Data/Map/Strict/Internal.hs

    {-# LANGUAGE CPP #-}
    {-# LANGUAGE BangPatterns #-}
    #if __GLASGOW_HASKELL__ >= 703
    {-# LANGUAGE Trustworthy #-}
    #endif

    #include "containers.h"

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Map.Strict.Internal
    -- Copyright   :  (c) Daan Leijen 2002
    --                (c) Andriy Palamarchuk 2008
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Portability :  portable
    --
    -- = WARNING
    --
    -- This module is considered __internal__.
    --
    -- The Package Versioning Policy __does not apply__.
    --
    -- This contents of this module may change __in any way whatsoever__
    -- and __without any warning__ between minor versions of this package.
    --
    -- Authors importing this module are expected to track development
    -- closely.
    --
    -- = Description
    --
    -- An efficient implementation of ordered maps from keys to values
    -- (dictionaries).
    --
    -- API of this module is strict in both the keys and the values.
    -- If you need value-lazy maps, use "Data.Map.Lazy" instead.
    -- The 'Map' type is shared between the lazy and strict modules,
    -- meaning that the same 'Map' value can be passed to functions in
    -- both modules (although that is rarely needed).
    --
    -- These modules are intended to be imported qualified, to avoid name
    -- clashes with Prelude functions, e.g.
    --
    -- >  import qualified Data.Map.Strict 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.
    --
    --  Bounds for 'union', 'intersection', and 'difference' are as given
    --  by
    --
    --    * Guy Blelloch, Daniel Ferizovic, and Yihan Sun,
    --      \"/Just Join for Parallel Ordered Sets/\",
    --      <https://arxiv.org/abs/1602.02120v3>.
    --
    -- 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'.
    --
    -- /Warning/: The size of the map must not exceed @maxBound::Int@. Violation of
    -- this condition is not detected and if the size limit is exceeded, its
    -- behaviour is undefined.
    --
    -- Operation comments contain the operation time complexity in
    -- the Big-O notation (<http://en.wikipedia.org/wiki/Big_O_notation>).
    --
    -- Be aware that the 'Functor', 'Traversable' and 'Data' instances
    -- are the same as for the "Data.Map.Lazy" module, so if they are used
    -- on strict maps, the resulting maps will be lazy.
    -----------------------------------------------------------------------------

    -- See the notes at the beginning of Data.Map.Internal.

    module Data.Map.Strict.Internal
        (
        -- * Strictness properties
        -- $strictness

        -- * 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
        , alterF

        -- * Combine

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

        -- ** Difference
        , difference
        , differenceWith
        , differenceWithKey

        -- ** Intersection
        , intersection
        , intersectionWith
        , intersectionWithKey

        -- ** General combining function
        , SimpleWhenMissing
        , SimpleWhenMatched
        , merge
        , runWhenMatched
        , runWhenMissing

        -- *** @WhenMatched@ tactics
        , zipWithMaybeMatched
        , zipWithMatched

        -- *** @WhenMissing@ tactics
        , mapMaybeMissing
        , dropMissing
        , preserveMissing
        , mapMissing
        , filterMissing

        -- ** Applicative general combining function
        , WhenMissing (..)
        , WhenMatched (..)
        , mergeA

        -- *** @WhenMatched@ tactics
        -- | The tactics described for 'merge' work for
        -- 'mergeA' as well. Furthermore, the following
        -- are available.
        , zipWithMaybeAMatched
        , zipWithAMatched

        -- *** @WhenMissing@ tactics
        -- | The tactics described for 'merge' work for
        -- 'mergeA' as well. Furthermore, the following
        -- are available.
        , traverseMaybeMissing
        , traverseMissing
        , filterAMissing

        -- *** Covariant maps for tactics
        , mapWhenMissing
        , mapWhenMatched

        -- ** Deprecated general combining function

        , mergeWithKey

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

        -- * Folds
        , foldr
        , foldl
        , foldrWithKey
        , foldlWithKey
        , foldMapWithKey

        -- ** 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
        , fromDescList
        , fromDescListWith
        , fromDescListWithKey
        , fromDistinctDescList

        -- * Filter
        , filter
        , filterWithKey
        , restrictKeys
        , withoutKeys
        , partition
        , partitionWithKey
        , takeWhileAntitone
        , dropWhileAntitone
        , spanAntitone

        , mapMaybe
        , mapMaybeWithKey
        , mapEither
        , mapEitherWithKey

        , split
        , splitLookup
        , splitRoot

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

        -- * Indexed
        , lookupIndex
        , findIndex
        , elemAt
        , updateAt
        , deleteAt
        , take
        , drop
        , splitAt

