif

hedgeDiff :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a c -> Map a b
hedgeDiff _ _ Tip _ = Tip
hedgeDiff blo bhi (Bin _ kx x l r) Tip = join kx x (filterGt blo l) (filterLt bhi r)
hedgeDiff blo bhi t (Bin _ kx _ l r) = merge (hedgeDiff blo bmi (trim blo bmi t) l)
 (hedgeDiff bmi bhi (trim bmi bhi t) r)
 where bmi = JustS kx
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE hedgeDiff #-}
#endif

-- | /O(n+m)/. Difference with a combining function.
-- When two equal keys are
-- encountered, the combining function is applied to the values of these keys.
-- If it returns 'Nothing', the element is discarded (proper set difference). If
-- it returns (@'Just' y@), the element is updated with a new value @y@.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
--
-- > let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
-- > differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
-- > == singleton 3 "b:B"

differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
differenceWith f m1 m2
 = differenceWithKey (\_ x y -> f x y) m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE differenceWith #-}
#endif

-- | /O(n+m)/. Difference with a combining function. When two equal keys are
-- encountered, the combining function is applied to the key and both values.
-- If it returns 'Nothing', the element is discarded (proper set difference). If
-- it returns (@'Just' y@), the element is updated with a new value @y@.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
--
-- > let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
-- > differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
-- > == singleton 3 "3:b|B"

differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
differenceWithKey f t1 t2 = mergeWithKey f id (const Tip) t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE differenceWithKey #-}
#endif


{--------------------------------------------------------------------
 Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. Intersection of two maps.
-- Return data in the first map for the keys existing in both maps.
-- (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).
--
-- > intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"

intersection :: Ord k => Map k a -> Map k b -> Map k a
intersection Tip _ = Tip
intersection _ Tip = Tip
intersection t1 t2 = hedgeInt NothingS NothingS t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE intersection #-}
#endif

hedgeInt :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k b -> Map k a
hedgeInt _ _ _ Tip = Tip
hedgeInt _ _ Tip _ = Tip
hedgeInt blo bhi (Bin _ kx x l r) t2 = let l' = hedgeInt blo bmi l (trim blo bmi t2)
 r' = hedgeInt bmi bhi r (trim bmi bhi t2)
 in if kx `member` t2 then join kx x l' r' else merge l' r'
 where bmi = JustS kx
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE hedgeInt #-}
#endif

-- | /O(n+m)/. Intersection with a combining function.
--
-- > intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"

intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c
intersectionWith f m1 m2
 = intersectionWithKey (\_ x y -> f x y) m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE intersectionWith #-}
#endif

-- | /O(n+m)/. Intersection with a combining function.
-- Intersection is more efficient on (bigset \``intersection`\` smallset).
--
-- > let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
-- > intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"


intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c
intersectionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) (const Tip) (const Tip) t1 t2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE intersectionWithKey #-}
#endif


{--------------------------------------------------------------------
 MergeWithKey
--------------------------------------------------------------------}

-- | /O(n+m)/. A high-performance universal combining function. This function
-- is used to define 'unionWith', 'unionWithKey', 'differenceWith',
-- 'differenceWithKey', 'intersectionWith', 'intersectionWithKey' and can be
-- used to define other custom combine functions.
--
-- Please make sure you know what is going on when using 'mergeWithKey',
-- otherwise you can be surprised by unexpected code growth or even
-- corruption of the data structure.
--
-- When 'mergeWithKey' is given three arguments, it is inlined to the call
-- site. You should therefore use 'mergeWithKey' only to define your custom
-- combining functions. For example, you could define 'unionWithKey',
-- 'differenceWithKey' and 'intersectionWithKey' as
--
-- > myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
-- > myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
-- > myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2
--
-- When calling @'mergeWithKey' combine only1 only2@, a function combining two
-- 'IntMap's is created, such that
--
-- * if a key is present in both maps, it is passed with both corresponding
-- values to the @combine@ function. Depending on the result, the key is either
-- present in the result with specified value, or is left out;
--
-- * a nonempty subtree present only in the first map is passed to @only1@ and
-- the output is added to the result;
--
-- * a nonempty subtree present only in the second map is passed to @only2@ and
-- the output is added to the result.
--
-- The @only1@ and @only2@ methods /must return a map with a subset (possibly empty) of the keys of the given map/.
-- The values can be modified arbitrarily. Most common variants of @only1@ and
-- @only2@ are 'id' and @'const' 'empty'@, but for example @'map' f@ or
-- @'filterWithKey' f@ could be used for any @f@.

mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c)
 -> Map k a -> Map k b -> Map k c
mergeWithKey f g1 g2 = go
 where
 go Tip t2 = g2 t2
 go t1 Tip = g1 t1
 go t1 t2 = hedgeMerge NothingS NothingS t1 t2

 hedgeMerge _ _ t1 Tip = g1 t1
 hedgeMerge blo bhi Tip (Bin _ kx x l r) = g2 $ join kx x (filterGt blo l) (filterLt bhi r)
 hedgeMerge blo bhi (Bin _ kx x l r) t2 = let l' = hedgeMerge blo bmi l (trim blo bmi t2)
 (found, trim_t2) = trimLookupLo kx bhi t2
 r' = hedgeMerge bmi bhi r trim_t2
 in case found of
 Nothing -> case g1 (singleton kx x) of
 Tip -> merge l' r'
 (Bin _ _ x' Tip Tip) -> join kx x' l' r'
 _ -> error "mergeWithKey: Given function only1 does not fulfil required conditions (see documentation)"
 Just x2 -> case f kx x x2 of
 Nothing -> merge l' r'
 Just x' -> join kx x' l' r'
 where bmi = JustS kx
{-# INLINE mergeWithKey #-}

