{-# LANGUAGE CPP #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE PatternGuards #-}
#if __GLASGOW_HASKELL__
{-# LANGUAGE MagicHash, DeriveDataTypeable, StandaloneDeriving #-}
{-# LANGUAGE ScopedTypeVariables #-}
#endif
#if !defined(TESTING) && defined(__GLASGOW_HASKELL__)
{-# LANGUAGE Trustworthy #-}
#endif
#if __GLASGOW_HASKELL__ >= 708
{-# LANGUAGE TypeFamilies #-}
#endif

{-# OPTIONS_HADDOCK not-home #-}

#include "containers.h"

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.IntMap.Internal
-- Copyright   :  (c) Daan Leijen 2002
--                (c) Andriy Palamarchuk 2008
--                (c) wren romano 2016
-- License     :  BSD-style
-- Maintainer  :  libraries@haskell.org
-- Portability :  portable
--
-- = WARNING
--
-- This module is considered __internal__.
--
-- The Package Versioning Policy __does not apply__.
--
-- The contents of this module may change __in any way whatsoever__
-- and __without any warning__ between minor versions of this package.
--
-- Authors importing this module are expected to track development
-- closely.
--
-- = Description
--
-- This defines the data structures and core (hidden) manipulations
-- on representations.
--
-- @since 0.5.9
-----------------------------------------------------------------------------

-- [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.Internal (
    -- * Map type
      IntMap(..), Key          -- instance Eq,Show

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

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

    -- * Construction
    , empty
    , singleton

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

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

    -- * Combine

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

    -- ** Difference
    , difference
    , differenceWith
    , differenceWithKey

    -- ** Intersection
    , intersection
    , intersectionWith
    , intersectionWithKey

    -- ** General combining function
    , SimpleWhenMissing
    , SimpleWhenMatched
    , runWhenMatched
    , runWhenMissing
    , merge
    -- *** @WhenMatched@ tactics
    , zipWithMaybeMatched
    , zipWithMatched
    -- *** @WhenMissing@ tactics
    , mapMaybeMissing
    , dropMissing
    , preserveMissing
    , mapMissing
    , filterMissing

    -- ** Applicative general combining function
    , WhenMissing (..)
    , WhenMatched (..)
    , mergeA
    -- *** @WhenMatched@ tactics
    -- | The tactics described for 'merge' work for
    -- 'mergeA' as well. Furthermore, the following
    -- are available.
    , zipWithMaybeAMatched
    , zipWithAMatched
    -- *** @WhenMissing@ tactics
    -- | The tactics described for 'merge' work for
    -- 'mergeA' as well. Furthermore, the following
    -- are available.
    , traverseMaybeMissing
    , traverseMissing
    , filterAMissing

    -- ** Deprecated general combining function
    , mergeWithKey
    , mergeWithKey'

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

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

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

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

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

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

    -- * 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

    -- * Debugging
    , showTree
    , showTreeWith

    -- * Internal types
    , Mask, Prefix, Nat

    -- * Utility
    , natFromInt
    , intFromNat
    , link
    , bin
    , binCheckLeft
    , binCheckRight
    , zero
    , nomatch
    , match
    , mask
    , maskW
    , shorter
    , branchMask
    , highestBitMask

    -- * Used by "IntMap.Merge.Lazy" and "IntMap.Merge.Strict"
    , mapWhenMissing
    , mapWhenMatched
    , lmapWhenMissing
    , contramapFirstWhenMatched
    , contramapSecondWhenMatched
    , mapGentlyWhenMissing
    , mapGentlyWhenMatched
    ) where

#if MIN_VERSION_base(4,8,0)
import Data.Functor.Identity (Identity (..))
import Control.Applicative (liftA2)
#else
import Control.Applicative (Applicative(pure, (<*>)), (<$>), liftA2)
import Data.Monoid (Monoid(..))
import Data.Traversable (Traversable(traverse))
import Data.Word (Word)
#endif
#if MIN_VERSION_base(4,9,0)
import Data.Semigroup (Semigroup(stimes))
#endif
#if !(MIN_VERSION_base(4,11,0)) && MIN_VERSION_base(4,9,0)
import Data.Semigroup (Semigroup((<>)))
#endif
#if MIN_VERSION_base(4,9,0)
import Data.Semigroup (stimesIdempotentMonoid)
import Data.Functor.Classes
#endif

import Control.DeepSeq (NFData(rnf))
import Data.Bits
import qualified Data.Foldable as Foldable
#if !MIN_VERSION_base(4,8,0)
import Data.Foldable (Foldable())
#endif
import Data.Maybe (fromMaybe)
import Data.Typeable
import Prelude hiding (lookup, map, filter, foldr, foldl, null)

import Data.IntSet.Internal (Key)
import qualified Data.IntSet.Internal as IntSet
import Utils.Containers.Internal.BitUtil
import Utils.Containers.Internal.StrictPair

