content: {-# LANGUAGE BangPatterns #-}
  {-# LANGUAGE Trustworthy #-}
  {-# OPTIONS_HADDOCK not-home #-}
  
  -----------------------------------------------------------------------------
  -- |
  -- Module      :  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 "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 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 "Map.Lazy" module, so if they are used
  -- on strict maps, the resulting maps will be lazy.
  -----------------------------------------------------------------------------
  
  -- See the notes at the beginning of Map.Internal.
  
  module Map.Strict.Internal
      (
      -- * Strictness properties
      -- $strictness
  
      -- * Map type
      Map(..)
      , L.Size
  
      -- * 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
      , valid
      ) where
  
  import Prelude hiding (lookup,map,filter,foldr,foldl,null,take,drop,splitAt)
  
  import Map.Internal
    ( Map (..)
    , AreWeStrict (..)
    , WhenMissing (..)
    , WhenMatched (..)
    , runWhenMatched
    , runWhenMissing
    , SimpleWhenMissing
    , SimpleWhenMatched
    , preserveMissing
    , dropMissing
    , filterMissing
    , filterAMissing
    , merge
    , mergeA
    , (!)
    , (!?)
    , (\\)
    , assocs
    , atKeyImpl
    , atKeyPlain
    , 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 Map.Internal.Debug (valid)
  
  import Control.Applicative (Const (..), liftA3)
  import qualified Set.Internal as Set
  import qualified Map.Internal as L
  import Data.Bits (shiftL, shiftR)
  import Data.Coerce
  import Data.Functor.Identity (Identity (..))
  
  import Internal.StrictFold
  import Internal.StrictPair
  import Key
  
  -- $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 :: a -> Key -> Map 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
  
  {--------------------------------------------------------------------
    Construction
  --------------------------------------------------------------------}
  
  -- | /O(1)/. A map with a single element.
  --
  -- > singleton 1 'a'        == fromList [(1, 'a')]
  -- > size (singleton 1 'a') == 1
  
  singleton :: Key -> a -> Map 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 :: Key -> a -> Map a -> Map a
  insert = go
    where
      go :: Key -> a -> Map a -> Map 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
  
  -- | /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 :: (a -> a -> a) -> Key -> a -> Map a -> Map a
  insertWith = go
    where
      go :: (a -> a -> a) -> Key -> a -> Map a -> Map 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
  
  insertWithR :: (a -> a -> a) -> Key -> a -> Map a -> Map a
  insertWithR = go
    where
      go :: (a -> a -> a) -> Key -> a -> Map a -> Map 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
  
  -- | /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 :: (Key -> a -> a -> a) -> Key -> a -> Map a -> Map a
  insertWithKey = go
    where
      go :: (Key -> a -> a -> a) -> Key -> a -> Map a -> Map 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
  
  insertWithKeyR :: (Key -> a -> a -> a) -> Key -> a -> Map a -> Map a
  insertWithKeyR = go
    where
      go :: (Key -> a -> a -> a) -> Key -> a -> Map a -> Map 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
  
  -- | /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 :: (Key -> a -> a -> a) -> Key -> a -> Map a -> (Maybe a, Map a)
  insertLookupWithKey f0 kx0 x0 t0 = toPair $ go f0 kx0 x0 t0
    where
      go ::  (Key -> a -> a -> a) -> Key -> a -> Map a -> StrictPair (Maybe a) (Map 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)
  
  {--------------------------------------------------------------------
    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 :: (a -> a) -> Key -> Map a -> Map a
  adjust f = adjustWithKey (\_ x -> f x)
  
  -- | /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 :: (Key -> a -> a) -> Key -> Map a -> Map a
  adjustWithKey = go
    where
      go :: (Key -> a -> a) -> Key -> Map a -> Map 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
  
  -- | /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 :: (a -> Maybe a) -> Key -> Map a -> Map a
  update f = updateWithKey (\_ x -> f x)
  