{--------------------------------------------------------------------
 Submap
--------------------------------------------------------------------}
-- | /O(n+m)/.
-- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
--
isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
isSubmapOf m1 m2 = isSubmapOfBy (==) m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE isSubmapOf #-}
#endif

{- | /O(n+m)/.
 The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if
 all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when
 applied to their respective values. For example, the following
 expressions are all 'True':

 > isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])

 But the following are all 'False':

 > isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
 > isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
 > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])


-}
isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool
isSubmapOfBy f t1 t2
 = (size t1 <= size t2) && (submap' f t1 t2)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE isSubmapOfBy #-}
#endif

submap' :: Ord a => (b -> c -> Bool) -> Map a b -> Map a c -> Bool
submap' _ Tip _ = True
submap' _ _ Tip = False
submap' f (Bin _ kx x l r) t
 = case found of
 Nothing -> False
 Just y -> f x y && submap' f l lt && submap' f r gt
 where
 (lt,found,gt) = splitLookup kx t
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE submap' #-}
#endif

-- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
-- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool
isProperSubmapOf m1 m2
 = isProperSubmapOfBy (==) m1 m2
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE isProperSubmapOf #-}
#endif

{- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
 The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when
 @m1@ and @m2@ are not equal,
 all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
 applied to their respective values. For example, the following
 expressions are all 'True':

 > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
 > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])

 But the following are all 'False':

 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
 > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
 > isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])


-}
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool
isProperSubmapOfBy f t1 t2
 = (size t1 < size t2) && (submap' f t1 t2)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE isProperSubmapOfBy #-}
#endif

{--------------------------------------------------------------------
 Filter and partition
--------------------------------------------------------------------}
-- | /O(n)/. Filter all values that satisfy the predicate.
--
-- > filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
-- > filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty
-- > filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty

filter :: (a -> Bool) -> Map k a -> Map k a
filter p m
 = filterWithKey (\_ x -> p x) m

-- | /O(n)/. Filter all keys\/values that satisfy the predicate.
--
-- > filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"

filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a
filterWithKey _ Tip = Tip
filterWithKey p (Bin _ kx x l r)
 | p kx x = join kx x (filterWithKey p l) (filterWithKey p r)
 | otherwise = merge (filterWithKey p l) (filterWithKey p r)

-- | /O(n)/. Partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
--
-- > partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
-- > partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
-- > partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])

partition :: (a -> Bool) -> Map k a -> (Map k a,Map k a)
partition p m
 = partitionWithKey (\_ x -> p x) m

-- | /O(n)/. Partition the map according to a predicate. The first
-- map contains all elements that satisfy the predicate, the second all
-- elements that fail the predicate. See also 'split'.
--
-- > partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
-- > partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
-- > partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])

partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a,Map k a)
partitionWithKey _ Tip = (Tip,Tip)
partitionWithKey p (Bin _ kx x l r)
 | p kx x = (join kx x l1 r1,merge l2 r2)
 | otherwise = (merge l1 r1,join kx x l2 r2)
 where
 (l1,l2) = partitionWithKey p l
 (r1,r2) = partitionWithKey p r

-- | /O(n)/. Map values and collect the 'Just' results.
--
-- > let f x = if x == "a" then Just "new a" else Nothing
-- > mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"

mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b
mapMaybe f = mapMaybeWithKey (\_ x -> f x)

-- | /O(n)/. Map keys\/values and collect the 'Just' results.
--
-- > let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing
-- > mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"

mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b
mapMaybeWithKey _ Tip = Tip
mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of
 Just y -> join kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
 Nothing -> merge (mapMaybeWithKey f l) (mapMaybeWithKey f r)

-- | /O(n)/. Map values and separate the 'Left' and 'Right' results.
--
-- > let f a = if a < "c" then Left a else Right a
-- > mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
-- > == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
-- >
-- > mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
-- > == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])

mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)
mapEither f m
 = mapEitherWithKey (\_ x -> f x) m

-- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.
--
-- > let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
-- > mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
-- > == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
-- >
-- > mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
-- > == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])

mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)
mapEitherWithKey _ Tip = (Tip, Tip)
mapEitherWithKey f (Bin _ kx x l r) = case f kx x of
 Left y -> (join kx y l1 r1, merge l2 r2)
 Right z -> (merge l1 r1, join kx z l2 r2)
 where
 (l1,l2) = mapEitherWithKey f l
 (r1,r2) = mapEitherWithKey f r

{--------------------------------------------------------------------
 Mapping
--------------------------------------------------------------------}
-- | /O(n)/. Map a function over all values in the map.
--
-- > map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

map :: (a -> b) -> Map k a -> Map k b
map _ Tip = Tip
map f (Bin sx kx x l r) = Bin sx kx (f x) (map f l) (map f r)

-- | /O(n)/. Map a function over all values in the map.
--
-- > let f key x = (show key) ++ ":" ++ x
-- > mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]

mapWithKey :: (k -> a -> b) -> Map k a -> Map k b
mapWithKey _ Tip = Tip
mapWithKey f (Bin sx kx x l r) = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)

-- | /O(n)/.
-- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
-- That is, behaves exactly like a regular 'traverse' except that the traversing
-- function also has access to the key associated with a value.
--
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing
{-# INLINE traverseWithKey #-}
traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)
traverseWithKey f = go
 where
 go Tip = pure Tip
 go (Bin s k v l r)
 = flip (Bin s k) <$> go l <*> f k v <*> go r

-- | /O(n)/. The function 'mapAccum' threads an accumulating
-- argument through the map in ascending order of keys.
--
-- > let f a b = (a ++ b, b ++ "X")
-- > mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])

mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccum f a m
 = mapAccumWithKey (\a' _ x' -> f a' x') a m

-- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating
-- argument through the map in ascending order of keys.
--
-- > let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
-- > mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])

mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumWithKey f a t
 = mapAccumL f a t

