{-# LANGUAGE CPP #-}
#if __GLASGOW_HASKELL__
{-# LANGUAGE MagicHash, DeriveDataTypeable, StandaloneDeriving #-}
#endif
#if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703
{-# LANGUAGE Trustworthy #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.IntMap.Base
-- Copyright   :  (c) Daan Leijen 2002
--                (c) Andriy Palamarchuk 2008
-- License     :  BSD-style
-- Maintainer  :  libraries@haskell.org
-- Stability   :  provisional
-- Portability :  portable
--
-- This defines the data structures and core (hidden) manipulations
-- on representations.
-----------------------------------------------------------------------------

-- [Note: INLINE bit fiddling]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- It is essential that the bit fiddling functions like mask, zero, branchMask
-- etc are inlined. If they do not, the memory allocation skyrockets. The GHC
-- usually gets it right, but it is disastrous if it does not. Therefore we
-- explicitly mark these functions INLINE.


-- [Note: Local 'go' functions and capturing]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- Care must be taken when using 'go' function which captures an argument.
-- Sometimes (for example when the argument is passed to a data constructor,
-- as in insert), GHC heap-allocates more than necessary. Therefore C-- code
-- must be checked for increased allocation when creating and modifying such
-- functions.


-- [Note: Order of constructors]
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-- The order of constructors of IntMap matters when considering performance.
-- Currently in GHC 7.0, when type has 3 constructors, they are matched from
-- the first to the last -- the best performance is achieved when the
-- constructors are ordered by frequency.
-- On GHC 7.0, reordering constructors from Nil | Tip | Bin to Bin | Tip | Nil
-- improves the benchmark by circa 10%.

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

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

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

    -- * Construction
    , empty
    , singleton

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

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

    -- * Combine

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

    -- ** Difference
    , difference
    , differenceWith
    , differenceWithKey

    -- ** Intersection
    , intersection
    , intersectionWith
    , intersectionWithKey

    -- ** Universal combining function
    , mergeWithKey
    , mergeWithKey'

    -- * Traversal
    -- ** Map
    , map
    , mapWithKey
    , traverseWithKey
    , 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

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

    , mapMaybe
    , mapMaybeWithKey
    , mapEither
    , mapEitherWithKey

    , split
    , splitLookup

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

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

    -- * Debugging
    , showTree
    , showTreeWith

    -- * Internal types
    , Mask, Prefix, Nat

    -- * Utility
    , natFromInt
    , intFromNat
    , join
    , bin
    , zero
    , nomatch
    , match
    , mask
    , maskW
    , shorter
    , branchMask
    , highestBitMask
    , foldlStrict
    ) where

import Control.Applicative (Applicative(pure, (<*>)), (<$>))
import Control.DeepSeq (NFData(rnf))
import Control.Monad (liftM)
import Data.Bits
import qualified Data.Foldable as Foldable
import Data.Maybe (fromMaybe)
import Data.Monoid (Monoid(..))
import Data.Traversable (Traversable(traverse))
import Data.Typeable
import Data.Word (Word)
import Prelude hiding (lookup, map, filter, foldr, foldl, null)

import Data.BitUtil
import Data.IntSet.Base (Key)
import qualified Data.IntSet.Base as IntSet
import Data.StrictPair

#if __GLASGOW_HASKELL__
import Data.Data (Data(..), Constr, mkConstr, constrIndex, Fixity(Prefix),
                  DataType, mkDataType)
import GHC.Exts (build)
import Text.Read
#endif

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

-- A "Nat" is a natural machine word (an unsigned Int)
type Nat = Word

natFromInt :: Key -> Nat
natFromInt = fromIntegral
{-# INLINE natFromInt #-}

intFromNat :: Nat -> Key
intFromNat = fromIntegral
{-# INLINE intFromNat #-}

{--------------------------------------------------------------------
  Types
--------------------------------------------------------------------}


-- | A map of integers to values @a@.

-- See Note: Order of constructors
data IntMap a = Bin {-# UNPACK #-} !Prefix
                    {-# UNPACK #-} !Mask
                    !(IntMap a)
                    !(IntMap a)
              | Tip {-# UNPACK #-} !Key a
              | Nil

type Prefix = Int
type Mask   = Int

{--------------------------------------------------------------------
  Operators
--------------------------------------------------------------------}

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

(!) :: IntMap a -> Key -> a
m ! k = find k m

-- | Same as 'difference'.
(\\) :: IntMap a -> IntMap b -> IntMap a
m1 \\ m2 = difference m1 m2

infixl 9 \\{-This comment teaches CPP correct behaviour -}

{--------------------------------------------------------------------
  Types
--------------------------------------------------------------------}

instance Monoid (IntMap a) where
    mempty  = empty
    mappend = union
    mconcat = unions

instance Foldable.Foldable IntMap where
  fold t = go t
    where go Nil = mempty
          go (Tip _ v) = v
          go (Bin _ _ l r) = go l `mappend` go r
  {-# INLINABLE fold #-}
  foldr = foldr
  {-# INLINE foldr #-}
  foldl = foldl
  {-# INLINE foldl #-}
  foldMap f t = go t
    where go Nil = mempty
          go (Tip _ v) = f v
          go (Bin _ _ l r) = go l `mappend` go r
  {-# INLINE foldMap #-}

instance Traversable IntMap where
    traverse f = traverseWithKey (\_ -> f)
    {-# INLINE traverse #-}

instance NFData a => NFData (IntMap a) where
    rnf Nil = ()
    rnf (Tip _ v) = rnf v
    rnf (Bin _ _ l r) = rnf l `seq` rnf r