#if __GLASGOW_HASKELL__
import Data.Data (Data(..), Constr, mkConstr, constrIndex, Fixity(Prefix),
                  DataType, mkDataType)
import GHC.Exts (build)
#if !MIN_VERSION_base(4,8,0)
import Data.Functor ((<$))
#endif
#if __GLASGOW_HASKELL__ >= 708
import qualified GHC.Exts as GHCExts
#endif
import Text.Read
#endif
import qualified Control.Category as Category
#if __GLASGOW_HASKELL__ >= 709
import Data.Coerce
#endif


-- 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)
-- Fields:
--   prefix: The most significant bits shared by all keys in this Bin.
--   mask: The switching bit to determine if a key should follow the left
--         or right subtree of a 'Bin'.
-- Invariant: Nil is never found as a child of Bin.
-- Invariant: The Mask is a power of 2. It is the largest bit position at which
--            two keys of the map differ.
-- Invariant: Prefix is the common high-order bits that all elements share to
--            the left of the Mask bit.
-- Invariant: In (Bin prefix mask left right), left consists of the elements that
--            don't have the mask bit set; right is all the elements that do.
              | Tip {-# UNPACK #-} !Key a
              | Nil

type Prefix = Int
type Mask   = Int


-- Some stuff from "Data.IntSet.Internal", for 'restrictKeys' and
-- 'withoutKeys' to use.
type IntSetPrefix = Int
type IntSetBitMap = Word

bitmapOf :: Int -> IntSetBitMap
bitmapOf x = shiftLL 1 (x .&. IntSet.suffixBitMask)
{-# INLINE bitmapOf #-}

{--------------------------------------------------------------------
  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

-- | /O(min(n,W))/. Find the value at a key.
-- Returns 'Nothing' when the element can not be found.
--
-- > fromList [(5,'a'), (3,'b')] !? 1 == Nothing
-- > fromList [(5,'a'), (3,'b')] !? 5 == Just 'a'
--
-- @since 0.5.11

(!?) :: IntMap a -> Key -> Maybe a
(!?) m k = lookup 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
    mconcat = unions
#if !(MIN_VERSION_base(4,9,0))
    mappend = union
#else
    mappend = (<>)

-- | @since 0.5.7
instance Semigroup (IntMap a) where
    (<>)    = union
    stimes  = stimesIdempotentMonoid
#endif

-- | Folds in order of increasing key.
instance Foldable.Foldable IntMap where
  fold = go
    where go Nil = mempty
          go (Tip _ v) = v
          go (Bin _ m l r)
            | m < 0     = go r `mappend` go l
            | otherwise = 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 _ m l r)
            | m < 0     = go r `mappend` go l
            | otherwise = go l `mappend` go r
  {-# INLINE foldMap #-}
  foldl' = foldl'
  {-# INLINE foldl' #-}
  foldr' = foldr'
  {-# INLINE foldr' #-}
#if MIN_VERSION_base(4,8,0)
  length = size
  {-# INLINE length #-}
  null   = null
  {-# INLINE null #-}
  toList = elems -- NB: Foldable.toList /= IntMap.toList
  {-# INLINE toList #-}
  elem = go
    where go !_ Nil = False
          go x (Tip _ y) = x == y
          go x (Bin _ _ l r) = go x l || go x r
  {-# INLINABLE elem #-}
  maximum = start
    where start Nil = error "Data.Foldable.maximum (for Data.IntMap): empty map"
          start (Tip _ y) = y
          start (Bin _ _ l r) = go (start l) r

          go !m Nil = m
          go m (Tip _ y) = max m y
          go m (Bin _ _ l r) = go (go m l) r
  {-# INLINABLE maximum #-}
  minimum = start
    where start Nil = error "Data.Foldable.minimum (for Data.IntMap): empty map"
          start (Tip _ y) = y
          start (Bin _ _ l r) = go (start l) r

          go !m Nil = m
          go m (Tip _ y) = min m y
          go m (Bin _ _ l r) = go (go m l) r
  {-# INLINABLE minimum #-}
  sum = foldl' (+) 0
  {-# INLINABLE sum #-}
  product = foldl' (*) 1
  {-# INLINABLE product #-}
#endif

