src/Data/IntMap/Strict/Internal.hs

    {-# LANGUAGE CPP #-}
    {-# LANGUAGE BangPatterns #-}

    #include "containers.h"

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.IntMap.Strict.Internal
    -- Copyright   :  (c) Daan Leijen 2002
    --                (c) Andriy Palamarchuk 2008
    -- License     :  BSD-style
    -- Maintainer  :  libraries@haskell.org
    -- Portability :  portable
    --
    --
    -- = Finite Int Maps (strict interface)
    --
    -- The @'IntMap' v@ type represents a finite map (sometimes called a dictionary)
    -- from key of type @Int@ to values of type @v@.
    --
    -- Each function in this module is careful to force values before installing
    -- them in an 'IntMap'. This is usually more efficient when laziness is not
    -- necessary. When laziness /is/ required, use the functions in
    -- "Data.IntMap.Lazy".
    --
    -- In particular, the functions in this module obey the following law:
    --
    --  - If all values stored in all maps in the arguments are in WHNF, then all
    --    values stored in all maps in the results will be in WHNF once those maps
    --    are evaluated.
    --
    -- For a walkthrough of the most commonly used functions see the
    -- <https://haskell-containers.readthedocs.io/en/latest/map.html maps introduction>.
    --
    -- This module is intended to be imported qualified, to avoid name clashes with
    -- Prelude functions:
    --
    -- > import Data.IntMap.Strict (IntMap)
    -- > import qualified Data.IntMap.Strict as IntMap
    --
    -- Note that the implementation is generally /left-biased/. Functions that take
    -- two maps as arguments and combine them, such as `union` and `intersection`,
    -- prefer the values in the first argument to those in the second.
    --
    --
    -- == Detailed performance information
    --
    -- The amortized running time is given for each operation, with /n/ referring to
    -- the number of entries in the map and /W/ referring to the number of bits in
    -- an 'Int' (32 or 64).
    --
    -- Benchmarks comparing "Data.IntMap.Strict" with other dictionary
    -- implementations can be found at https://github.com/haskell-perf/dictionaries.
    --
    --
    -- == Warning
    --
    -- The 'IntMap' type is shared between the lazy and strict modules, meaning that
    -- the same 'IntMap' value can be passed to functions in both modules. This
    -- means that the 'Functor', 'Traversable' and 'Data.Data.Data' instances are
    -- the same as for the "Data.IntMap.Lazy" module, so if they are used the
    -- resulting map may contain suspended values (thunks).
    --
    --
    -- == Implementation
    --
    -- The implementation is based on /big-endian patricia trees/.  This data
    -- structure performs especially well on binary operations like 'union' and
    -- 'intersection'. Additionally, benchmarks show that it is also (much) faster
    -- on insertions and deletions when compared to a generic size-balanced map
    -- implementation (see "Data.Map").
    --
    --    * Chris Okasaki and Andy Gill,  \"/Fast Mergeable Integer Maps/\",
    --      Workshop on ML, September 1998, pages 77-86,
    --      <http://citeseer.ist.psu.edu/okasaki98fast.html>
    --
    --    * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
    --      Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
    --      October 1968, pages 514-534.
    --
    -----------------------------------------------------------------------------

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

    module Data.IntMap.Strict.Internal (
        -- * Map type
    #if !defined(TESTING)
        IntMap, Key          -- instance Eq,Show
    #else
        IntMap(..), Key          -- instance Eq,Show
    #endif

        -- * Construction
        , empty
        , singleton
        , fromSet

        -- ** From Unordered Lists
        , fromList
        , fromListWith
        , fromListWithKey

        -- ** From Ascending Lists
        , fromAscList
        , fromAscListWith
        , fromAscListWithKey
        , fromDistinctAscList

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

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

        -- * Query
        -- ** Lookup
        , lookup
        , (!?)
        , (!)
        , findWithDefault
        , member
        , notMember
        , lookupLT
        , lookupGT
        , lookupLE
        , lookupGE

        -- ** Size
        , null
        , size

        -- * Combine

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

        -- ** Difference
        , difference
        , (\\)
        , differenceWith
        , differenceWithKey