#if __GLASGOW_HASKELL__

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

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

instance Data a => Data (IntMap a) where
  gfoldl f z im = z fromList `f` (toList im)
  toConstr _     = fromListConstr
  gunfold k z c  = case constrIndex c of
    1 -> k (z fromList)
    _ -> error "gunfold"
  dataTypeOf _   = intMapDataType
  dataCast1 f    = gcast1 f

fromListConstr :: Constr
fromListConstr = mkConstr intMapDataType "fromList" [] Prefix

intMapDataType :: DataType
intMapDataType = mkDataType "Data.IntMap.Base.IntMap" [fromListConstr]

#endif

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

null :: IntMap a -> Bool
null Nil = True
null _   = False
{-# INLINE null #-}

-- | /O(n)/. Number of elements in the map.
--
-- > size empty                                   == 0
-- > size (singleton 1 'a')                       == 1
-- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
size :: IntMap a -> Int
size t
  = case t of
      Bin _ _ l r -> size l + size r
      Tip _ _ -> 1
      Nil     -> 0

-- | /O(min(n,W))/. Is the key a member of the map?
--
-- > member 5 (fromList [(5,'a'), (3,'b')]) == True
-- > member 1 (fromList [(5,'a'), (3,'b')]) == False

-- See Note: Local 'go' functions and capturing]
member :: Key -> IntMap a -> Bool
member k = k `seq` go
  where
    go (Bin p m l r) | nomatch k p m = False
                     | zero k m  = go l
                     | otherwise = go r
    go (Tip kx _) = k == kx
    go Nil = False

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

notMember :: Key -> IntMap a -> Bool
notMember k m = not $ member k m

-- | /O(min(n,W))/. Lookup the value at a key in the map. See also 'Data.Map.lookup'.

-- See Note: Local 'go' functions and capturing]
lookup :: Key -> IntMap a -> Maybe a
lookup k = k `seq` go
  where
    go (Bin p m l r) | nomatch k p m = Nothing
                     | zero k m  = go l
                     | otherwise = go r
    go (Tip kx x) | k == kx   = Just x
                  | otherwise = Nothing
    go Nil = Nothing


-- See Note: Local 'go' functions and capturing]
find :: Key -> IntMap a -> a
find k = k `seq` go
  where
    go (Bin p m l r) | nomatch k p m = not_found
                     | zero k m  = go l
                     | otherwise = go r
    go (Tip kx x) | k == kx   = x
                  | otherwise = not_found
    go Nil = not_found

    not_found = error ("IntMap.!: key " ++ show k ++ " is not an element of the map")

-- | /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 Note: Local 'go' functions and capturing]
findWithDefault :: a -> Key -> IntMap a -> a
findWithDefault def k = k `seq` 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

-- | /O(log n)/. Find largest key smaller than the given one and return the
-- corresponding (key, value) pair.
--
-- > lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
-- > lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')

-- See Note: Local 'go' functions and capturing.
lookupLT :: Key -> IntMap a -> Maybe (Key, a)
lookupLT k t = k `seq` case t of
    Bin _ m l r | m < 0 -> if k >= 0 then go r l else go Nil r
    _ -> go Nil t
  where
    go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMax def else unsafeFindMax r
                         | zero k m  = go def l
                         | otherwise = go l r
    go def (Tip ky y) | k <= ky   = unsafeFindMax def
                      | otherwise = Just (ky, y)
    go def Nil = unsafeFindMax def

-- | /O(log n)/. Find smallest key greater than the given one and return the
-- corresponding (key, value) pair.
--
-- > lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
-- > lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing

-- See Note: Local 'go' functions and capturing.
lookupGT :: Key -> IntMap a -> Maybe (Key, a)
lookupGT k t = k `seq` case t of
    Bin _ m l r | m < 0 -> if k >= 0 then go Nil l else go l r
    _ -> go Nil t
  where
    go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMin l else unsafeFindMin def
                         | zero k m  = go r l
                         | otherwise = go def r
    go def (Tip ky y) | k >= ky   = unsafeFindMin def
                      | otherwise = Just (ky, y)
    go def Nil = unsafeFindMin def

-- | /O(log n)/. Find largest key smaller or equal to the given one and return
-- the corresponding (key, value) pair.
--
-- > lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
-- > lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
-- > lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')

-- See Note: Local 'go' functions and capturing.
lookupLE :: Key -> IntMap a -> Maybe (Key, a)
lookupLE k t = k `seq` case t of
    Bin _ m l r | m < 0 -> if k >= 0 then go r l else go Nil r
    _ -> go Nil t
  where
    go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMax def else unsafeFindMax r
                         | zero k m  = go def l
                         | otherwise = go l r
    go def (Tip ky y) | k < ky    = unsafeFindMax def
                      | otherwise = Just (ky, y)
    go def Nil = unsafeFindMax def