-- | /O(n)/. The function 'mapAccumL' threads an accumulating
-- argument through the map in ascending order of keys.
mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumL _ a Tip = (a,Tip)
mapAccumL f a (Bin sx kx x l r) =
 let (a1,l') = mapAccumL f a l
 (a2,x') = f a1 kx x
 (a3,r') = mapAccumL f a2 r
 in (a3,Bin sx kx x' l' r')

-- | /O(n)/. The function 'mapAccumR' threads an accumulating
-- argument through the map in descending order of keys.
mapAccumRWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)
mapAccumRWithKey _ a Tip = (a,Tip)
mapAccumRWithKey f a (Bin sx kx x l r) =
 let (a1,r') = mapAccumRWithKey f a r
 (a2,x') = f a1 kx x
 (a3,l') = mapAccumRWithKey f a2 l
 in (a3,Bin sx kx x' l' r')

-- | /O(n*log n)/.
-- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
--
-- The size of the result may be smaller if @f@ maps two or more distinct
-- keys to the same new key. In this case the value at the greatest of the
-- original keys is retained.
--
-- > mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]
-- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
-- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"

mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a
mapKeys f = fromList . foldrWithKey (\k x xs -> (f k, x) : xs) []
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE mapKeys #-}
#endif

-- | /O(n*log n)/.
-- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.
--
-- The size of the result may be smaller if @f@ maps two or more distinct
-- keys to the same new key. In this case the associated values will be
-- combined using @c@.
--
-- > mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
-- > mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"

mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a
mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE mapKeysWith #-}
#endif


-- | /O(n)/.
-- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@
-- is strictly monotonic.
-- That is, for any values @x@ and @y@, if @x@ < @y@ then @f x@ < @f y@.
-- /The precondition is not checked./
-- Semi-formally, we have:
--
-- > and [x < y ==> f x < f y | x <- ls, y <- ls]
-- > ==> mapKeysMonotonic f s == mapKeys f s
-- > where ls = keys s
--
-- This means that @f@ maps distinct original keys to distinct resulting keys.
-- This function has better performance than 'mapKeys'.
--
-- > mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
-- > valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True
-- > valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False

mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a
mapKeysMonotonic _ Tip = Tip
mapKeysMonotonic f (Bin sz k x l r) =
 Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)

{--------------------------------------------------------------------
 Folds
--------------------------------------------------------------------}

-- | /O(n)/. Fold the values in the map using the given right-associative
-- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'elems'@.
--
-- For example,
--
-- > elems map = foldr (:) [] map
--
-- > let f a len = len + (length a)
-- > foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
foldr :: (a -> b -> b) -> b -> Map k a -> b
foldr f z = go z
 where
 go z' Tip = z'
 go z' (Bin _ _ x l r) = go (f x (go z' r)) l
{-# INLINE foldr #-}

-- | /O(n)/. A strict version of 'foldr'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Map k a -> b
foldr' f z = go z
 where
 STRICT_1_OF_2(go)
 go z' Tip = z'
 go z' (Bin _ _ x l r) = go (f x (go z' r)) l
{-# INLINE foldr' #-}

-- | /O(n)/. Fold the values in the map using the given left-associative
-- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'elems'@.
--
-- For example,
--
-- > elems = reverse . foldl (flip (:)) []
--
-- > let f len a = len + (length a)
-- > foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
foldl :: (a -> b -> a) -> a -> Map k b -> a
foldl f z = go z
 where
 go z' Tip = z'
 go z' (Bin _ _ x l r) = go (f (go z' l) x) r
{-# INLINE foldl #-}

-- | /O(n)/. A strict version of 'foldl'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Map k b -> a
foldl' f z = go z
 where
 STRICT_1_OF_2(go)
 go z' Tip = z'
 go z' (Bin _ _ x l r) = go (f (go z' l) x) r
{-# INLINE foldl' #-}

-- | /O(n)/. Fold the keys and values in the map using the given right-associative
-- binary operator, such that
-- @'foldrWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
--
-- For example,
--
-- > keys map = foldrWithKey (\k x ks -> k:ks) [] map
--
-- > let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
-- > foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
foldrWithKey f z = go z
 where
 go z' Tip = z'
 go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l
{-# INLINE foldrWithKey #-}

-- | /O(n)/. A strict version of 'foldrWithKey'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b
foldrWithKey' f z = go z
 where
 STRICT_1_OF_2(go)
 go z' Tip = z'
 go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l
{-# INLINE foldrWithKey' #-}

-- | /O(n)/. Fold the keys and values in the map using the given left-associative
-- binary operator, such that
-- @'foldlWithKey' f z == 'Prelude.foldl' (\\z' (kx, x) -> f z' kx x) z . 'toAscList'@.
--
-- For example,
--
-- > keys = reverse . foldlWithKey (\ks k x -> k:ks) []
--
-- > let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
-- > foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a
foldlWithKey f z = go z
 where
 go z' Tip = z'
 go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r
{-# INLINE foldlWithKey #-}

-- | /O(n)/. A strict version of 'foldlWithKey'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a
foldlWithKey' f z = go z
 where
 STRICT_1_OF_2(go)
 go z' Tip = z'
 go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r
{-# INLINE foldlWithKey' #-}

{--------------------------------------------------------------------
 List variations
--------------------------------------------------------------------}
-- | /O(n)/.
-- Return all elements of the map in the ascending order of their keys.
-- Subject to list fusion.
--
-- > elems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
-- > elems empty == []

elems :: Map k a -> [a]
elems = foldr (:) []

-- | /O(n)/. Return all keys of the map in ascending order. Subject to list
-- fusion.
--
-- > keys (fromList [(5,"a"), (3,"b")]) == [3,5]
-- > keys empty == []

keys :: Map k a -> [k]
keys = foldrWithKey (\k _ ks -> k : ks) []