-- | Traverses in order of increasing key.
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.Internal.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 = go 0
  where
    go !acc (Bin _ _ l r) = go (go acc l) r
    go acc (Tip _ _) = 1 + acc
    go acc Nil = acc

-- | /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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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

{--------------------------------------------------------------------
  Disjoint
--------------------------------------------------------------------}
-- | /O(n+m)/. Check whether the key sets of two maps are disjoint
-- (i.e. their 'intersection' is empty).
--
-- > disjoint (fromList [(2,'a')]) (fromList [(1,()), (3,())])   == True
-- > disjoint (fromList [(2,'a')]) (fromList [(1,'a'), (2,'b')]) == False
-- > disjoint (fromList [])        (fromList [])                 == True
--
-- > disjoint a b == null (intersection a b)
--
-- @since 0.6.2.1
disjoint :: IntMap a -> IntMap b -> Bool
disjoint Nil _ = True
disjoint _ Nil = True
disjoint (Tip kx _) ys = notMember kx ys
disjoint xs (Tip ky _) = notMember ky xs
disjoint t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
  | shorter m1 m2 = disjoint1
  | shorter m2 m1 = disjoint2
  | p1 == p2      = disjoint l1 l2 && disjoint r1 r2
  | otherwise     = True
  where
    disjoint1 | nomatch p2 p1 m1 = True
              | zero p2 m1       = disjoint l1 t2
              | otherwise        = disjoint r1 t2
    disjoint2 | nomatch p1 p2 m2 = True
              | zero p1 m2       = disjoint t1 l2
              | otherwise        = disjoint t1 r2

{--------------------------------------------------------------------
  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@(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)
insert k x t@(Tip ky _)
  | k==ky         = Tip k x
  | otherwise     = link k (Tip k x) ky t
insert k x 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@(Bin p m l r)
  | nomatch k p m = link 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)
insertWithKey f k x t@(Tip ky y)
  | k == ky       = Tip k (f k x y)
  | otherwise     = link k (Tip k x) ky t
insertWithKey _ k x 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@(Bin p m l r)
  | nomatch k p m = (Nothing,link 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')
insertLookupWithKey f k x t@(Tip ky y)
  | k == ky       = (Just y,Tip k (f k x y))
  | otherwise     = (Nothing,link k (Tip k x) ky t)
insertLookupWithKey _ k x 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@(Bin p m l r)
  | nomatch k p m = t
  | zero k m      = binCheckLeft p m (delete k l) r
  | otherwise     = binCheckRight p m l (delete k r)
delete k t@(Tip ky _)
  | k == ky       = Nil
  | otherwise     = t
delete _k 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 !k t@(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)
adjustWithKey f k t@(Tip ky y)
  | k == ky       = Tip ky (f k y)
  | otherwise     = t
adjustWithKey _ _ 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@(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)
updateWithKey f k t@(Tip ky y)
  | k == ky       = case (f k y) of
                      Just y' -> Tip ky y'
                      Nothing -> Nil
  | otherwise     = t
updateWithKey _ _ 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@(Bin p m l r)
  | nomatch k p m = (Nothing,t)
  | zero k m      = let !(found,l') = updateLookupWithKey f k l
                    in (found,binCheckLeft p m l' r)
  | otherwise     = let !(found,r') = updateLookupWithKey f k r
                    in (found,binCheckRight p m l r')
updateLookupWithKey f k t@(Tip ky y)
  | k==ky         = case (f k y) of
                      Just y' -> (Just y,Tip ky y')
                      Nothing -> (Just y,Nil)
  | otherwise     = (Nothing,t)
updateLookupWithKey _ _ 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@(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)
alter f k t@(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 -> Tip ky y
alter f k 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 stolen 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.
--
-- > 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 :: Foldable f => f (IntMap a) -> IntMap a
unions xs
  = Foldable.foldl' 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 :: Foldable f => (a->a->a) -> f (IntMap a) -> IntMap a
unionsWith f ts
  = Foldable.foldl' (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