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

        -- * Debugging
        , showTree
        , showTreeWith
        , valid
        ) where

    import Prelude hiding (lookup,map,filter,foldr,foldl,null,take,drop,splitAt)

    import Data.Map.Internal
      ( Map (..)
      , AreWeStrict (..)
      , WhenMissing (..)
      , WhenMatched (..)
      , runWhenMatched
      , runWhenMissing
      , SimpleWhenMissing
      , SimpleWhenMatched
      , preserveMissing
      , dropMissing
      , filterMissing
      , filterAMissing
      , merge
      , mergeA
      , (!)
      , (!?)
      , (\\)
      , assocs
      , atKeyImpl
    #if MIN_VERSION_base(4,8,0)
      , atKeyPlain
    #endif
      , balance
      , balanceL
      , balanceR
      , elemAt
      , elems
      , empty
      , delete
      , deleteAt
      , deleteFindMax
      , deleteFindMin
      , deleteMin
      , deleteMax
      , difference
      , drop
      , dropWhileAntitone
      , filter
      , filterWithKey
      , findIndex
      , findMax
      , findMin
      , foldl
      , foldl'
      , foldlWithKey
      , foldlWithKey'
      , foldMapWithKey
      , foldr
      , foldr'
      , foldrWithKey
      , foldrWithKey'
      , glue
      , insertMax
      , intersection
      , isProperSubmapOf
      , isProperSubmapOfBy
      , isSubmapOf
      , isSubmapOfBy
      , keys
      , keysSet
      , link
      , lookup
      , lookupGE
      , lookupGT
      , lookupIndex
      , lookupLE
      , lookupLT
      , lookupMin
      , lookupMax
      , mapKeys
      , mapKeysMonotonic
      , maxView
      , maxViewWithKey
      , member
      , link2
      , minView
      , minViewWithKey
      , notMember
      , null
      , partition
      , partitionWithKey
      , restrictKeys
      , size
      , spanAntitone
      , split
      , splitAt
      , splitLookup
      , splitRoot
      , take
      , takeWhileAntitone
      , toList
      , toAscList
      , toDescList
      , union
      , unions
      , withoutKeys )

    import Data.Map.Internal.DeprecatedShowTree (showTree, showTreeWith)
    import Data.Map.Internal.Debug (valid)

    import Control.Applicative (Const (..), liftA3)
    #if !MIN_VERSION_base(4,8,0)
    import Control.Applicative (Applicative (..), (<$>))
    #endif
    import qualified Data.Set.Internal as Set
    import qualified Data.Map.Internal as L
    import Utils.Containers.Internal.StrictFold
    import Utils.Containers.Internal.StrictPair

    import Data.Bits (shiftL, shiftR)
    #if __GLASGOW_HASKELL__ >= 709
    import Data.Coerce
    #endif

    #if __GLASGOW_HASKELL__ && MIN_VERSION_base(4,8,0)
    import Data.Functor.Identity (Identity (..))
    #endif

    -- $strictness
    --
    -- This module satisfies the following strictness properties:
    --
    -- 1. Key arguments are evaluated to WHNF;
    --
    -- 2. Keys and values are evaluated to WHNF before they are stored in
    --    the map.
    --
    -- Here's an example illustrating the first property:
    --
    -- > delete undefined m  ==  undefined
    --
    -- Here are some examples that illustrate the second property:
    --
    -- > map (\ v -> undefined) m  ==  undefined      -- m is not empty
    -- > mapKeys (\ k -> undefined) m  ==  undefined  -- m is not empty

    -- [Note: Pointer equality for sharing]
    --
    -- We use pointer equality to enhance sharing between the arguments
    -- of some functions and their results. Notably, we use it
    -- for insert, delete, union, intersection, and difference. We do
    -- *not* use it for functions, like insertWith, unionWithKey,
    -- intersectionWith, etc., that allow the user to modify the elements.
    -- While we *could* do so, we would only get sharing under fairly
    -- narrow conditions and at a relatively high cost. It does not seem
    -- worth the price.