-- | /O(n)/. An alias for 'toAscList'. Return all key\/value pairs in the map
-- in ascending key order. Subject to list fusion.
--
-- > assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
-- > assocs empty == []

assocs :: Map k a -> [(k,a)]
assocs m
 = toAscList m

-- | /O(n)/. The set of all keys of the map.
--
-- > keysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
-- > keysSet empty == Data.Set.empty

keysSet :: Map k a -> Set.Set k
keysSet Tip = Set.Tip
keysSet (Bin sz kx _ l r) = Set.Bin sz kx (keysSet l) (keysSet r)

-- | /O(n)/. Build a map from a set of keys and a function which for each key
-- computes its value.
--
-- > fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
-- > fromSet undefined Data.Set.empty == empty

fromSet :: (k -> a) -> Set.Set k -> Map k a
fromSet _ Set.Tip = Tip
fromSet f (Set.Bin sz x l r) = Bin sz x (f x) (fromSet f l) (fromSet f r)

{--------------------------------------------------------------------
 Lists
 use [foldlStrict] to reduce demand on the control-stack
--------------------------------------------------------------------}
-- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.
-- If the list contains more than one value for the same key, the last value
-- for the key is retained.
--
-- > fromList [] == empty
-- > fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
-- > fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]

fromList :: Ord k => [(k,a)] -> Map k a
fromList xs
 = foldlStrict ins empty xs
 where
 ins t (k,x) = insert k x t
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromList #-}
#endif

-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.
--
-- > fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
-- > fromListWith (++) [] == empty

fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a
fromListWith f xs
 = fromListWithKey (\_ x y -> f x y) xs
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromListWith #-}
#endif

-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.
--
-- > let f k a1 a2 = (show k) ++ a1 ++ a2
-- > fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
-- > fromListWithKey f [] == empty

fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
fromListWithKey f xs
 = foldlStrict ins empty xs
 where
 ins t (k,x) = insertWithKey f k x t
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromListWithKey #-}
#endif

-- | /O(n)/. Convert the map to a list of key\/value pairs. Subject to list fusion.
--
-- > toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
-- > toList empty == []

toList :: Map k a -> [(k,a)]
toList = toAscList

-- | /O(n)/. Convert the map to a list of key\/value pairs where the keys are
-- in ascending order. Subject to list fusion.
--
-- > toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]

toAscList :: Map k a -> [(k,a)]
toAscList = foldrWithKey (\k x xs -> (k,x):xs) []

-- | /O(n)/. Convert the map to a list of key\/value pairs where the keys
-- are in descending order. Subject to list fusion.
--
-- > toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]

toDescList :: Map k a -> [(k,a)]
toDescList = foldlWithKey (\xs k x -> (k,x):xs) []

-- List fusion for the list generating functions.
#if __GLASGOW_HASKELL__
-- The foldrFB and foldlFB are fold{r,l}WithKey equivalents, used for list fusion.
-- They are important to convert unfused methods back, see mapFB in prelude.
foldrFB :: (k -> a -> b -> b) -> b -> Map k a -> b
foldrFB = foldrWithKey
{-# INLINE[0] foldrFB #-}
foldlFB :: (a -> k -> b -> a) -> a -> Map k b -> a
foldlFB = foldlWithKey
{-# INLINE[0] foldlFB #-}

-- Inline assocs and toList, so that we need to fuse only toAscList.
{-# INLINE assocs #-}
{-# INLINE toList #-}

-- The fusion is enabled up to phase 2 included. If it does not succeed,
-- convert in phase 1 the expanded elems,keys,to{Asc,Desc}List calls back to
-- elems,keys,to{Asc,Desc}List. In phase 0, we inline fold{lr}FB (which were
-- used in a list fusion, otherwise it would go away in phase 1), and let compiler
-- do whatever it wants with elems,keys,to{Asc,Desc}List -- it was forbidden to
-- inline it before phase 0, otherwise the fusion rules would not fire at all.
{-# NOINLINE[0] elems #-}
{-# NOINLINE[0] keys #-}
{-# NOINLINE[0] toAscList #-}
{-# NOINLINE[0] toDescList #-}
{-# RULES "Map.elems" [~1] forall m . elems m = build (\c n -> foldrFB (\_ x xs -> c x xs) n m) #-}
{-# RULES "Map.elemsBack" [1] foldrFB (\_ x xs -> x : xs) [] = elems #-}
{-# RULES "Map.keys" [~1] forall m . keys m = build (\c n -> foldrFB (\k _ xs -> c k xs) n m) #-}
{-# RULES "Map.keysBack" [1] foldrFB (\k _ xs -> k : xs) [] = keys #-}
{-# RULES "Map.toAscList" [~1] forall m . toAscList m = build (\c n -> foldrFB (\k x xs -> c (k,x) xs) n m) #-}
{-# RULES "Map.toAscListBack" [1] foldrFB (\k x xs -> (k, x) : xs) [] = toAscList #-}
{-# RULES "Map.toDescList" [~1] forall m . toDescList m = build (\c n -> foldlFB (\xs k x -> c (k,x) xs) n m) #-}
{-# RULES "Map.toDescListBack" [1] foldlFB (\xs k x -> (k, x) : xs) [] = toDescList #-}
#endif

{--------------------------------------------------------------------
 Building trees from ascending/descending lists can be done in linear time.