-- | /O(log n)/. Find smallest key greater or equal to the given one and return
-- the corresponding (key, value) pair.
--
-- > lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
-- > lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
-- > lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing

-- See Note: Local 'go' functions and capturing.
lookupGE :: Key -> IntMap a -> Maybe (Key, a)
lookupGE k t = k `seq` case t of
    Bin _ m l r | m < 0 -> if k >= 0 then go Nil l else go l r
    _ -> go Nil t
  where
    go def (Bin p m l r) | nomatch k p m = if k < p then unsafeFindMin l else unsafeFindMin def
                         | zero k m  = go r l
                         | otherwise = go def r
    go def (Tip ky y) | k > ky    = unsafeFindMin def
                      | otherwise = Just (ky, y)
    go def Nil = unsafeFindMin def


-- Helper function for lookupGE and lookupGT. It assumes that if a Bin node is
-- given, it has m > 0.
unsafeFindMin :: IntMap a -> Maybe (Key, a)
unsafeFindMin Nil = Nothing
unsafeFindMin (Tip ky y) = Just (ky, y)
unsafeFindMin (Bin _ _ l _) = unsafeFindMin l

-- Helper function for lookupLE and lookupLT. It assumes that if a Bin node is
-- given, it has m > 0.
unsafeFindMax :: IntMap a -> Maybe (Key, a)
unsafeFindMax Nil = Nothing
unsafeFindMax (Tip ky y) = Just (ky, y)
unsafeFindMax (Bin _ _ _ r) = unsafeFindMax r

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

empty :: IntMap a
empty
  = Nil
{-# INLINE empty #-}

-- | /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 = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> join 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     -> join 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"

insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a
insertWithKey f k x t = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> join k (Tip 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     -> join k (Tip k x) ky t
    Nil -> Tip 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 f k x t = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> (Nothing,join k (Tip k x) p t)
      | zero k m      -> let (found,l') = insertLookupWithKey f k x l in (found,Bin p m l' r)
      | otherwise     -> let (found,r') = insertLookupWithKey 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,join k (Tip k x) ky t)
    Nil -> (Nothing,Tip k x)


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

delete :: Key -> IntMap a -> IntMap a
delete k t = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> t
      | zero k m      -> bin p m (delete k l) r
      | otherwise     -> bin p m l (delete k r)
    Tip ky _
      | k==ky         -> Nil
      | otherwise     -> t
    Nil -> Nil

-- | /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
  = updateWithKey (\k' x -> Just (f k' 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 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 = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> t
      | zero k m      -> bin p m (updateWithKey f k l) r
      | otherwise     -> bin 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 f k t = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> (Nothing,t)
      | zero k m      -> let (found,l') = updateLookupWithKey f k l in (found,bin p m l' r)
      | otherwise     -> let (found,r') = updateLookupWithKey f k r in (found,bin 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 = k `seq`
  case t of
    Bin p m l r
      | nomatch k p m -> case f Nothing of
                           Nothing -> t
                           Just x -> join k (Tip k x) p t
      | zero k m      -> bin p m (alter f k l) r
      | otherwise     -> bin 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 -> join k (Tip k x) ky t
                           Nothing -> Tip ky y
    Nil               -> case f Nothing of
                           Just x -> Tip k x
                           Nothing -> Nil


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

unions :: [IntMap a] -> IntMap a
unions xs
  = foldlStrict union empty xs

-- | 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 :: (a->a->a) -> [IntMap a] -> IntMap a
unionsWith f ts
  = foldlStrict (unionWith f) empty ts

-- | /O(n+m)/. The (left-biased) union of two maps.
-- It prefers the first map when duplicate keys are encountered,
-- i.e. (@'union' == 'unionWith' 'const'@).
--
-- > union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]

union :: IntMap a -> IntMap a -> IntMap a
union m1 m2
  = mergeWithKey' Bin const id id m1 m2

-- | /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 between two maps (based on keys).
--
-- > difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"

difference :: IntMap a -> IntMap b -> IntMap a
difference m1 m2
  = mergeWithKey (\_ _ _ -> Nothing) id (const Nil) m1 m2

-- | /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 (left-biased) intersection of two maps (based on keys).
--
-- > intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"

intersection :: IntMap a -> IntMap b -> IntMap a
intersection m1 m2
  = mergeWithKey' bin const (const Nil) (const Nil) m1 m2

-- | /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 #-}

-- Slightly more general version of mergeWithKey. It differs in the following:
--
-- * the combining function operates on maps instead of keys and values. The
--   reason is to enable sharing in union, difference and intersection.
--
-- * mergeWithKey' is given an equivalent of bin. The reason is that in union*,
--   Bin constructor can be used, because we know both subtrees are nonempty.