    {--------------------------------------------------------------------
      Query
    --------------------------------------------------------------------}

    -- | /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'

    -- See Map.Internal.Note: Local 'go' functions and capturing
    findWithDefault :: Ord k => a -> k -> Map k a -> a
    findWithDefault def k = k `seq` go
      where
        go Tip = def
        go (Bin _ kx x l r) = case compare k kx of
          LT -> go l
          GT -> go r
          EQ -> x
    #if __GLASGOW_HASKELL__
    {-# INLINABLE findWithDefault #-}
    #else
    {-# INLINE findWithDefault #-}
    #endif

    {--------------------------------------------------------------------
      Construction
    --------------------------------------------------------------------}

    -- | /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 = x `seq` 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 Map.Internal.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
        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__
    {-# INLINABLE insert #-}
    #else
    {-# INLINE insert #-}
    #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 = go
      where
        go :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
        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 -> let !y' = f x y in Bin sy kx y' l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertWith #-}
    #else
    {-# INLINE insertWith #-}
    #endif

    insertWithR :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
    insertWithR = go
      where
        go :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
        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 -> let !y' = f y x in Bin sy ky y' l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertWithR #-}
    #else
    {-# INLINE insertWithR #-}
    #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 Map.Internal.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
        -- Forcing `kx` may look redundant, but it's possible `compare` will
        -- be lazy.
        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 -> let !x' = f kx x y
                      in Bin sy kx x' l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertWithKey #-}
    #else
    {-# INLINE insertWithKey #-}
    #endif

    insertWithKeyR :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
    insertWithKeyR = go
      where
        go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a
        -- Forcing `kx` may look redundant, but it's possible `compare` will
        -- be lazy.
        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 -> let !y' = f ky y x
                      in Bin sy ky y' l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertWithKeyR #-}
    #else
    {-# INLINE insertWithKeyR #-}
    #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 Map.Internal.Note: Type of local 'go' function
    insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a
                        -> (Maybe a, Map k a)
    insertLookupWithKey f0 kx0 x0 t0 = toPair $ go f0 kx0 x0 t0
      where
        go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> StrictPair (Maybe a) (Map k a)
        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 -> let x' = f kx x y
                      in x' `seq` (Just y :*: Bin sy kx x' l r)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE insertLookupWithKey #-}
    #else
    {-# INLINE insertLookupWithKey #-}
    #endif

    {--------------------------------------------------------------------
      Deletion
    --------------------------------------------------------------------}

    -- | /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__
    {-# 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 = go
      where
        go :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a
        go _ !_ Tip = Tip
        go f k (Bin sx kx x l r) =
            case compare k kx of
               LT -> Bin sx kx x (go f k l) r
               GT -> Bin sx kx x l (go f k r)
               EQ -> Bin sx kx x' l r
                 where !x' = f kx x
    #if __GLASGOW_HASKELL__
    {-# 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__
    {-# 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 Map.Internal.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
        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' -> x' `seq` Bin sx kx x' l r
                       Nothing -> glue l r
    #if __GLASGOW_HASKELL__
    {-# 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 Map.Internal.Note: Type of local 'go' function
    updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)
    updateLookupWithKey f0 k0 t0 = toPair $ go f0 k0 t0
     where
       go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> StrictPair (Maybe a) (Map k a)
       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' -> x' `seq` (Just x' :*: Bin sx kx x' l r)
                           Nothing -> (Just x :*: glue l r)
    #if __GLASGOW_HASKELL__
    {-# 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 Map.Internal.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
        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' -> x' `seq` Bin sx kx x' l r
                           Nothing -> glue l r
    #if __GLASGOW_HASKELL__
    {-# INLINABLE alter #-}
    #else
    {-# INLINE alter #-}
    #endif

    -- | /O(log n)/. The expression (@'alterF' f k map@) alters the value @x@ at @k@, or absence thereof.
    -- 'alterF' can be used to inspect, insert, delete, or update a value in a 'Map'.
    -- In short: @'lookup' k \<$\> 'alterF' f k m = f ('lookup' k m)@.
    --
    -- Example:
    --
    -- @
    -- interactiveAlter :: Int -> Map Int String -> IO (Map Int String)
    -- interactiveAlter k m = alterF f k m where
    --   f Nothing -> do
    --      putStrLn $ show k ++
    --          " was not found in the map. Would you like to add it?"
    --      getUserResponse1 :: IO (Maybe String)
    --   f (Just old) -> do
    --      putStrLn "The key is currently bound to " ++ show old ++
    --          ". Would you like to change or delete it?"
    --      getUserresponse2 :: IO (Maybe String)
    -- @
    --
    -- 'alterF' is the most general operation for working with an individual
    -- key that may or may not be in a given map. When used with trivial
    -- functors like 'Identity' and 'Const', it is often slightly slower than
    -- more specialized combinators like 'lookup' and 'insert'. However, when
    -- the functor is non-trivial and key comparison is not particularly cheap,
    -- it is the fastest way.
    --
    -- Note on rewrite rules:
    --
    -- This module includes GHC rewrite rules to optimize 'alterF' for
    -- the 'Const' and 'Identity' functors. In general, these rules
    -- improve performance. The sole exception is that when using
    -- 'Identity', deleting a key that is already absent takes longer
    -- than it would without the rules. If you expect this to occur
    -- a very large fraction of the time, you might consider using a
    -- private copy of the 'Identity' type.
    --
    -- Note: 'alterF' is a flipped version of the 'at' combinator from
    -- 'Control.Lens.At'.
    alterF :: (Functor f, Ord k)
           => (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)
    alterF f k m = atKeyImpl Strict k f m