 Note that if [xs] is ascending that:
 fromAscList xs == fromList xs
 fromAscListWith f xs == fromListWith f xs
--------------------------------------------------------------------}
-- | /O(n)/. Build a map from an ascending list in linear time.
-- /The precondition (input list is ascending) is not checked./
--
-- > fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
-- > fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
-- > valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
-- > valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False

fromAscList :: Eq k => [(k,a)] -> Map k a
fromAscList xs
 = fromAscListWithKey (\_ x _ -> x) xs
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromAscList #-}
#endif

-- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.
-- /The precondition (input list is ascending) is not checked./
--
-- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
-- > valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
-- > valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False

fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
fromAscListWith f xs
 = fromAscListWithKey (\_ x y -> f x y) xs
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromAscListWith #-}
#endif

-- | /O(n)/. Build a map from an ascending list in linear time with a
-- combining function for equal keys.
-- /The precondition (input list is ascending) is not checked./
--
-- > let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
-- > fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
-- > valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
-- > valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False

fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
fromAscListWithKey f xs
 = fromDistinctAscList (combineEq f xs)
 where
 -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
 combineEq _ xs'
 = case xs' of
 [] -> []
 [x] -> [x]
 (x:xx) -> combineEq' x xx

 combineEq' z [] = [z]
 combineEq' z@(kz,zz) (x@(kx,xx):xs')
 | kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs'
 | otherwise = z:combineEq' x xs'
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE fromAscListWithKey #-}
#endif


-- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.
-- /The precondition is not checked./
--
-- > fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
-- > valid (fromDistinctAscList [(3,"b"), (5,"a")]) == True
-- > valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False

fromDistinctAscList :: [(k,a)] -> Map k a
fromDistinctAscList xs
 = create const (length xs) xs
 where
 -- 1) use continuations so that we use heap space instead of stack space.
 -- 2) special case for n==5 to create bushier trees.
 create c 0 xs' = c Tip xs'
 create c 5 xs' = case xs' of
 ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)
 -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx
 _ -> error "fromDistinctAscList create"
 create c n xs' = seq nr $ create (createR nr c) nl xs'
 where nl = n `div` 2
 nr = n - nl - 1

 createR n c l ((k,x):ys) = create (createB l k x c) n ys
 createR _ _ _ [] = error "fromDistinctAscList createR []"
 createB l k x c r zs = c (bin k x l r) zs


{--------------------------------------------------------------------
 Utility functions that return sub-ranges of the original
 tree. Some functions take a `Maybe value` as an argument to
 allow comparisons against infinite values. These are called `blow`
 (Nothing is -\infty) and `bhigh` (here Nothing is +\infty).
 We use MaybeS value, which is a Maybe strict in the Just case.

 [trim blow bhigh t] A tree that is either empty or where [x > blow]
 and [x < bhigh] for the value [x] of the root.
 [filterGt blow t] A tree where for all values [k]. [k > blow]
 [filterLt bhigh t] A tree where for all values [k]. [k < bhigh]

 [split k t] Returns two trees [l] and [r] where all keys
 in [l] are <[k] and all keys in [r] are >[k].
 [splitLookup k t] Just like [split] but also returns whether [k]
 was found in the tree.
--------------------------------------------------------------------}

data MaybeS a = NothingS | JustS !a

{--------------------------------------------------------------------
 [trim blo bhi t] trims away all subtrees that surely contain no
 values between the range [blo] to [bhi]. The returned tree is either
 empty or the key of the root is between @blo@ and @bhi@.
--------------------------------------------------------------------}
trim :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k a
trim NothingS NothingS t = t
trim (JustS lk) NothingS t = greater lk t where greater lo (Bin _ k _ _ r) | k <= lo = greater lo r
 greater _ t' = t'
trim NothingS (JustS hk) t = lesser hk t where lesser hi (Bin _ k _ l _) | k >= hi = lesser hi l
 lesser _ t' = t'
trim (JustS lk) (JustS hk) t = middle lk hk t where middle lo hi (Bin _ k _ _ r) | k <= lo = middle lo hi r
 middle lo hi (Bin _ k _ l _) | k >= hi = middle lo hi l
 middle _ _ t' = t'
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE trim #-}
#endif

-- Helper function for 'mergeWithKey'. The @'trimLookupLo' lk hk t@ performs both
-- @'trim' (JustS lk) hk t@ and @'lookup' lk t@.

-- See Note: Type of local 'go' function
trimLookupLo :: Ord k => k -> MaybeS k -> Map k a -> (Maybe a, Map k a)
trimLookupLo lk NothingS t = greater lk t
 where greater :: Ord k => k -> Map k a -> (Maybe a, Map k a)
 greater lo t'@(Bin _ kx x l r) = case compare lo kx of LT -> lookup lo l `strictPair` t'
 EQ -> (Just x, r)
 GT -> greater lo r
 greater _ Tip = (Nothing, Tip)
trimLookupLo lk (JustS hk) t = middle lk hk t
 where middle :: Ord k => k -> k -> Map k a -> (Maybe a, Map k a)
 middle lo hi t'@(Bin _ kx x l r) = case compare lo kx of LT | kx < hi -> lookup lo l `strictPair` t'
 | otherwise -> middle lo hi l
 EQ -> Just x `strictPair` lesser hi r
 GT -> middle lo hi r
 middle _ _ Tip = (Nothing, Tip)

 lesser :: Ord k => k -> Map k a -> Map k a
 lesser hi (Bin _ k _ l _) | k >= hi = lesser hi l
 lesser _ t' = t'
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE trimLookupLo #-}
#endif


{--------------------------------------------------------------------
 [filterGt b t] filter all keys >[b] from tree [t]
 [filterLt b t] filter all keys <[b] from tree [t]
--------------------------------------------------------------------}
filterGt :: Ord k => MaybeS k -> Map k v -> Map k v
filterGt NothingS t = t
filterGt (JustS b) t = filter' b t
 where filter' _ Tip = Tip
 filter' b' (Bin _ kx x l r) =
 case compare b' kx of LT -> join kx x (filter' b' l) r
 EQ -> r
 GT -> filter' b' r
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE filterGt #-}
#endif

filterLt :: Ord k => MaybeS k -> Map k v -> Map k v
filterLt NothingS t = t
filterLt (JustS b) t = filter' b t
 where filter' _ Tip = Tip
 filter' b' (Bin _ kx x l r) =
 case compare kx b' of LT -> join kx x l (filter' b' r)
 EQ -> l
 GT -> filter' b' l
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE filterLt #-}
#endif