        -- ** Intersection
        , intersection
        , intersectionWith
        , intersectionWithKey

        -- ** Disjoint
        , disjoint

        -- ** Universal 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

        -- ** Lists
        , toList

    -- ** Ordered lists
        , toAscList
        , toDescList

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

        , mapMaybe
        , mapMaybeWithKey
        , mapEither
        , mapEitherWithKey

        , split
        , splitLookup
        , splitRoot

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

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

    #ifdef __GLASGOW_HASKELL__
        -- * Debugging
        , showTree
        , showTreeWith
    #endif
        ) where

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

    import Data.Bits
    import qualified Data.IntMap.Internal as L
    import Data.IntMap.Internal
      ( IntMap (..)
      , Key
      , Prefix
      , Mask
      , mask
      , branchMask
      , shorter
      , nomatch
      , zero
      , natFromInt
      , intFromNat
      , bin
      , binCheckLeft
      , binCheckRight
      , link

      , (\\)
      , (!)
      , (!?)
      , empty
      , assocs
      , filter
      , filterWithKey
      , findMin
      , findMax
      , foldMapWithKey
      , foldr
      , foldl
      , foldr'
      , foldl'
      , foldlWithKey
      , foldrWithKey
      , foldlWithKey'
      , foldrWithKey'
      , keysSet
      , mergeWithKey'
      , delete
      , deleteMin
      , deleteMax
      , deleteFindMax
      , deleteFindMin
      , difference
      , elems
      , intersection
      , disjoint
      , isProperSubmapOf
      , isProperSubmapOfBy
      , isSubmapOf
      , isSubmapOfBy
      , lookup
      , lookupLE
      , lookupGE
      , lookupLT
      , lookupGT
      , lookupMin
      , lookupMax
      , minView
      , maxView
      , minViewWithKey
      , maxViewWithKey
      , keys
      , mapKeys
      , mapKeysMonotonic
      , member
      , notMember
      , null
      , partition
      , partitionWithKey
      , restrictKeys
      , size
      , split
      , splitLookup
      , splitRoot
      , toAscList
      , toDescList
      , toList
      , union
      , unions
      , withoutKeys
      )
    #ifdef __GLASGOW_HASKELL__
    import Data.IntMap.Internal.DeprecatedDebug (showTree, showTreeWith)
    #endif
    import qualified Data.IntSet.Internal as IntSet
    import Utils.Containers.Internal.BitUtil
    import Utils.Containers.Internal.StrictPair
    #if !MIN_VERSION_base(4,8,0)
    import Data.Functor((<$>))
    #endif
    import Control.Applicative (Applicative (..), liftA2)
    import qualified Data.Foldable as Foldable
    #if !MIN_VERSION_base(4,8,0)
    import Data.Foldable (Foldable())
    #endif

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

    -- | /O(min(n,W))/. The expression @('findWithDefault' def k map)@
    -- returns the value at key @k@ or returns @def@ when the key is not an
    -- element of the map.
    --
    -- > findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
    -- > findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'

    -- See IntMap.Internal.Note: Local 'go' functions and capturing]
    findWithDefault :: a -> Key -> IntMap a -> a
    findWithDefault def !k = go
      where
        go (Bin p m l r) | nomatch k p m = def
                         | zero k m  = go l
                         | otherwise = go r
        go (Tip kx x) | k == kx   = x
                      | otherwise = def
        go Nil = def

    {--------------------------------------------------------------------
      Construction
    --------------------------------------------------------------------}
    -- | /O(1)/. A map of one element.
    --
    -- > singleton 1 'a'        == fromList [(1, 'a')]
    -- > size (singleton 1 'a') == 1

    singleton :: Key -> a -> IntMap a
    singleton k !x
      = Tip k x
    {-# INLINE singleton #-}