    #ifndef __GLASGOW_HASKELL__
    {-# INLINE alterF #-}
    #else
    {-# INLINABLE [2] alterF #-}

    -- We can save a little time by recognizing the special case of
    -- `Control.Applicative.Const` and just doing a lookup.
    {-# RULES
    "alterF/Const" forall k (f :: Maybe a -> Const b (Maybe a)) . alterF f k = \m -> Const . getConst . f $ lookup k m
     #-}
    #if MIN_VERSION_base(4,8,0)
    -- base 4.8 and above include Data.Functor.Identity, so we can
    -- save a pretty decent amount of time by handling it specially.
    {-# RULES
    "alterF/Identity" forall k f . alterF f k = atKeyIdentity k f
     #-}

    atKeyIdentity :: Ord k => k -> (Maybe a -> Identity (Maybe a)) -> Map k a -> Identity (Map k a)
    atKeyIdentity k f t = Identity $ atKeyPlain Strict k (coerce f) t
    {-# INLINABLE atKeyIdentity #-}
    #endif
    #endif

    {--------------------------------------------------------------------
      Indexing
    --------------------------------------------------------------------}

    -- | /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' -> x' `seq` Bin sx kx x' l r
                  Nothing -> glue l r
          where
            sizeL = size l

    {--------------------------------------------------------------------
      Minimal, Maximal
    --------------------------------------------------------------------}

    -- | /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' -> x' `seq` 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' -> x' `seq` Bin sx kx x' l Tip
    updateMaxWithKey f (Bin _ kx x l r)    = balanceL kx x l (updateMaxWithKey f r)

    {--------------------------------------------------------------------
      Union.
    --------------------------------------------------------------------}

    -- | 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__
    {-# INLINABLE unionsWith #-}
    #endif

    {--------------------------------------------------------------------
      Union with a combining function
    --------------------------------------------------------------------}
    -- | /O(m*log(n\/m + 1)), m <= n/. Union with a combining function.
    --
    -- > 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 t1 Tip = t1
    unionWith f t1 (Bin _ k x Tip Tip) = insertWithR f k x t1
    unionWith f (Bin _ k x Tip Tip) t2 = insertWith f k x t2
    unionWith _f Tip t2 = t2
    unionWith f (Bin _ k1 x1 l1 r1) t2 = case splitLookup k1 t2 of
      (l2, mb, r2) -> link k1 x1' (unionWith f l1 l2) (unionWith f r1 r2)
        where !x1' = maybe x1 (f x1) mb
    #if __GLASGOW_HASKELL__
    {-# INLINABLE unionWith #-}
    #endif

    -- | /O(m*log(n\/m + 1)), m <= n/.
    -- Union with a combining function.
    --
    -- > 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 Tip = t1
    unionWithKey f t1 (Bin _ k x Tip Tip) = insertWithKeyR f k x t1
    unionWithKey f (Bin _ k x Tip Tip) t2 = insertWithKey f k x t2
    unionWithKey _f Tip t2 = t2
    unionWithKey f (Bin _ k1 x1 l1 r1) t2 = case splitLookup k1 t2 of
      (l2, mb, r2) -> link k1 x1' (unionWithKey f l1 l2) (unionWithKey f r1 r2)
        where !x1' = maybe x1 (f k1 x1) mb
    #if __GLASGOW_HASKELL__
    {-# INLINABLE unionWithKey #-}
    #endif

    {--------------------------------------------------------------------
      Difference
    --------------------------------------------------------------------}

    -- | /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@.
    --
    -- > 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 = merge preserveMissing dropMissing (zipWithMaybeMatched $ \_ x1 x2 -> f x1 x2)
    #if __GLASGOW_HASKELL__
    {-# 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@.
    --
    -- > 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 = merge preserveMissing dropMissing (zipWithMaybeMatched f)
    #if __GLASGOW_HASKELL__
    {-# INLINABLE differenceWithKey #-}
    #endif