mergeWithKey' :: (Prefix -> Mask -> IntMap c -> IntMap c -> IntMap c)
              -> (IntMap a -> IntMap b -> IntMap c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c)
              -> IntMap a -> IntMap b -> IntMap c
mergeWithKey' bin' f g1 g2 = go
  where
    go t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = merge1
      | shorter m2 m1  = merge2
      | p1 == p2       = bin' p1 m1 (go l1 l2) (go r1 r2)
      | otherwise      = maybe_join p1 (g1 t1) p2 (g2 t2)
      where
        merge1 | nomatch p2 p1 m1  = maybe_join p1 (g1 t1) p2 (g2 t2)
               | zero p2 m1        = bin' p1 m1 (go l1 t2) (g1 r1)
               | otherwise         = bin' p1 m1 (g1 l1) (go r1 t2)
        merge2 | nomatch p1 p2 m2  = maybe_join p1 (g1 t1) p2 (g2 t2)
               | zero p1 m2        = bin' p2 m2 (go t1 l2) (g2 r2)
               | otherwise         = bin' p2 m2 (g2 l2) (go t1 r2)

    go t1'@(Bin _ _ _ _) t2'@(Tip k2' _) = merge t2' k2' t1'
      where merge t2 k2 t1@(Bin p1 m1 l1 r1) | nomatch k2 p1 m1 = maybe_join p1 (g1 t1) k2 (g2 t2)
                                             | zero k2 m1 = bin' p1 m1 (merge t2 k2 l1) (g1 r1)
                                             | otherwise  = bin' p1 m1 (g1 l1) (merge t2 k2 r1)
            merge t2 k2 t1@(Tip k1 _) | k1 == k2 = f t1 t2
                                      | otherwise = maybe_join k1 (g1 t1) k2 (g2 t2)
            merge t2 _  Nil = g2 t2

    go t1@(Bin _ _ _ _) Nil = g1 t1

    go t1'@(Tip k1' _) t2' = merge t1' k1' t2'
      where merge t1 k1 t2@(Bin p2 m2 l2 r2) | nomatch k1 p2 m2 = maybe_join k1 (g1 t1) p2 (g2 t2)
                                             | zero k1 m2 = bin' p2 m2 (merge t1 k1 l2) (g2 r2)
                                             | otherwise  = bin' p2 m2 (g2 l2) (merge t1 k1 r2)
            merge t1 k1 t2@(Tip k2 _) | k1 == k2 = f t1 t2
                                      | otherwise = maybe_join k1 (g1 t1) k2 (g2 t2)
            merge t1 _  Nil = g1 t1

    go Nil t2 = g2 t2

    maybe_join _ Nil _ t2 = t2
    maybe_join _ t1 _ Nil = t1
    maybe_join p1 t1 p2 t2 = join p1 t1 p2 t2
    {-# INLINE maybe_join #-}
{-# INLINE mergeWithKey' #-}

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

-- | /O(min(n,W))/. 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 -> bin p m l (go f r)
            _ -> go f t
  where
    go f' (Bin p m l r) = bin 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(min(n,W))/. 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 -> bin p m (go f l) r
            _ -> go f t
  where
    go f' (Bin p m l r) = bin 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(min(n,W))/. Retrieves the maximal (key,value) pair of the map, and
-- the map stripped of that element, or 'Nothing' if passed an empty map.
--
-- > maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
-- > maxViewWithKey empty == Nothing

maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
maxViewWithKey t =
  case t of Nil -> Nothing
            Bin p m l r | m < 0 -> case go l of (result, l') -> Just (result, bin p m l' r)
            _ -> Just (go t)
  where
    go (Bin p m l r) = case go r of (result, r') -> (result, bin p m l r')
    go (Tip k y) = ((k, y), Nil)
    go Nil = error "maxViewWithKey Nil"

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

minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)
minViewWithKey t =
  case t of Nil -> Nothing
            Bin p m l r | m < 0 -> case go r of (result, r') -> Just (result, bin p m l r')
            _ -> Just (go t)
  where
    go (Bin p m l r) = case go l of (result, l') -> (result, bin p m l' r)
    go (Tip k y) = ((k, y), Nil)
    go Nil = error "minViewWithKey Nil"

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

-- Similar to the Arrow instance.
first :: (a -> c) -> (a, b) -> (c, b)
first f (x,y) = (f x,y)

-- | /O(min(n,W))/. Retrieves the maximal key of the map, and the map
-- stripped of that element, or 'Nothing' if passed an empty map.
maxView :: IntMap a -> Maybe (a, IntMap a)
maxView t = liftM (first snd) (maxViewWithKey t)

-- | /O(min(n,W))/. Retrieves the minimal key of the map, and the map
-- stripped of that element, or 'Nothing' if passed an empty map.
minView :: IntMap a -> Maybe (a, IntMap a)
minView t = liftM (first snd) (minViewWithKey t)

-- | /O(min(n,W))/. Delete and find the maximal element.
deleteFindMax :: IntMap a -> ((Key, a), IntMap a)
deleteFindMax = fromMaybe (error "deleteFindMax: empty map has no maximal element") . maxViewWithKey

-- | /O(min(n,W))/. Delete and find the minimal element.
deleteFindMin :: IntMap a -> ((Key, a), IntMap a)
deleteFindMin = fromMaybe (error "deleteFindMin: empty map has no minimal element") . minViewWithKey