  -- | /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 :: (Key -> a -> Maybe a) -> Key -> Map a -> Map a
  updateWithKey = go
    where
      go :: (Key -> a -> Maybe a) -> Key -> Map a -> Map 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
  
  -- | /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 ::  (Key -> a -> Maybe a) -> Key -> Map a -> (Maybe a,Map a)
  updateLookupWithKey f0 k0 t0 = toPair $ go f0 k0 t0
   where
     go ::  (Key -> a -> Maybe a) -> Key -> Map a -> StrictPair (Maybe a) (Map 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)
  
  -- | /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 ::  (Maybe a -> Maybe a) -> Key -> Map a -> Map a
  alter = go
    where
      go ::  (Maybe a -> Maybe a) -> Key -> Map a -> Map 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
  
  -- | /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 => (Maybe a -> f (Maybe a)) -> Key -> Map a -> f (Map a)
  alterF f k m = atKeyImpl Strict k f m
  
  {-# 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
   #-}
  -- 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 ::  Key -> (Maybe a -> Identity (Maybe a)) -> Map a -> Identity (Map a)
  atKeyIdentity k f t = Identity $ atKeyPlain Strict k (coerce f) t
  {-# INLINABLE atKeyIdentity #-}
  
  {--------------------------------------------------------------------
    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 :: (Key -> a -> Maybe a) -> Int -> Map a -> Map 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 a -> Map 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 a -> Map 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 :: (Key -> a -> Maybe a) -> Map a -> Map 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 :: (Key -> a -> Maybe a) -> Map a -> Map 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 ::  (a->a->a) -> [Map a] -> Map a
  unionsWith f ts = foldlStrict (unionWith f) empty ts
  
  {--------------------------------------------------------------------
    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 ::  (a -> a -> a) -> Map a -> Map a -> Map 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
  
  -- | /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 ::  (Key -> a -> a -> a) -> Map a -> Map a -> Map 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
  
  {--------------------------------------------------------------------
    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 ::  (a -> b -> Maybe a) -> Map a -> Map b -> Map a
  differenceWith f = merge preserveMissing dropMissing (zipWithMaybeMatched $ \_ x1 x2 -> f x1 x2)
  
  -- | /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 ::  (Key -> a -> b -> Maybe a) -> Map a -> Map b -> Map a
  differenceWithKey f = merge preserveMissing dropMissing (zipWithMaybeMatched f)
  
  
  {--------------------------------------------------------------------
    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 ::  (a -> b -> c) -> Map a -> Map b -> Map 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
  
  -- | /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 ::  (Key -> a -> b -> c) -> Map a -> Map b -> Map 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
  
  -- | Map covariantly over a @'WhenMissing' f x@.
  mapWhenMissing :: Functor f => (a -> b) -> WhenMissing f x a -> WhenMissing f 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 x y a -> WhenMatched f 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 :: (Key -> x -> y -> Maybe z)
  --                     -> SimpleWhenMatched x y z
  -- @
  zipWithMaybeMatched :: Applicative f
                      => (Key -> x -> y -> Maybe z)
                      -> WhenMatched f 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
                       => (Key -> x -> y -> f (Maybe z))
                       -> WhenMatched f 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
                  => (Key -> x -> y -> f z)
                  -> WhenMatched f 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 :: (Key -> x -> y -> z)
  --                -> SimpleWhenMatched x y z
  -- @
  zipWithMatched :: Applicative f
                 => (Key -> x -> y -> z) -> WhenMatched f 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 :: (Key -> x -> Maybe y) -> SimpleWhenMissing x y
  -- @
  --
  -- prop> mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))
  --
  -- but @mapMaybeMissing@ uses fewer unnecessary 'Applicative' operations.
  mapMaybeMissing :: Applicative f => (Key -> x -> Maybe y) -> WhenMissing f 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 :: (Key -> x -> y) -> SimpleWhenMissing x y
  -- @
  --
  -- prop> mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)
  --
  -- but @mapMissing@ is somewhat faster.
  mapMissing :: Applicative f => (Key -> x -> y) -> WhenMissing f 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
                       => (Key -> x -> f (Maybe y)) -> WhenMissing f 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
                       => (Key -> x -> f y) -> WhenMissing f 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 'Map.Merge.Strict.merge' or
  -- '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 :: (Key -> a -> b -> Maybe c)
               -> (Map a -> Map c)
               -> (Map b -> Map c)
               -> Map a -> Map b -> Map 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 a -> Map b
  mapMaybe f = mapMaybeWithKey (\_ x -> f x)
  