    {--------------------------------------------------------------------
      Intersection
    --------------------------------------------------------------------}

    -- | /O(m*log(n\/m + 1)), m <= n/. 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 Tip _ = Tip
    intersectionWith _f _ Tip = Tip
    intersectionWith f (Bin _ k x1 l1 r1) t2 = case mb of
        Just x2 -> let !x1' = f x1 x2 in link k x1' l1l2 r1r2
        Nothing -> link2 l1l2 r1r2
      where
        !(l2, mb, r2) = splitLookup k t2
        !l1l2 = intersectionWith f l1 l2
        !r1r2 = intersectionWith f r1 r2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE intersectionWith #-}
    #endif

    -- | /O(m*log(n\/m + 1)), m <= n/. Intersection with a combining function.
    --
    -- > 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 Tip _ = Tip
    intersectionWithKey _f _ Tip = Tip
    intersectionWithKey f (Bin _ k x1 l1 r1) t2 = case mb of
        Just x2 -> let !x1' = f k x1 x2 in link k x1' l1l2 r1r2
        Nothing -> link2 l1l2 r1r2
      where
        !(l2, mb, r2) = splitLookup k t2
        !l1l2 = intersectionWithKey f l1 l2
        !r1r2 = intersectionWithKey f r1 r2
    #if __GLASGOW_HASKELL__
    {-# INLINABLE intersectionWithKey #-}
    #endif

    -- | Map covariantly over a @'WhenMissing' f k x@.
    mapWhenMissing :: Functor f => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b
    mapWhenMissing f q = WhenMissing
      { missingSubtree = fmap (map f) . missingSubtree q
      , missingKey = \k x -> fmap (forceMaybe . fmap f) $ missingKey q k x}

    -- | Map covariantly over a @'WhenMatched' f k x y@.
    mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b
    mapWhenMatched f q = WhenMatched
      { matchedKey = \k x y -> fmap (forceMaybe . fmap f) $ runWhenMatched q k x y }

    -- | When a key is found in both maps, apply a function to the
    -- key and values and maybe use the result in the merged map.
    --
    -- @
    -- zipWithMaybeMatched :: (k -> x -> y -> Maybe z)
    --                     -> SimpleWhenMatched k x y z
    -- @
    zipWithMaybeMatched :: Applicative f
                        => (k -> x -> y -> Maybe z)
                        -> WhenMatched f k x y z
    zipWithMaybeMatched f = WhenMatched $
      \k x y -> pure $! forceMaybe $! f k x y
    {-# INLINE zipWithMaybeMatched #-}

    -- | When a key is found in both maps, apply a function to the
    -- key and values, perform the resulting action, and maybe use
    -- the result in the merged map.
    --
    -- This is the fundamental 'WhenMatched' tactic.
    zipWithMaybeAMatched :: Applicative f
                         => (k -> x -> y -> f (Maybe z))
                         -> WhenMatched f k x y z
    zipWithMaybeAMatched f = WhenMatched $
      \ k x y -> forceMaybe <$> f k x y
    {-# INLINE zipWithMaybeAMatched #-}

    -- | When a key is found in both maps, apply a function to the
    -- key and values to produce an action and use its result in the merged map.
    zipWithAMatched :: Applicative f
                    => (k -> x -> y -> f z)
                    -> WhenMatched f k x y z
    zipWithAMatched f = WhenMatched $
      \ k x y -> (Just $!) <$> f k x y
    {-# INLINE zipWithAMatched #-}

    -- | When a key is found in both maps, apply a function to the
    -- key and values and use the result in the merged map.
    --
    -- @
    -- zipWithMatched :: (k -> x -> y -> z)
    --                -> SimpleWhenMatched k x y z
    -- @
    zipWithMatched :: Applicative f
                   => (k -> x -> y -> z) -> WhenMatched f k x y z
    zipWithMatched f = WhenMatched $
      \k x y -> pure $! Just $! f k x y
    {-# INLINE zipWithMatched #-}

    -- | Map over the entries whose keys are missing from the other map,
    -- optionally removing some. This is the most powerful 'SimpleWhenMissing'
    -- tactic, but others are usually more efficient.
    --
    -- @
    -- mapMaybeMissing :: (k -> x -> Maybe y) -> SimpleWhenMissing k x y
    -- @
    --
    -- prop> mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))
    --
    -- but @mapMaybeMissing@ uses fewer unnecessary 'Applicative' operations.
    mapMaybeMissing :: Applicative f => (k -> x -> Maybe y) -> WhenMissing f k x y
    mapMaybeMissing f = WhenMissing
      { missingSubtree = \m -> pure $! mapMaybeWithKey f m
      , missingKey = \k x -> pure $! forceMaybe $! f k x }
    {-# INLINE mapMaybeMissing #-}