    {--------------------------------------------------------------------
      Insert
    --------------------------------------------------------------------}
    -- | /O(min(n,W))/. Insert a new key\/value pair in the map.
    -- If the key is already present in the map, the associated value is
    -- replaced with the supplied value, i.e. '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'

    insert :: Key -> a -> IntMap a -> IntMap a
    insert !k !x t =
      case t of
        Bin p m l r
          | nomatch k p m -> link k (Tip k x) p t
          | zero k m      -> Bin p m (insert k x l) r
          | otherwise     -> Bin p m l (insert k x r)
        Tip ky _
          | k==ky         -> Tip k x
          | otherwise     -> link k (Tip k x) ky t
        Nil -> Tip k x

    -- right-biased insertion, used by 'union'
    -- | /O(min(n,W))/. Insert with a combining function.
    -- @'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 @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 -> IntMap a -> IntMap a
    insertWith f k x t
      = insertWithKey (\_ x' y' -> f x' y') k x t

    -- | /O(min(n,W))/. Insert with a combining function.
    -- @'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 @f key new_value old_value@.
    --
    -- > 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"
    --
    -- If the key exists in the map, this function is lazy in @value@ but strict
    -- in the result of @f@.

    insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
    insertWithKey f !k x t =
      case t of
        Bin p m l r
          | nomatch k p m -> link k (singleton k x) p t
          | zero k m      -> Bin p m (insertWithKey f k x l) r
          | otherwise     -> Bin p m l (insertWithKey f k x r)
        Tip ky y
          | k==ky         -> Tip k $! f k x y
          | otherwise     -> link k (singleton k x) ky t
        Nil -> singleton k x

    -- | /O(min(n,W))/. 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")])

    insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)
    insertLookupWithKey f0 !k0 x0 t0 = toPair $ go f0 k0 x0 t0
      where
        go f k x t =
          case t of
            Bin p m l r
              | nomatch k p m -> Nothing :*: link k (singleton k x) p t
              | zero k m      -> let (found :*: l') = go f k x l in (found :*: Bin p m l' r)
              | otherwise     -> let (found :*: r') = go f k x r in (found :*: Bin p m l r')
            Tip ky y
              | k==ky         -> (Just y :*: (Tip k $! f k x y))
              | otherwise     -> (Nothing :*: link k (singleton k x) ky t)
            Nil -> Nothing :*: (singleton k x)

    {--------------------------------------------------------------------
      Deletion
    --------------------------------------------------------------------}
    -- | /O(min(n,W))/. Adjust a value at a specific key. 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 -> IntMap a -> IntMap a
    adjust f k m
      = adjustWithKey (\_ x -> f x) k m

    -- | /O(min(n,W))/. 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 -> IntMap a -> IntMap a
    adjustWithKey f !k t =
      case t of
        Bin p m l r
          | nomatch k p m -> t
          | zero k m      -> Bin p m (adjustWithKey f k l) r
          | otherwise     -> Bin p m l (adjustWithKey f k r)
        Tip ky y
          | k==ky         -> Tip ky $! f k y
          | otherwise     -> t
        Nil -> Nil

    -- | /O(min(n,W))/. 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 -> IntMap a -> IntMap a
    update f
      = updateWithKey (\_ x -> f x)

    -- | /O(min(n,W))/. The expression (@'update' 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"

    updateWithKey ::  (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a
    updateWithKey f !k t =
      case t of
        Bin p m l r
          | nomatch k p m -> t
          | zero k m      -> binCheckLeft p m (updateWithKey f k l) r
          | otherwise     -> binCheckRight p m l (updateWithKey f k r)
        Tip ky y
          | k==ky         -> case f k y of
                               Just !y' -> Tip ky y'
                               Nothing -> Nil
          | otherwise     -> t
        Nil -> Nil

    -- | /O(min(n,W))/. Lookup and update.
    -- The function returns original value, if it is updated.
    -- This is different behavior than 'Data.Map.updateLookupWithKey'.
    -- 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 "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")

    updateLookupWithKey ::  (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a,IntMap a)
    updateLookupWithKey f0 !k0 t0 = toPair $ go f0 k0 t0
      where
        go f k t =
          case t of
            Bin p m l r
              | nomatch k p m -> (Nothing :*: t)
              | zero k m      -> let (found :*: l') = go f k l in (found :*: binCheckLeft p m l' r)
              | otherwise     -> let (found :*: r') = go f k r in (found :*: binCheckRight p m l r')
            Tip ky y
              | k==ky         -> case f k y of
                                   Just !y' -> (Just y :*: Tip ky y')
                                   Nothing  -> (Just y :*: Nil)
              | otherwise     -> (Nothing :*: t)
            Nil -> (Nothing :*: Nil)