-- | /O(min(n,W))/. The minimal key of the map.
findMin :: IntMap a -> (Key, a)
findMin Nil = error $ "findMin: empty map has no minimal element"
findMin (Tip k v) = (k,v)
findMin (Bin _ m l r)
  |   m < 0   = go r
  | otherwise = go l
    where go (Tip k v)      = (k,v)
          go (Bin _ _ l' _) = go l'
          go Nil            = error "findMax Nil"

-- | /O(min(n,W))/. The maximal key of the map.
findMax :: IntMap a -> (Key, a)
findMax Nil = error $ "findMax: empty map has no maximal element"
findMax (Tip k v) = (k,v)
findMax (Bin _ m l r)
  |   m < 0   = go l
  | otherwise = go r
    where go (Tip k v)      = (k,v)
          go (Bin _ _ _ r') = go r'
          go Nil            = error "findMax Nil"

-- | /O(min(n,W))/. Delete the minimal key. Returns an empty map if the map is empty.
--
-- Note that this is a change of behaviour for consistency with 'Data.Map.Map' &#8211;
-- versions prior to 0.5 threw an error if the 'IntMap' was already empty.
deleteMin :: IntMap a -> IntMap a
deleteMin = maybe Nil snd . minView

-- | /O(min(n,W))/. Delete the maximal key. Returns an empty map if the map is empty.
--
-- Note that this is a change of behaviour for consistency with 'Data.Map.Map' &#8211;
-- versions prior to 0.5 threw an error if the 'IntMap' was already empty.
deleteMax :: IntMap a -> IntMap a
deleteMax = maybe Nil snd . maxView


{--------------------------------------------------------------------
  Submap
--------------------------------------------------------------------}
-- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).
-- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).
isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
isProperSubmapOf m1 m2
  = isProperSubmapOfBy (==) m1 m2

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

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

 But the following are all 'False':

  > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
  > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
  > isProperSubmapOfBy (<)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
-}
isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
isProperSubmapOfBy predicate t1 t2
  = case submapCmp predicate t1 t2 of
      LT -> True
      _  -> False

submapCmp :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Ordering
submapCmp predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
  | shorter m1 m2  = GT
  | shorter m2 m1  = submapCmpLt
  | p1 == p2       = submapCmpEq
  | otherwise      = GT  -- disjoint
  where
    submapCmpLt | nomatch p1 p2 m2  = GT
                | zero p1 m2        = submapCmp predicate t1 l2
                | otherwise         = submapCmp predicate t1 r2
    submapCmpEq = case (submapCmp predicate l1 l2, submapCmp predicate r1 r2) of
                    (GT,_ ) -> GT
                    (_ ,GT) -> GT
                    (EQ,EQ) -> EQ
                    _       -> LT

submapCmp _         (Bin _ _ _ _) _  = GT
submapCmp predicate (Tip kx x) (Tip ky y)
  | (kx == ky) && predicate x y = EQ
  | otherwise                   = GT  -- disjoint
submapCmp predicate (Tip k x) t
  = case lookup k t of
     Just y | predicate x y -> LT
     _                      -> GT -- disjoint
submapCmp _    Nil Nil = EQ
submapCmp _    Nil _   = LT

-- | /O(n+m)/. Is this a submap?
-- Defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).
isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool
isSubmapOf m1 m2
  = isSubmapOfBy (==) m1 m2

{- | /O(n+m)/.
 The expression (@'isSubmapOfBy' f m1 m2@) returns 'True' if
 all keys in @m1@ are in @m2@, and when @f@ returns 'True' when
 applied to their respective values. For example, the following
 expressions are all 'True':

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

 But the following are all 'False':

  > isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
  > isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
  > isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
-}
isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool
isSubmapOfBy predicate t1@(Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
  | shorter m1 m2  = False
  | shorter m2 m1  = match p1 p2 m2 && (if zero p1 m2 then isSubmapOfBy predicate t1 l2
                                                      else isSubmapOfBy predicate t1 r2)
  | otherwise      = (p1==p2) && isSubmapOfBy predicate l1 l2 && isSubmapOfBy predicate r1 r2
isSubmapOfBy _         (Bin _ _ _ _) _ = False
isSubmapOfBy predicate (Tip k x) t     = case lookup k t of
                                         Just y  -> predicate x y
                                         Nothing -> False
isSubmapOfBy _         Nil _           = True

{--------------------------------------------------------------------
  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 t
  = case t of
      Bin p m l r -> Bin p m (map f l) (map f r)
      Tip k x     -> Tip k (f x)
      Nil         -> Nil

-- | /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

-- | /O(n)/.
-- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
-- That is, behaves exactly like a regular 'traverse' except that the traversing
-- function also has access to the key associated with a value.
--
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)
traverseWithKey f = go
  where
    go Nil = pure Nil
    go (Tip k v) = Tip k <$> f k v
    go (Bin p m l r) = Bin p m <$> go l <*> 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 -> 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.
mapAccumL :: (a -> Key -> b -> (a,c)) -> a -> IntMap b -> (a,IntMap c)
mapAccumL f a t
  = case t of
      Bin p m l r -> let (a1,l') = mapAccumL f a l
                         (a2,r') = mapAccumL 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 f a t
  = case t of
      Bin p m l r -> let (a1,r') = mapAccumRWithKey f a r
                         (a2,l') = mapAccumRWithKey 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*min(n,W))/.
-- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.
--
-- The size of the result may be smaller if @f@ maps two or more distinct
-- keys to the same new key.  In this case the value at the greatest of the
-- original keys is retained.
--
-- > mapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
-- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
-- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"

mapKeys :: (Key->Key) -> IntMap a -> IntMap a
mapKeys f = fromList . foldrWithKey (\k x xs -> (f k, x) : xs) []