    -- | Map over the entries whose keys are missing from the other map.
    --
    -- @
    -- mapMissing :: (k -> x -> y) -> SimpleWhenMissing k x y
    -- @
    --
    -- prop> mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)
    --
    -- but @mapMissing@ is somewhat faster.
    mapMissing :: Applicative f => (k -> x -> y) -> WhenMissing f k x y
    mapMissing f = WhenMissing
      { missingSubtree = \m -> pure $! mapWithKey f m
      , missingKey = \k x -> pure $! Just $! f k x }
    {-# INLINE mapMissing #-}

    -- | Traverse over the entries whose keys are missing from the other map,
    -- optionally producing values to put in the result.
    -- This is the most powerful 'WhenMissing' tactic, but others are usually
    -- more efficient.
    traverseMaybeMissing :: Applicative f
                         => (k -> x -> f (Maybe y)) -> WhenMissing f k x y
    traverseMaybeMissing f = WhenMissing
      { missingSubtree = traverseMaybeWithKey f
      , missingKey = \k x -> forceMaybe <$> f k x }
    {-# INLINE traverseMaybeMissing #-}

    -- | Traverse over the entries whose keys are missing from the other map.
    traverseMissing :: Applicative f
                         => (k -> x -> f y) -> WhenMissing f k x y
    traverseMissing f = WhenMissing
      { missingSubtree = traverseWithKey f
      , missingKey = \k x -> (Just $!) <$> f k x }
    {-# INLINE traverseMissing #-}

    forceMaybe :: Maybe a -> Maybe a
    forceMaybe Nothing = Nothing
    forceMaybe m@(Just !_) = m
    {-# INLINE forceMaybe #-}

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

    -- | /O(n+m)/. An unsafe universal combining function.
    --
    -- WARNING: This function can produce corrupt maps and its results
    -- may depend on the internal structures of its inputs. Users should
    -- prefer 'Data.Map.Merge.Strict.merge' or
    -- 'Data.Map.Merge.Strict.mergeA'.
    --
    -- When 'mergeWithKey' is given three arguments, it is inlined to the call
    -- site. You should therefore use 'mergeWithKey' only to define 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
    -- 'Map'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 (Bin _ kx x l1 r1) t2 =
          case found of
            Nothing -> case g1 (singleton kx x) of
                         Tip -> link2 l' r'
                         (Bin _ _ x' Tip Tip) -> link kx x' l' r'
                         _ -> error "mergeWithKey: Given function only1 does not fulfill required conditions (see documentation)"
            Just x2 -> case f kx x x2 of
                         Nothing -> link2 l' r'
                         Just x' -> link kx x' l' r'
          where
            (l2, found, r2) = splitLookup kx t2
            l' = go l1 l2
            r' = go r1 r2
    {-# INLINE mergeWithKey #-}

    {--------------------------------------------------------------------
      Filter and partition
    --------------------------------------------------------------------}

    -- | /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  -> y `seq` link kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)
      Nothing -> link2 (mapMaybeWithKey f l) (mapMaybeWithKey f r)

    -- | /O(n)/. Traverse keys\/values and collect the 'Just' results.
    --
    -- @since 0.5.8

    traverseMaybeWithKey :: Applicative f
                         => (k -> a -> f (Maybe b)) -> Map k a -> f (Map k b)
    traverseMaybeWithKey = go
      where
        go _ Tip = pure Tip
        go f (Bin _ kx x Tip Tip) = maybe Tip (\ !x' -> Bin 1 kx x' Tip Tip) <$> f kx x
        go f (Bin _ kx x l r) = liftA3 combine (go f l) (f kx x) (go f r)
          where
            combine !l' mx !r' = case mx of
              Nothing -> link2 l' r'
              Just !x' -> link kx x' l' 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 f0 t0 = toPair $ go f0 t0
      where
        go _ Tip = (Tip :*: Tip)
        go f (Bin _ kx x l r) = case f kx x of
          Left y  -> y `seq` (link kx y l1 r1 :*: link2 l2 r2)
          Right z -> z `seq` (link2 l1 r1 :*: link kx z l2 r2)
         where
            (l1 :*: l2) = go f l
            (r1 :*: r2) = go 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 f = go
      where
        go Tip = Tip
        go (Bin sx kx x l r) = let !x' = f x in Bin sx kx x' (go l) (go r)
    -- We use `go` to let `map` inline. This is important if `f` is a constant
    -- function.