{--------------------------------------------------------------------
 Split
--------------------------------------------------------------------}
-- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where
-- the keys in @map1@ are smaller than @k@ and the keys in @map2@ larger than @k@.
-- Any key equal to @k@ is found in neither @map1@ nor @map2@.
--
-- > split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
-- > split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
-- > split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
-- > split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
-- > split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)

split :: Ord k => k -> Map k a -> (Map k a,Map k a)
split k t = k `seq`
 case t of
 Tip -> (Tip, Tip)
 Bin _ kx x l r -> case compare k kx of
 LT -> let (lt,gt) = split k l in (lt,join kx x gt r)
 GT -> let (lt,gt) = split k r in (join kx x l lt,gt)
 EQ -> (l,r)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE split #-}
#endif

-- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just
-- like 'split' but also returns @'lookup' k map@.
--
-- > splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
-- > splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
-- > splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
-- > splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
-- > splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)

splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)
splitLookup k t = k `seq`
 case t of
 Tip -> (Tip,Nothing,Tip)
 Bin _ kx x l r -> case compare k kx of
 LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join kx x gt r)
 GT -> let (lt,z,gt) = splitLookup k r in (join kx x l lt,z,gt)
 EQ -> (l,Just x,r)
#if __GLASGOW_HASKELL__ >= 700
{-# INLINABLE splitLookup #-}
#endif

{--------------------------------------------------------------------
 Utility functions that maintain the balance properties of the tree.
 All constructors assume that all values in [l] < [k] and all values
 in [r] > [k], and that [l] and [r] are valid trees.

 In order of sophistication:
 [Bin sz k x l r] The type constructor.
 [bin k x l r] Maintains the correct size, assumes that both [l]
 and [r] are balanced with respect to each other.
 [balance k x l r] Restores the balance and size.
 Assumes that the original tree was balanced and
 that [l] or [r] has changed by at most one element.
 [join k x l r] Restores balance and size.

 Furthermore, we can construct a new tree from two trees. Both operations
 assume that all values in [l] < all values in [r] and that [l] and [r]
 are valid:
 [glue l r] Glues [l] and [r] together. Assumes that [l] and
 [r] are already balanced with respect to each other.
 [merge l r] Merges two trees and restores balance.

 Note: in contrast to Adam's paper, we use (<=) comparisons instead
 of (<) comparisons in [join], [merge] and [balance].
 Quickcheck (on [difference]) showed that this was necessary in order
 to maintain the invariants. It is quite unsatisfactory that I haven't
 been able to find out why this is actually the case! Fortunately, it
 doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
 Join
--------------------------------------------------------------------}
join :: k -> a -> Map k a -> Map k a -> Map k a
join kx x Tip r = insertMin kx x r
join kx x l Tip = insertMax kx x l
join kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)
 | delta*sizeL < sizeR = balanceL kz z (join kx x l lz) rz
 | delta*sizeR < sizeL = balanceR ky y ly (join kx x ry r)
 | otherwise = bin kx x l r


-- insertMin and insertMax don't perform potentially expensive comparisons.
insertMax,insertMin :: k -> a -> Map k a -> Map k a
insertMax kx x t
 = case t of
 Tip -> singleton kx x
 Bin _ ky y l r
 -> balanceR ky y l (insertMax kx x r)

insertMin kx x t
 = case t of
 Tip -> singleton kx x
 Bin _ ky y l r
 -> balanceL ky y (insertMin kx x l) r

{--------------------------------------------------------------------
 [merge l r]: merges two trees.
--------------------------------------------------------------------}
merge :: Map k a -> Map k a -> Map k a
merge Tip r = r
merge l Tip = l
merge l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)
 | delta*sizeL < sizeR = balanceL ky y (merge l ly) ry
 | delta*sizeR < sizeL = balanceR kx x lx (merge rx r)
 | otherwise = glue l r

{--------------------------------------------------------------------
 [glue l r]: glues two trees together.
 Assumes that [l] and [r] are already balanced with respect to each other.
--------------------------------------------------------------------}
glue :: Map k a -> Map k a -> Map k a
glue Tip r = r
glue l Tip = l
glue l r
 | size l > size r = let ((km,m),l') = deleteFindMax l in balanceR km m l' r
 | otherwise = let ((km,m),r') = deleteFindMin r in balanceL km m l r'


-- | /O(log n)/. Delete and find the minimal element.
--
-- > deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
-- > deleteFindMin Error: can not return the minimal element of an empty map

deleteFindMin :: Map k a -> ((k,a),Map k a)
deleteFindMin t
 = case t of
 Bin _ k x Tip r -> ((k,x),r)
 Bin _ k x l r -> let (km,l') = deleteFindMin l in (km,balanceR k x l' r)
 Tip -> (error "Map.deleteFindMin: can not return the minimal element of an empty map", Tip)

-- | /O(log n)/. Delete and find the maximal element.
--
-- > deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
-- > deleteFindMax empty Error: can not return the maximal element of an empty map

deleteFindMax :: Map k a -> ((k,a),Map k a)
deleteFindMax t
 = case t of
 Bin _ k x l Tip -> ((k,x),l)
 Bin _ k x l r -> let (km,r') = deleteFindMax r in (km,balanceL k x l r')
 Tip -> (error "Map.deleteFindMax: can not return the maximal element of an empty map", Tip)


{--------------------------------------------------------------------
 [balance l x r] balances two trees with value x.
 The sizes of the trees should balance after decreasing the
 size of one of them. (a rotation).

 [delta] is the maximal relative difference between the sizes of
 two trees, it corresponds with the [w] in Adams' paper.
 [ratio] is the ratio between an outer and inner sibling of the
 heavier subtree in an unbalanced setting. It determines
 whether a double or single rotation should be performed
 to restore balance. It is corresponds with the inverse
 of $\alpha$ in Adam's article.