  -- | /O(n)/. Mapeys\/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 :: (Key -> a -> Maybe b) -> Map a -> Map 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
                       => (Key -> a -> f (Maybe b)) -> Map a -> f (Map 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 a -> (Map b, Map c)
  mapEither f m
    = mapEitherWithKey (\_ x -> f x) m
  
  -- | /O(n)/. Mapeys\/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 :: (Key -> a -> Either b c) -> Map a -> (Map b, Map 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 a -> Map 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.
  
  {-# 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
   #-}
  
  -- | /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 :: (Key -> a -> b) -> Map a -> Map 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)
  
  {-# 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
   #-}
  
  -- | /O(n)/.
  -- @'traverseWithKey' f m == 'fromList' <$> 'traverse' (\(k, v) -> (\v' -> v' `seq` (k,v')) <$> f k v) ('toList' m)@
  -- That is, it behaves much like a regular 'traverse' except that the traversing
  -- function also has access to the key associated with a value and the values are
  -- forced before they are installed in the result map.
  --
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
  -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
  traverseWithKey :: Applicative t => (Key -> a -> t b) -> Map a -> t (Map b)
  traverseWithKey f = go
    where
      go Tip = pure Tip
      go (Bin 1 k v _ _) = (\ !v' -> Bin 1 k v' Tip Tip) <$> 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 b -> (a,Map 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 -> Key -> b -> (a,c)) -> a -> Map b -> (a,Map 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 -> Key -> b -> (a,c)) -> a -> Map b -> (a,Map 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 -> Key -> b -> (a,c)) -> a -> Map b -> (a,Map 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@. The value at the greater of the two original keys
  -- is used as the first argument to @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 :: (a -> a -> a) -> (Key -> Key) -> Map a -> Map a
  mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []
  
  {--------------------------------------------------------------------
    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 :: (Key -> a) -> Set.Set -> Map 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 ::  [(Key,a)] -> Map 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)
  
  -- | /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 ::  (a -> a -> a) -> [(Key,a)] -> Map a
  fromListWith f xs = fromListWithKey (\_ x y -> f x y) xs
  {-# INLINABLE fromListWith #-}
  
  -- | /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 ::  (Key -> a -> a -> a) -> [(Key,a)] -> Map a
  fromListWithKey f xs = foldlStrict ins empty xs where
    ins t (k,x) = insertWithKey f k x t
  
  {--------------------------------------------------------------------
    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 :: [(Key,a)] -> Map a
  fromAscList xs = fromAscListWithKey (\_ x _ -> x) xs
  
  -- | /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 :: [(Key,a)] -> Map a
  fromDescList xs = fromDescListWithKey (\_ x _ -> x) xs
  
  -- | /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 :: (a -> a -> a) -> [(Key,a)] -> Map a
  fromAscListWith f xs = fromAscListWithKey (\_ x y -> f x y) xs
  
  -- | /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 :: (a -> a -> a) -> [(Key,a)] -> Map a
  fromDescListWith f xs = fromDescListWithKey (\_ x y -> f x y) xs
  
  -- | /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 :: (Key -> a -> a -> a) -> [(Key,a)] -> Map 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'
  
  -- | /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 :: (Key -> a -> a -> a) -> [(Key,a)] -> Map 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'
  
  -- | /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 :: [(Key,a)] -> Map 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 :: [(Key,a)] -> Map 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)
  