    #ifdef __GLASGOW_HASKELL__
    {-# NOINLINE [1] map #-}
    {-# RULES
    "map/map" forall f g xs . map f (map g xs) = map (\x -> f $! g x) xs
    "map/mapL" forall f g xs . map f (L.map g xs) = map (\x -> f (g x)) xs
     #-}
    #endif

    -- | /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) =
      let x' = f kx x
      in x' `seq` Bin sx kx x' (mapWithKey f l) (mapWithKey f r)

    #ifdef __GLASGOW_HASKELL__
    {-# NOINLINE [1] mapWithKey #-}
    {-# RULES
    "mapWithKey/mapWithKey" forall f g xs . mapWithKey f (mapWithKey g xs) =
      mapWithKey (\k a -> f k $! g k a) xs
    "mapWithKey/mapWithKeyL" forall f g xs . mapWithKey f (L.mapWithKey g xs) =
      mapWithKey (\k a -> f k (g k a)) xs
    "mapWithKey/map" forall f g xs . mapWithKey f (map g xs) =
      mapWithKey (\k a -> f k $! g a) xs
    "mapWithKey/mapL" forall f g xs . mapWithKey f (L.map g xs) =
      mapWithKey (\k a -> f k (g a)) xs
    "map/mapWithKey" forall f g xs . map f (mapWithKey g xs) =
      mapWithKey (\k a -> f $! g k a) xs
    "map/mapWithKeyL" forall f g xs . map f (L.mapWithKey g xs) =
      mapWithKey (\k a -> f (g k a)) xs
     #-}
    #endif

    -- | /O(n)/.
    -- @'traverseWithKey' f m == 'fromList' <$> 'traverse' (\(k, v) -> (\v' -> v' `seq` (k,v')) <$> f k v) ('toList' m)@
    -- That is, it behaves much like a regular 'traverse' except that the traversing
    -- function also has access to the key associated with a value and the values are
    -- forced before they are installed in the result map.
    --
    -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
    -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
    traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)
    traverseWithKey f = go
      where
        go Tip = pure Tip
        go (Bin 1 k v _ _) = (\ !v' -> Bin 1 k v' Tip Tip) <$> f k v
        go (Bin s k v l r) = liftA3 (\ l' !v' r' -> Bin s k v' l' r') (go l) (f k v) (go r)
    {-# INLINE traverseWithKey #-}

    -- | /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 x' `seq` (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 x' `seq` (a3,Bin sx kx x' l' r')

    -- | /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__
    {-# INLINABLE mapKeysWith #-}
    #endif

    {--------------------------------------------------------------------
      Conversions
    --------------------------------------------------------------------}

    -- | /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) = case f x of v -> v `seq` Bin sz x v (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.
    --
    -- If the keys of the list are ordered, linear-time implementation is used,
    -- with the performance equal to 'fromDistinctAscList'.
    --
    -- > 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")]

    -- For some reason, when 'singleton' is used in fromList or in
    -- create, it is not inlined, so we inline it manually.
    fromList :: Ord k => [(k,a)] -> Map k a
    fromList [] = Tip
    fromList [(kx, x)] = x `seq` Bin 1 kx x Tip Tip
    fromList ((kx0, x0) : xs0) | not_ordered kx0 xs0 = x0 `seq` fromList' (Bin 1 kx0 x0 Tip Tip) xs0
                               | otherwise = x0 `seq` go (1::Int) (Bin 1 kx0 x0 Tip Tip) xs0
      where
        not_ordered _ [] = False
        not_ordered kx ((ky,_) : _) = kx >= ky
        {-# INLINE not_ordered #-}

        fromList' t0 xs = foldlStrict ins t0 xs
          where ins t (k,x) = insert k x t

        go !_ t [] = t
        go _ t [(kx, x)] = x `seq` insertMax kx x t
        go s l xs@((kx, x) : xss) | not_ordered kx xss = fromList' l xs
                                  | otherwise = case create s xss of
                                      (r, ys, []) -> x `seq` go (s `shiftL` 1) (link kx x l r) ys
                                      (r, _,  ys) -> x `seq` fromList' (link kx x l r) ys