-- | /O(n*min(n,W))/.
-- @'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) []

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

mapKeysMonotonic :: (Key->Key) -> IntMap a -> IntMap a
mapKeysMonotonic f = fromDistinctAscList . foldrWithKey (\k x xs -> (f k, x) : xs) []

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

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

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

filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a
filterWithKey predicate t
  = case t of
      Bin p m l r
        -> bin p m (filterWithKey predicate l) (filterWithKey predicate r)
      Tip k x
        | predicate k x -> t
        | otherwise     -> Nil
      Nil -> Nil

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

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

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

partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a,IntMap a)
partitionWithKey predicate0 t0 = toPair $ go predicate0 t0
  where
    go predicate t
      = case t of
          Bin p m l r
            -> let (l1 :*: l2) = go predicate l
                   (r1 :*: r2) = go predicate r
               in bin p m l1 r1 :*: bin p m l2 r2
          Tip k x
            | predicate k x -> (t :*: Nil)
            | otherwise     -> (Nil :*: t)
          Nil -> (Nil :*: Nil)

-- | /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)

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

split :: Key -> IntMap a -> (IntMap a, IntMap a)
split k t =
  case t of
      Bin _ m l r
          | m < 0 -> if k >= 0 -- handle negative numbers.
                     then case go k l of (lt :*: gt) -> let lt' = union r lt 
                                                        in lt' `seq` (lt', gt)
                     else case go k r of (lt :*: gt) -> let gt' = union gt l
                                                        in gt' `seq` (lt, gt')
      _ -> case go k t of
          (lt :*: gt) -> (lt, gt)
  where
    go k' t'@(Bin p m l r) | nomatch k' p m = if k' > p then t' :*: Nil else Nil :*: t'
                           | zero k' m = case go k' l of (lt :*: gt) -> lt :*: union gt r
                           | otherwise = case go k' r of (lt :*: gt) -> union l lt :*: gt
    go k' t'@(Tip ky _) | k' > ky   = (t' :*: Nil)
                        | k' < ky   = (Nil :*: t')
                        | otherwise = (Nil :*: Nil)
    go _ Nil = (Nil :*: Nil)

-- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot
-- key was found in the original map.
--
-- > splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
-- > splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
-- > splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
-- > splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
-- > splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)

splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)
splitLookup k t =
  case t of
      Bin _ m l r
          | m < 0 -> if k >= 0 -- handle negative numbers.
                     then case go k l of
                         (lt, fnd, gt) -> let lt' = union r lt
                                          in lt' `seq` (lt', fnd, gt)
                     else case go k r of
                         (lt, fnd, gt) -> let gt' = union gt l
                                          in gt' `seq` (lt, fnd, gt')
      _ -> go k t
  where
    go k' t'@(Bin p m l r)
        | nomatch k' p m = if k' > p then (t', Nothing, Nil) else (Nil, Nothing, t')
        | zero k' m      = case go k' l of
            (lt, fnd, gt) -> let gt' = union gt r in gt' `seq` (lt, fnd, gt')
        | otherwise      = case go k' r of
            (lt, fnd, gt) -> let lt' = union l lt in lt' `seq` (lt', fnd, gt)
    go k' t'@(Tip ky y) | k' > ky   = (t', Nothing, Nil)
                        | k' < ky   = (Nil, Nothing, t')
                        | otherwise = (Nil, Just y, Nil)
    go _ Nil = (Nil, Nothing, Nil)

{--------------------------------------------------------------------
  Fold
--------------------------------------------------------------------}
-- | /O(n)/. Fold the values in the map using the given right-associative
-- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'elems'@.
--
-- For example,
--
-- > elems map = foldr (:) [] map
--
-- > let f a len = len + (length a)
-- > foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
foldr :: (a -> b -> b) -> b -> IntMap a -> b
foldr f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                        | otherwise -> go (go z r) l
            _ -> go z t
  where
    go z' Nil           = z'
    go z' (Tip _ x)     = f x z'
    go z' (Bin _ _ l r) = go (go z' r) l
{-# INLINE foldr #-}

-- | /O(n)/. A strict version of 'foldr'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> IntMap a -> b
foldr' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                        | otherwise -> go (go z r) l
            _ -> go z t
  where
    STRICT_1_OF_2(go)
    go z' Nil           = z'
    go z' (Tip _ x)     = f x z'
    go z' (Bin _ _ l r) = go (go z' r) l
{-# INLINE foldr' #-}

-- | /O(n)/. Fold the values in the map using the given left-associative
-- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'elems'@.
--
-- For example,
--
-- > elems = reverse . foldl (flip (:)) []
--
-- > let f len a = len + (length a)
-- > foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
foldl :: (a -> b -> a) -> a -> IntMap b -> a
foldl f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                        | otherwise -> go (go z l) r
            _ -> go z t
  where
    go z' Nil           = z'
    go z' (Tip _ x)     = f z' x
    go z' (Bin _ _ l r) = go (go z' l) r
{-# INLINE foldl #-}