 Note that according to the Adam's paper:
 - [delta] should be larger than 4.646 with a [ratio] of 2.
 - [delta] should be larger than 3.745 with a [ratio] of 1.534.

 But the Adam's paper is erroneous:
 - It can be proved that for delta=2 and delta>=5 there does
 not exist any ratio that would work.
 - Delta=4.5 and ratio=2 does not work.

 That leaves two reasonable variants, delta=3 and delta=4,
 both with ratio=2.

 - A lower [delta] leads to a more 'perfectly' balanced tree.
 - A higher [delta] performs less rebalancing.

 In the benchmarks, delta=3 is faster on insert operations,
 and delta=4 has slightly better deletes. As the insert speedup
 is larger, we currently use delta=3.

--------------------------------------------------------------------}
delta,ratio :: Int
delta = 3
ratio = 2

-- The balance function is equivalent to the following:
--
-- balance :: k -> a -> Map k a -> Map k a -> Map k a
-- balance k x l r
-- | sizeL + sizeR <= 1 = Bin sizeX k x l r
-- | sizeR > delta*sizeL = rotateL k x l r
-- | sizeL > delta*sizeR = rotateR k x l r
-- | otherwise = Bin sizeX k x l r
-- where
-- sizeL = size l
-- sizeR = size r
-- sizeX = sizeL + sizeR + 1
--
-- rotateL :: a -> b -> Map a b -> Map a b -> Map a b
-- rotateL k x l r@(Bin _ _ _ ly ry) | size ly < ratio*size ry = singleL k x l r
-- | otherwise = doubleL k x l r
--
-- rotateR :: a -> b -> Map a b -> Map a b -> Map a b
-- rotateR k x l@(Bin _ _ _ ly ry) r | size ry < ratio*size ly = singleR k x l r
-- | otherwise = doubleR k x l r
--
-- singleL, singleR :: a -> b -> Map a b -> Map a b -> Map a b
-- singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
-- singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)
--
-- doubleL, doubleR :: a -> b -> Map a b -> Map a b -> Map a b
-- doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4) = bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)
-- doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)
--
-- It is only written in such a way that every node is pattern-matched only once.

balance :: k -> a -> Map k a -> Map k a -> Map k a
balance k x l r = case l of
 Tip -> case r of
 Tip -> Bin 1 k x Tip Tip
 (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r
 (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr
 (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)
 (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))
 | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr
 | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)

 (Bin ls lk lx ll lr) -> case r of
 Tip -> case (ll, lr) of
 (Tip, Tip) -> Bin 2 k x l Tip
 (Tip, (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)
 ((Bin _ _ _ _ _), Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)
 ((Bin lls _ _ _ _), (Bin lrs lrk lrx lrl lrr))
 | lrs < ratio*lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)
 | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)
 (Bin rs rk rx rl rr)
 | rs > delta*ls -> case (rl, rr) of
 (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)
 | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr
 | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
 (_, _) -> error "Failure in Data.Map.balance"
 | ls > delta*rs -> case (ll, lr) of
 (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)
 | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)
 | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)
 (_, _) -> error "Failure in Data.Map.balance"
 | otherwise -> Bin (1+ls+rs) k x l r
{-# NOINLINE balance #-}

-- Functions balanceL and balanceR are specialised versions of balance.
-- balanceL only checks whether the left subtree is too big,
-- balanceR only checks whether the right subtree is too big.

-- balanceL is called when left subtree might have been inserted to or when
-- right subtree might have been deleted from.
balanceL :: k -> a -> Map k a -> Map k a -> Map k a
balanceL k x l r = case r of
 Tip -> case l of
 Tip -> Bin 1 k x Tip Tip
 (Bin _ _ _ Tip Tip) -> Bin 2 k x l Tip
 (Bin _ lk lx Tip (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)
 (Bin _ lk lx ll@(Bin _ _ _ _ _) Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)
 (Bin ls lk lx ll@(Bin lls _ _ _ _) lr@(Bin lrs lrk lrx lrl lrr))
 | lrs < ratio*lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)
 | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)

 (Bin rs _ _ _ _) -> case l of
 Tip -> Bin (1+rs) k x Tip r

 (Bin ls lk lx ll lr)
 | ls > delta*rs -> case (ll, lr) of
 (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)
 | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)
 | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)
 (_, _) -> error "Failure in Data.Map.balanceL"
 | otherwise -> Bin (1+ls+rs) k x l r
{-# NOINLINE balanceL #-}

-- balanceR is called when right subtree might have been inserted to or when
-- left subtree might have been deleted from.
balanceR :: k -> a -> Map k a -> Map k a -> Map k a
balanceR k x l r = case l of
 Tip -> case r of
 Tip -> Bin 1 k x Tip Tip
 (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r
 (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr
 (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)
 (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))
 | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr
 | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)

 (Bin ls _ _ _ _) -> case r of
 Tip -> Bin (1+ls) k x l Tip

 (Bin rs rk rx rl rr)
 | rs > delta*ls -> case (rl, rr) of
 (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)
 | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr
 | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)
 (_, _) -> error "Failure in Data.Map.balanceR"
 | otherwise -> Bin (1+ls+rs) k x l r
{-# NOINLINE balanceR #-}


{--------------------------------------------------------------------
 The bin constructor maintains the size of the tree
--------------------------------------------------------------------}
bin :: k -> a -> Map k a -> Map k a -> Map k a
bin k x l r
 = Bin (size l + size r + 1) k x l r
{-# INLINE bin #-}


{--------------------------------------------------------------------
 Eq converts the tree to a list. In a lazy setting, this
 actually seems one of the faster methods to compare two trees
 and it is certainly the simplest :-)
--------------------------------------------------------------------}
instance (Eq k,Eq a) => Eq (Map k a) where
 t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)