        -- The create is returning a triple (tree, xs, ys). Both xs and ys
        -- represent not yet processed elements and only one of them can be nonempty.
        -- If ys is nonempty, the keys in ys are not ordered with respect to tree
        -- and must be inserted using fromList'. Otherwise the keys have been
        -- ordered so far.
        create !_ [] = (Tip, [], [])
        create s xs@(xp : xss)
          | s == 1 = case xp of (kx, x) | not_ordered kx xss -> x `seq` (Bin 1 kx x Tip Tip, [], xss)
                                        | otherwise -> x `seq` (Bin 1 kx x Tip Tip, xss, [])
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_, [], _) -> res
                          (l, [(ky, y)], zs) -> y `seq` (insertMax ky y l, [], zs)
                          (l, ys@((ky, y):yss), _) | not_ordered ky yss -> (l, [], ys)
                                                   | otherwise -> case create (s `shiftR` 1) yss of
                                                       (r, zs, ws) -> y `seq` (link ky y l r, zs, ws)
    #if __GLASGOW_HASKELL__
    {-# 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__
    {-# 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__
    {-# INLINABLE fromListWithKey #-}
    #endif

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

      Note that if [xs] is ascending then:
        fromAscList xs       == fromList xs
        fromAscListWith f xs == fromListWith f xs

      If [xs] is descending then:
        fromDescList xs       == fromList xs
        fromDescListWith 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__
    {-# INLINABLE fromAscList #-}
    #endif

    -- | /O(n)/. Build a map from a descending list in linear time.
    -- /The precondition (input list is descending) is not checked./
    --
    -- > fromDescList [(5,"a"), (3,"b")]          == fromList [(3, "b"), (5, "a")]
    -- > fromDescList [(5,"a"), (5,"b"), (3,"a")] == fromList [(3, "b"), (5, "b")]
    -- > valid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True
    -- > valid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False
    fromDescList :: Eq k => [(k,a)] -> Map k a
    fromDescList xs
      = fromDescListWithKey (\_ x _ -> x) xs
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromDescList #-}
    #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__
    {-# INLINABLE fromAscListWith #-}
    #endif

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

    fromDescListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a
    fromDescListWith f xs
      = fromDescListWithKey (\_ x y -> f x y) xs
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromDescListWith #-}
    #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 yy `seq` combineEq' (kx,yy) xs'
        | otherwise = z:combineEq' x xs'
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromAscListWithKey #-}
    #endif

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

    fromDescListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a
    fromDescListWithKey f xs
      = fromDistinctDescList (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 yy `seq` combineEq' (kx,yy) xs'
        | otherwise = z:combineEq' x xs'
    #if __GLASGOW_HASKELL__
    {-# INLINABLE fromDescListWithKey #-}
    #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

    -- For some reason, when 'singleton' is used in fromDistinctAscList or in
    -- create, it is not inlined, so we inline it manually.
    fromDistinctAscList :: [(k,a)] -> Map k a
    fromDistinctAscList [] = Tip
    fromDistinctAscList ((kx0, x0) : xs0) = x0 `seq` go (1::Int) (Bin 1 kx0 x0 Tip Tip) xs0
      where
        go !_ t [] = t
        go s l ((kx, x) : xs) =
          case create s xs of
            (r :*: ys) -> x `seq` let !t' = link kx x l r
                               in go (s `shiftL` 1) t' ys

        create !_ [] = (Tip :*: [])
        create s xs@(x' : xs')
          | s == 1 = case x' of (kx, x) -> x `seq` (Bin 1 kx x Tip Tip :*: xs')
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_ :*: []) -> res
                          (l :*: (ky, y):ys) -> case create (s `shiftR` 1) ys of
                            (r :*: zs) -> y `seq` (link ky y l r :*: zs)

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

    -- For some reason, when 'singleton' is used in fromDistinctDescList or in
    -- create, it is not inlined, so we inline it manually.
    fromDistinctDescList :: [(k,a)] -> Map k a
    fromDistinctDescList [] = Tip
    fromDistinctDescList ((kx0, x0) : xs0) = x0 `seq` go (1::Int) (Bin 1 kx0 x0 Tip Tip) xs0
      where
        go !_ t [] = t
        go s r ((kx, x) : xs) =
          case create s xs of
            (l :*: ys) -> x `seq` let !t' = link kx x l r
                                  in go (s `shiftL` 1) t' ys

        create !_ [] = (Tip :*: [])
        create s xs@(x' : xs')
          | s == 1 = case x' of (kx, x) -> x `seq` (Bin 1 kx x Tip Tip :*: xs')
          | otherwise = case create (s `shiftR` 1) xs of
                          res@(_ :*: []) -> res
                          (r :*: (ky, y):ys) -> case create (s `shiftR` 1) ys of
                            (l :*: zs) -> y `seq` (link ky y l r :*: zs)