-- | /O(n)/. A strict version of 'foldl'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> IntMap b -> a
foldl' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                        | otherwise -> go (go z l) r
            _ -> go z t
  where
    STRICT_1_OF_2(go)
    go z' Nil           = z'
    go z' (Tip _ x)     = f z' x
    go z' (Bin _ _ l r) = go (go z' l) r
{-# INLINE foldl' #-}

-- | /O(n)/. Fold the keys and values in the map using the given right-associative
-- binary operator, such that
-- @'foldrWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.
--
-- For example,
--
-- > keys map = foldrWithKey (\k x ks -> k:ks) [] map
--
-- > let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
-- > foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
foldrWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b
foldrWithKey f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                        | otherwise -> go (go z r) l
            _ -> go z t
  where
    go z' Nil           = z'
    go z' (Tip kx x)    = f kx x z'
    go z' (Bin _ _ l r) = go (go z' r) l
{-# INLINE foldrWithKey #-}

-- | /O(n)/. A strict version of 'foldrWithKey'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldrWithKey' :: (Key -> a -> b -> b) -> b -> IntMap a -> b
foldrWithKey' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z l) r -- put negative numbers before
                        | otherwise -> go (go z r) l
            _ -> go z t
  where
    STRICT_1_OF_2(go)
    go z' Nil           = z'
    go z' (Tip kx x)    = f kx x z'
    go z' (Bin _ _ l r) = go (go z' r) l
{-# INLINE foldrWithKey' #-}

-- | /O(n)/. Fold the keys and values in the map using the given left-associative
-- binary operator, such that
-- @'foldlWithKey' f z == 'Prelude.foldl' (\\z' (kx, x) -> f z' kx x) z . 'toAscList'@.
--
-- For example,
--
-- > keys = reverse . foldlWithKey (\ks k x -> k:ks) []
--
-- > let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
-- > foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
foldlWithKey :: (a -> Key -> b -> a) -> a -> IntMap b -> a
foldlWithKey f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                        | otherwise -> go (go z l) r
            _ -> go z t
  where
    go z' Nil           = z'
    go z' (Tip kx x)    = f z' kx x
    go z' (Bin _ _ l r) = go (go z' l) r
{-# INLINE foldlWithKey #-}

-- | /O(n)/. A strict version of 'foldlWithKey'. Each application of the operator is
-- evaluated before using the result in the next application. This
-- function is strict in the starting value.
foldlWithKey' :: (a -> Key -> b -> a) -> a -> IntMap b -> a
foldlWithKey' f z = \t ->      -- Use lambda t to be inlinable with two arguments only.
  case t of Bin _ m l r | m < 0 -> go (go z r) l -- put negative numbers before
                        | otherwise -> go (go z l) r
            _ -> go z t
  where
    STRICT_1_OF_2(go)
    go z' Nil           = z'
    go z' (Tip kx x)    = f z' kx x
    go z' (Bin _ _ l r) = go (go z' l) r
{-# INLINE foldlWithKey' #-}

-- | /O(n)/. Fold the keys and values in the map using the given monoid, such that
--
-- @'foldMapWithKey' f = 'Prelude.fold' . 'mapWithKey' f@
--
-- This can be an asymptotically faster than 'foldrWithKey' or 'foldlWithKey' for some monoids.
foldMapWithKey :: Monoid m => (Key -> a -> m) -> IntMap a -> m
foldMapWithKey f = go
  where
    go Nil           = mempty
    go (Tip kx x)    = f kx x
    go (Bin _ _ l r) = go l `mappend` go r
{-# INLINE foldMapWithKey #-}

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

elems :: IntMap a -> [a]
elems = foldr (:) []

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

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

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

assocs :: IntMap a -> [(Key,a)]
assocs = toAscList

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

keysSet :: IntMap a -> IntSet.IntSet
keysSet Nil = IntSet.Nil
keysSet (Tip kx _) = IntSet.singleton kx
keysSet (Bin p m l r)
  | m .&. IntSet.suffixBitMask == 0 = IntSet.Bin p m (keysSet l) (keysSet r)
  | otherwise = IntSet.Tip (p .&. IntSet.prefixBitMask) (computeBm (computeBm 0 l) r)
  where STRICT_1_OF_2(computeBm)
        computeBm acc (Bin _ _ l' r') = computeBm (computeBm acc l') r'
        computeBm acc (Tip kx _) = acc .|. IntSet.bitmapOf kx
        computeBm _   Nil = error "Data.IntSet.keysSet: Nil"

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

toList :: IntMap a -> [(Key,a)]
toList = toAscList

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

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

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

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

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

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

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


-- | /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
  = foldlStrict 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,"c")] == fromList [(3, "ab"), (5, "cba")]
-- > 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'.
--
-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
-- > fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
-- > fromListWithKey f [] == empty

fromListWithKey :: (Key -> a -> a -> a) -> [(Key,a)] -> IntMap a
fromListWithKey f xs
  = foldlStrict 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./
--
-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
-- > fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]