    -- | /O(min(n,W))/. 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 an 'IntMap'.
    -- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@.
    alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a
    alter f !k t =
      case t of
        Bin p m l r
          | nomatch k p m -> case f Nothing of
                               Nothing -> t
                               Just !x  -> link k (Tip k x) p t
          | zero k m      -> binCheckLeft p m (alter f k l) r
          | otherwise     -> binCheckRight p m l (alter f k r)
        Tip ky y
          | k==ky         -> case f (Just y) of
                               Just !x -> Tip ky x
                               Nothing -> Nil
          | otherwise     -> case f Nothing of
                               Just !x -> link k (Tip k x) ky t
                               Nothing -> t
        Nil               -> case f Nothing of
                               Just !x -> Tip k x
                               Nothing -> Nil

    -- | /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 an 'IntMap'.  In short : @'lookup' k <$> 'alterF' f k m = f
    -- ('lookup' k m)@.
    --
    -- Example:
    --
    -- @
    -- interactiveAlter :: Int -> IntMap String -> IO (IntMap 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.

    -- Note: 'alterF' is a flipped version of the 'at' combinator from
    -- 'Control.Lens.At'.
    --
    -- @since 0.5.8

    alterF :: Functor f
           => (Maybe a -> f (Maybe a)) -> Key -> IntMap a -> f (IntMap a)
    -- This implementation was modified from 'Control.Lens.At'.
    alterF f k m = (<$> f mv) $ \fres ->
      case fres of
        Nothing -> maybe m (const (delete k m)) mv
        Just !v' -> insert k v' m
      where mv = lookup k m

    {--------------------------------------------------------------------
      Union
    --------------------------------------------------------------------}
    -- | The union of a list of maps, with a combining operation.
    --
    -- > unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
    -- >     == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]

    unionsWith :: Foldable f => (a->a->a) -> f (IntMap a) -> IntMap a
    unionsWith f ts
      = Foldable.foldl' (unionWith f) empty ts

    -- | /O(n+m)/. The 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) -> IntMap a -> IntMap a -> IntMap a
    unionWith f m1 m2
      = unionWithKey (\_ x y -> f x y) m1 m2

    -- | /O(n+m)/. The 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) -> IntMap a -> IntMap a -> IntMap a
    unionWithKey f m1 m2
      = mergeWithKey' Bin (\(Tip k1 x1) (Tip _k2 x2) -> Tip k1 $! f k1 x1 x2) id id m1 m2

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

    -- | /O(n+m)/. Difference with a combining function.
    --
    -- > 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) -> IntMap a -> IntMap b -> IntMap a
    differenceWith f m1 m2
      = differenceWithKey (\_ x y -> f x y) m1 m2

    -- | /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) -> IntMap a -> IntMap b -> IntMap a
    differenceWithKey f m1 m2
      = mergeWithKey f id (const Nil) m1 m2

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

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

    intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c
    intersectionWith f m1 m2
      = intersectionWithKey (\_ x y -> f x y) m1 m2

    -- | /O(n+m)/. The 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) -> IntMap a -> IntMap b -> IntMap c
    intersectionWithKey f m1 m2
      = mergeWithKey' bin (\(Tip k1 x1) (Tip _k2 x2) -> Tip k1 $! f k1 x1 x2) (const Nil) (const Nil) m1 m2