{--------------------------------------------------------------------
 Ord
--------------------------------------------------------------------}

instance (Ord k, Ord v) => Ord (Map k v) where
 compare m1 m2 = compare (toAscList m1) (toAscList m2)

{--------------------------------------------------------------------
 Functor
--------------------------------------------------------------------}
instance Functor (Map k) where
 fmap f m = map f m

instance Traversable (Map k) where
 traverse f = traverseWithKey (\_ -> f)

instance Foldable.Foldable (Map k) where
 fold Tip = mempty
 fold (Bin _ _ v l r) = Foldable.fold l `mappend` v `mappend` Foldable.fold r
 foldr = foldr
 foldl = foldl
 foldMap _ Tip = mempty
 foldMap f (Bin _ _ v l r) = Foldable.foldMap f l `mappend` f v `mappend` Foldable.foldMap f r

instance (NFData k, NFData a) => NFData (Map k a) where
 rnf Tip = ()
 rnf (Bin _ kx x l r) = rnf kx `seq` rnf x `seq` rnf l `seq` rnf r

{--------------------------------------------------------------------
 Read
--------------------------------------------------------------------}
instance (Ord k, Read k, Read e) => Read (Map k e) where
#ifdef __GLASGOW_HASKELL__
 readPrec = parens $ prec 10 $ do
 Ident "fromList" <- lexP
 xs <- readPrec
 return (fromList xs)

 readListPrec = readListPrecDefault
#else
 readsPrec p = readParen (p > 10) $ \ r -> do
 ("fromList",s) <- lex r
 (xs,t) <- reads s
 return (fromList xs,t)
#endif

{--------------------------------------------------------------------
 Show
--------------------------------------------------------------------}
instance (Show k, Show a) => Show (Map k a) where
 showsPrec d m = showParen (d > 10) $
 showString "fromList " . shows (toList m)

-- | /O(n)/. Show the tree that implements the map. The tree is shown
-- in a compressed, hanging format. See 'showTreeWith'.
showTree :: (Show k,Show a) => Map k a -> String
showTree m
 = showTreeWith showElem True False m
 where
 showElem k x = show k ++ ":=" ++ show x


{- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows
 the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is
 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
 @wide@ is 'True', an extra wide version is shown.

> Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t
> (4,())
> +--(2,())
> | +--(1,())
> | +--(3,())
> +--(5,())
>
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t
> (4,())
> |
> +--(2,())
> | |
> | +--(1,())
> | |
> | +--(3,())
> |
> +--(5,())
>
> Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t
> +--(5,())
> |
> (4,())
> |
> | +--(3,())
> | |
> +--(2,())
> |
> +--(1,())

-}
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
showTreeWith showelem hang wide t
 | hang = (showsTreeHang showelem wide [] t) ""
 | otherwise = (showsTree showelem wide [] [] t) ""

showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
showsTree showelem wide lbars rbars t
 = case t of
 Tip -> showsBars lbars . showString "|\n"
 Bin _ kx x Tip Tip
 -> showsBars lbars . showString (showelem kx x) . showString "\n"
 Bin _ kx x l r
 -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .
 showWide wide rbars .
 showsBars lbars . showString (showelem kx x) . showString "\n" .
 showWide wide lbars .
 showsTree showelem wide (withEmpty lbars) (withBar lbars) l

showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
showsTreeHang showelem wide bars t
 = case t of
 Tip -> showsBars bars . showString "|\n"
 Bin _ kx x Tip Tip
 -> showsBars bars . showString (showelem kx x) . showString "\n"
 Bin _ kx x l r
 -> showsBars bars . showString (showelem kx x) . showString "\n" .
 showWide wide bars .
 showsTreeHang showelem wide (withBar bars) l .
 showWide wide bars .
 showsTreeHang showelem wide (withEmpty bars) r

showWide :: Bool -> [String] -> String -> String
showWide wide bars
 | wide = showString (concat (reverse bars)) . showString "|\n"
 | otherwise = id

showsBars :: [String] -> ShowS
showsBars bars
 = case bars of
 [] -> id
 _ -> showString (concat (reverse (tail bars))) . showString node

node :: String
node = "+--"

withBar, withEmpty :: [String] -> [String]
withBar bars = "| ":bars
withEmpty bars = " ":bars

{--------------------------------------------------------------------
 Typeable
--------------------------------------------------------------------}

#include "Typeable.h"
INSTANCE_TYPEABLE2(Map,mapTc,"Map")

{--------------------------------------------------------------------
 Assertions
--------------------------------------------------------------------}
-- | /O(n)/. Test if the internal map structure is valid.
--
-- > valid (fromAscList [(3,"b"), (5,"a")]) == True
-- > valid (fromAscList [(5,"a"), (3,"b")]) == False

valid :: Ord k => Map k a -> Bool
valid t
 = balanced t && ordered t && validsize t

ordered :: Ord a => Map a b -> Bool
ordered t
 = bounded (const True) (const True) t
 where
 bounded lo hi t'
 = case t' of
 Tip -> True
 Bin _ kx _ l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r

-- | Exported only for "Debug.QuickCheck"
balanced :: Map k a -> Bool
balanced t
 = case t of
 Tip -> True
 Bin _ _ _ l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
 balanced l && balanced r

validsize :: Map a b -> Bool
validsize t
 = (realsize t == Just (size t))
 where
 realsize t'
 = case t' of
 Tip -> Just 0
 Bin sz _ _ l r -> case (realsize l,realsize r) of
 (Just n,Just m) | n+m+1 == sz -> Just sz
 _ -> Nothing

{--------------------------------------------------------------------
 Utilities
--------------------------------------------------------------------}
foldlStrict :: (a -> b -> a) -> a -> [b] -> a
foldlStrict f = go
 where
 go z [] = z
 go z (x:xs) = let z' = f z x in z' `seq` go z' xs
{-# INLINE foldlStrict #-}