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 (join 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


{--------------------------------------------------------------------
  Eq
--------------------------------------------------------------------}
instance Eq a => Eq (IntMap a) where
  t1 == t2  = equal t1 t2
  t1 /= t2  = nequal t1 t2

equal :: Eq a => IntMap a -> IntMap a -> Bool
equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
  = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
equal (Tip kx x) (Tip ky y)
  = (kx == ky) && (x==y)
equal Nil Nil = True
equal _   _   = False

nequal :: Eq a => IntMap a -> IntMap a -> Bool
nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
  = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
nequal (Tip kx x) (Tip ky y)
  = (kx /= ky) || (x/=y)
nequal Nil Nil = False
nequal _   _   = True

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

instance Ord a => Ord (IntMap a) where
    compare m1 m2 = compare (toList m1) (toList m2)

{--------------------------------------------------------------------
  Functor
--------------------------------------------------------------------}

instance Functor IntMap where
    fmap = map

{--------------------------------------------------------------------
  Show
--------------------------------------------------------------------}

instance Show a => Show (IntMap a) where
  showsPrec d m   = showParen (d > 10) $
    showString "fromList " . shows (toList m)

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

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

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

#include "Typeable.h"
INSTANCE_TYPEABLE1(IntMap,intMapTc,"IntMap")

{--------------------------------------------------------------------
  Helpers
--------------------------------------------------------------------}
{--------------------------------------------------------------------
  Join
--------------------------------------------------------------------}
join :: Prefix -> IntMap a -> Prefix -> IntMap a -> IntMap a
join p1 t1 p2 t2
  | zero p1 m = Bin p m t1 t2
  | otherwise = Bin p m t2 t1
  where
    m = branchMask p1 p2
    p = mask p1 m
{-# INLINE join #-}

{--------------------------------------------------------------------
  @bin@ assures that we never have empty trees within a tree.
--------------------------------------------------------------------}
bin :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a
bin _ _ l Nil = l
bin _ _ Nil r = r
bin p m l r   = Bin p m l r
{-# INLINE bin #-}


{--------------------------------------------------------------------
  Endian independent bit twiddling
--------------------------------------------------------------------}
zero :: Key -> Mask -> Bool
zero i m
  = (natFromInt i) .&. (natFromInt m) == 0
{-# INLINE zero #-}

nomatch,match :: Key -> Prefix -> Mask -> Bool
nomatch i p m
  = (mask i m) /= p
{-# INLINE nomatch #-}

match i p m
  = (mask i m) == p
{-# INLINE match #-}

mask :: Key -> Mask -> Prefix
mask i m
  = maskW (natFromInt i) (natFromInt m)
{-# INLINE mask #-}


{--------------------------------------------------------------------
  Big endian operations
--------------------------------------------------------------------}
maskW :: Nat -> Nat -> Prefix
maskW i m
  = intFromNat (i .&. (complement (m-1) `xor` m))
{-# INLINE maskW #-}

shorter :: Mask -> Mask -> Bool
shorter m1 m2
  = (natFromInt m1) > (natFromInt m2)
{-# INLINE shorter #-}

branchMask :: Prefix -> Prefix -> Mask
branchMask p1 p2
  = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
{-# INLINE branchMask #-}

{--------------------------------------------------------------------
  Utilities
--------------------------------------------------------------------}

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

{--------------------------------------------------------------------
  Debugging
--------------------------------------------------------------------}
-- | /O(n)/. Show the tree that implements the map. The tree is shown
-- in a compressed, hanging format.
showTree :: Show a => IntMap a -> String
showTree s
  = showTreeWith True False s


{- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
 the tree that implements the map. If @hang@ is
 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
 @wide@ is 'True', an extra wide version is shown.
-}
showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String
showTreeWith hang wide t
  | hang      = (showsTreeHang wide [] t) ""
  | otherwise = (showsTree wide [] [] t) ""

showsTree :: Show a => Bool -> [String] -> [String] -> IntMap a -> ShowS
showsTree wide lbars rbars t
  = case t of
      Bin p m l r
          -> showsTree wide (withBar rbars) (withEmpty rbars) r .
             showWide wide rbars .
             showsBars lbars . showString (showBin p m) . showString "\n" .
             showWide wide lbars .
             showsTree wide (withEmpty lbars) (withBar lbars) l
      Tip k x
          -> showsBars lbars . showString " " . shows k . showString ":=" . shows x . showString "\n"
      Nil -> showsBars lbars . showString "|\n"

showsTreeHang :: Show a => Bool -> [String] -> IntMap a -> ShowS
showsTreeHang wide bars t
  = case t of
      Bin p m l r
          -> showsBars bars . showString (showBin p m) . showString "\n" .
             showWide wide bars .
             showsTreeHang wide (withBar bars) l .
             showWide wide bars .
             showsTreeHang wide (withEmpty bars) r
      Tip k x
          -> showsBars bars . showString " " . shows k . showString ":=" . shows x . showString "\n"
      Nil -> showsBars bars . showString "|\n"

showBin :: Prefix -> Mask -> String
showBin _ _
  = "*" -- ++ show (p,m)

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

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

node :: String
node           = "+--"

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