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

    -- | /O(n+m)/. A high-performance universal combining function. Using
    -- 'mergeWithKey', all combining functions can be defined without any loss of
    -- efficiency (with exception of 'union', 'difference' and 'intersection',
    -- where sharing of some nodes is lost with 'mergeWithKey').
    --
    -- Please make sure you know what is going on when using 'mergeWithKey',
    -- otherwise you can be surprised by unexpected code growth or even
    -- corruption of the data structure.
    --
    -- When 'mergeWithKey' is given three arguments, it is inlined to the call
    -- site. You should therefore use 'mergeWithKey' only to define your custom
    -- combining functions. For example, you could define 'unionWithKey',
    -- 'differenceWithKey' and 'intersectionWithKey' as
    --
    -- > myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2
    -- > myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
    -- > myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2
    --
    -- When calling @'mergeWithKey' combine only1 only2@, a function combining two
    -- 'IntMap's is created, such that
    --
    -- * if a key is present in both maps, it is passed with both corresponding
    --   values to the @combine@ function. Depending on the result, the key is either
    --   present in the result with specified value, or is left out;
    --
    -- * a nonempty subtree present only in the first map is passed to @only1@ and
    --   the output is added to the result;
    --
    -- * a nonempty subtree present only in the second map is passed to @only2@ and
    --   the output is added to the result.
    --
    -- The @only1@ and @only2@ methods /must return a map with a subset (possibly empty) of the keys of the given map/.
    -- The values can be modified arbitrarily.  Most common variants of @only1@ and
    -- @only2@ are 'id' and @'const' 'empty'@, but for example @'map' f@ or
    -- @'filterWithKey' f@ could be used for any @f@.

    mergeWithKey :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c)
                 -> IntMap a -> IntMap b -> IntMap c
    mergeWithKey f g1 g2 = mergeWithKey' bin combine g1 g2
      where -- We use the lambda form to avoid non-exhaustive pattern matches warning.
            combine = \(Tip k1 x1) (Tip _k2 x2) -> case f k1 x1 x2 of Nothing -> Nil
                                                                      Just !x -> Tip k1 x
            {-# INLINE combine #-}
    {-# INLINE mergeWithKey #-}

    {--------------------------------------------------------------------
      Min\/Max
    --------------------------------------------------------------------}

    -- | /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) -> IntMap a -> IntMap a
    updateMinWithKey f t =
      case t of Bin p m l r | m < 0 -> binCheckRight p m l (go f r)
                _ -> go f t
      where
        go f' (Bin p m l r) = binCheckLeft p m (go f' l) r
        go f' (Tip k y) = case f' k y of
                            Just !y' -> Tip k y'
                            Nothing -> Nil
        go _ Nil = error "updateMinWithKey Nil"

    -- | /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) -> IntMap a -> IntMap a
    updateMaxWithKey f t =
      case t of Bin p m l r | m < 0 -> binCheckLeft p m (go f l) r
                _ -> go f t
      where
        go f' (Bin p m l r) = binCheckRight p m l (go f' r)
        go f' (Tip k y) = case f' k y of
                            Just !y' -> Tip k y'
                            Nothing -> Nil
        go _ Nil = error "updateMaxWithKey Nil"

    -- | /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) -> IntMap a -> IntMap a
    updateMax f = updateMaxWithKey (const f)

    -- | /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) -> IntMap a -> IntMap a
    updateMin f = updateMinWithKey (const f)

    {--------------------------------------------------------------------
      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) -> IntMap a -> IntMap b
    map f = go
      where
        go (Bin p m l r) = Bin p m (go l) (go r)
        go (Tip k x)     = Tip k $! f x
        go Nil           = Nil

    #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 :: (Key -> a -> b) -> IntMap a -> IntMap b
    mapWithKey f t
      = case t of
          Bin p m l r -> Bin p m (mapWithKey f l) (mapWithKey f r)
          Tip k x     -> Tip k $! f k x
          Nil         -> Nil

    #ifdef __GLASGOW_HASKELL__
    -- Pay close attention to strictness here. We need to force the
    -- intermediate result for map f . map g, and we need to refrain
    -- from forcing it for map f . L.map g, etc.
    --
    -- TODO Consider moving map and mapWithKey to IntMap.Internal so we can write
    -- non-orphan RULES for things like L.map f (map g xs). We'd need a new function
    -- for this, and we'd have to pay attention to simplifier phases. Something like
    --
    -- lsmap :: (b -> c) -> (a -> b) -> IntMap a -> IntMap c
    -- lsmap _ _ Nil = Nil
    -- lsmap f g (Tip k x) = let !gx = g x in Tip k (f gx)
    -- lsmap f g (Bin p m l r) = Bin p m (lsmap f g l) (lsmap f g 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
     #-}
    #endif

    -- | /O(n)/.
    -- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
    -- That is, behaves exactly like a regular 'traverse' except that the traversing
    -- function also has access to the key associated with a value.
    --
    -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
    -- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
    traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)
    traverseWithKey f = go
      where
        go Nil = pure Nil
        go (Tip k v) = (\ !v' -> Tip k v') <$> f k v
        go (Bin p m l r) = liftA2 (Bin p m) (go l) (go r)
    {-# INLINE traverseWithKey #-}

    -- | /O(n)/. Traverse keys\/values and collect the 'Just' results.
    traverseMaybeWithKey
      :: Applicative f => (Key -> a -> f (Maybe b)) -> IntMap a -> f (IntMap b)
    traverseMaybeWithKey f = go
        where
        go Nil           = pure Nil
        go (Tip k x)     = maybe Nil (Tip k $!) <$> f k x
        go (Bin p m l r) = liftA2 (bin p m) (go l) (go r)

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

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

    -- | /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 -> IntMap b -> (a,IntMap 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.  Strict in
    -- the accumulating argument and the both elements of the
    -- result of the function.
    mapAccumL :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
    mapAccumL f0 a0 t0 = toPair $ go f0 a0 t0
      where
        go f a t
          = case t of
              Bin p m l r -> let (a1 :*: l') = go f a l
                                 (a2 :*: r') = go f a1 r
                             in (a2 :*: Bin p m l' r')
              Tip k x     -> let !(a',!x') = f a k x in (a' :*: Tip k x')
              Nil         -> (a :*: Nil)

    -- | /O(n)/. The function @'mapAccumR'@ threads an accumulating
    -- argument through the map in descending order of keys.
    mapAccumRWithKey :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
    mapAccumRWithKey f0 a0 t0 = toPair $ go f0 a0 t0
      where
        go f a t
          = case t of
              Bin p m l r -> let (a1 :*: r') = go f a r
                                 (a2 :*: l') = go f a1 l
                             in (a2 :*: Bin p m l' r')
              Tip k x     -> let !(a',!x') = f a k x in (a' :*: Tip k x')
              Nil         -> (a :*: Nil)

    -- | /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 :: (a -> a -> a) -> (Key->Key) -> IntMap a -> IntMap a
    mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []

    {--------------------------------------------------------------------
      Filter
    --------------------------------------------------------------------}
    -- | /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) -> IntMap a -> IntMap 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 :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b
    mapMaybeWithKey f (Bin p m l r)
      = bin p m (mapMaybeWithKey f l) (mapMaybeWithKey f r)
    mapMaybeWithKey f (Tip k x) = case f k x of
      Just !y  -> Tip k y
      Nothing -> Nil
    mapMaybeWithKey _ Nil = Nil

    -- | /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) -> IntMap a -> (IntMap b, IntMap 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 :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)
    mapEitherWithKey f0 t0 = toPair $ go f0 t0
      where
        go f (Bin p m l r)
          = bin p m l1 r1 :*: bin p m l2 r2
          where
            (l1 :*: l2) = go f l
            (r1 :*: r2) = go f r
        go f (Tip k x) = case f k x of
          Left !y  -> (Tip k y :*: Nil)
          Right !z -> (Nil :*: Tip k z)
        go _ Nil = (Nil :*: Nil)

    {--------------------------------------------------------------------
      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.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
    -- > fromSet undefined Data.IntSet.empty == empty

    fromSet :: (Key -> a) -> IntSet.IntSet -> IntMap a
    fromSet _ IntSet.Nil = Nil
    fromSet f (IntSet.Bin p m l r) = Bin p m (fromSet f l) (fromSet f r)
    fromSet f (IntSet.Tip kx bm) = buildTree f kx bm (IntSet.suffixBitMask + 1)
      where -- This is slightly complicated, as we to convert the dense
            -- representation of IntSet into tree representation of IntMap.
            --
            -- We are given a nonzero bit mask 'bmask' of 'bits' bits with prefix 'prefix'.
            -- We split bmask into halves corresponding to left and right subtree.
            -- If they are both nonempty, we create a Bin node, otherwise exactly
            -- one of them is nonempty and we construct the IntMap from that half.
            buildTree g !prefix !bmask bits = case bits of
              0 -> Tip prefix $! g prefix
              _ -> case intFromNat ((natFromInt bits) `shiftRL` 1) of
                     bits2 | bmask .&. ((1 `shiftLL` bits2) - 1) == 0 ->
                               buildTree g (prefix + bits2) (bmask `shiftRL` bits2) bits2
                           | (bmask `shiftRL` bits2) .&. ((1 `shiftLL` bits2) - 1) == 0 ->
                               buildTree g prefix bmask bits2
                           | otherwise ->
                               Bin prefix bits2 (buildTree g prefix bmask bits2) (buildTree g (prefix + bits2) (bmask `shiftRL` bits2) bits2)

    {--------------------------------------------------------------------
      Lists
    --------------------------------------------------------------------}
    -- | /O(n*min(n,W))/. Create a map from a list of key\/value pairs.
    --
    -- > fromList [] == empty
    -- > fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
    -- > fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]

    fromList :: [(Key,a)] -> IntMap a
    fromList xs
      = Foldable.foldl' ins empty xs
      where
        ins t (k,x)  = insert k x t

    -- | /O(n*min(n,W))/. Create 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)] -> IntMap a
    fromListWith f xs
      = fromListWithKey (\_ x y -> f x y) xs

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

    fromListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
    fromListWithKey f xs
      = Foldable.foldl' ins empty xs
      where
        ins t (k,x) = insertWithKey f k x t

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

    fromAscList :: [(Key,a)] -> IntMap a
    fromAscList xs
      = fromAscListWithKey (\_ x _ -> x) xs

    -- | /O(n)/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order, with a combining function on equal keys.
    -- /The precondition (input list is ascending) is not checked./
    --
    -- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

    fromAscListWith :: (a -> a -> a) -> [(Key,a)] -> IntMap a
    fromAscListWith f xs
      = fromAscListWithKey (\_ x y -> f x y) xs

    -- | /O(n)/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order, with a combining function on equal keys.
    -- /The precondition (input list is ascending) is not checked./
    --
    -- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]

    fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
    fromAscListWithKey _ []         = Nil
    fromAscListWithKey f (x0 : xs0) = fromDistinctAscList (combineEq x0 xs0)
      where
        -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]
        combineEq z [] = [z]
        combineEq z@(kz,zz) (x@(kx,xx):xs)
          | kx==kz    = let !yy = f kx xx zz in combineEq (kx,yy) xs
          | otherwise = z:combineEq x xs

    -- | /O(n)/. Build a map from a list of key\/value pairs where
    -- the keys are in ascending order and all distinct.
    -- /The precondition (input list is strictly ascending) is not checked./
    --
    -- > fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]

    fromDistinctAscList :: [(Key,a)] -> IntMap a
    fromDistinctAscList []         = Nil
    fromDistinctAscList (z0 : zs0) = work z0 zs0 Nada
      where
        work (kx,!vx) []            stk = finish kx (Tip kx vx) stk
        work (kx,!vx) (z@(kz,_):zs) stk = reduce z zs (branchMask kx kz) kx (Tip kx vx) stk

        reduce :: (Key,a) -> [(Key,a)] -> Mask -> Prefix -> IntMap a -> Stack a -> IntMap a
        reduce z zs _ px tx Nada = work z zs (Push px tx Nada)
        reduce z zs m px tx stk@(Push py ty stk') =
            let mxy = branchMask px py
                pxy = mask px mxy
            in  if shorter m mxy
                     then reduce z zs m pxy (Bin pxy mxy ty tx) stk'
                     else work z zs (Push px tx stk)

        finish _  t  Nada = t
        finish px tx (Push py ty stk) = finish p (link py ty px tx) stk
            where m = branchMask px py
                  p = mask px m

    data Stack a = Push {-# UNPACK #-} !Prefix !(IntMap a) !(Stack a) | Nada