-- TODO(wrengr): re-verify that asymptotic bound
-- | /O(n+m)/. Remove all the keys in a given set from a map.
--
-- @
-- m \`withoutKeys\` s = 'filterWithKey' (\k _ -> k ``IntSet.notMember`` s) m
-- @
--
-- @since 0.5.8
withoutKeys :: IntMap a -> IntSet.IntSet -> IntMap a
withoutKeys t1@(Bin p1 m1 l1 r1) t2@(IntSet.Bin p2 m2 l2 r2)
    | shorter m1 m2  = difference1
    | shorter m2 m1  = difference2
    | p1 == p2       = bin p1 m1 (withoutKeys l1 l2) (withoutKeys r1 r2)
    | otherwise      = t1
    where
    difference1
        | nomatch p2 p1 m1  = t1
        | zero p2 m1        = binCheckLeft p1 m1 (withoutKeys l1 t2) r1
        | otherwise         = binCheckRight p1 m1 l1 (withoutKeys r1 t2)
    difference2
        | nomatch p1 p2 m2  = t1
        | zero p1 m2        = withoutKeys t1 l2
        | otherwise         = withoutKeys t1 r2
withoutKeys t1@(Bin p1 m1 _ _) (IntSet.Tip p2 bm2) =
    let minbit = bitmapOf p1
        lt_minbit = minbit - 1
        maxbit = bitmapOf (p1 .|. (m1 .|. (m1 - 1)))
        gt_maxbit = (-maxbit) `xor` maxbit
    -- TODO(wrengr): should we manually inline/unroll 'updatePrefix'
    -- and 'withoutBM' here, in order to avoid redundant case analyses?
    in updatePrefix p2 t1 $ withoutBM (bm2 .|. lt_minbit .|. gt_maxbit)
withoutKeys t1@(Bin _ _ _ _) IntSet.Nil = t1
withoutKeys t1@(Tip k1 _) t2
    | k1 `IntSet.member` t2 = Nil
    | otherwise = t1
withoutKeys Nil _ = Nil


updatePrefix
    :: IntSetPrefix -> IntMap a -> (IntMap a -> IntMap a) -> IntMap a
updatePrefix !kp t@(Bin p m l r) f
    | m .&. IntSet.suffixBitMask /= 0 =
        if p .&. IntSet.prefixBitMask == kp then f t else t
    | nomatch kp p m = t
    | zero kp m      = binCheckLeft p m (updatePrefix kp l f) r
    | otherwise      = binCheckRight p m l (updatePrefix kp r f)
updatePrefix kp t@(Tip kx _) f
    | kx .&. IntSet.prefixBitMask == kp = f t
    | otherwise = t
updatePrefix _ Nil _ = Nil


withoutBM :: IntSetBitMap -> IntMap a -> IntMap a
withoutBM 0 t = t
withoutBM bm (Bin p m l r) =
    let leftBits = bitmapOf (p .|. m) - 1
        bmL = bm .&. leftBits
        bmR = bm `xor` bmL -- = (bm .&. complement leftBits)
    in  bin p m (withoutBM bmL l) (withoutBM bmR r)
withoutBM bm t@(Tip k _)
    -- TODO(wrengr): need we manually inline 'IntSet.Member' here?
    | k `IntSet.member` IntSet.Tip (k .&. IntSet.prefixBitMask) bm = Nil
    | otherwise = t
withoutBM _ Nil = Nil


{--------------------------------------------------------------------
  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


-- TODO(wrengr): re-verify that asymptotic bound
-- | /O(n+m)/. The restriction of a map to the keys in a set.
--
-- @
-- m \`restrictKeys\` s = 'filterWithKey' (\k _ -> k ``IntSet.member`` s) m
-- @
--
-- @since 0.5.8
restrictKeys :: IntMap a -> IntSet.IntSet -> IntMap a
restrictKeys t1@(Bin p1 m1 l1 r1) t2@(IntSet.Bin p2 m2 l2 r2)
    | shorter m1 m2  = intersection1
    | shorter m2 m1  = intersection2
    | p1 == p2       = bin p1 m1 (restrictKeys l1 l2) (restrictKeys r1 r2)
    | otherwise      = Nil
    where
    intersection1
        | nomatch p2 p1 m1  = Nil
        | zero p2 m1        = restrictKeys l1 t2
        | otherwise         = restrictKeys r1 t2
    intersection2
        | nomatch p1 p2 m2  = Nil
        | zero p1 m2        = restrictKeys t1 l2
        | otherwise         = restrictKeys t1 r2
restrictKeys t1@(Bin p1 m1 _ _) (IntSet.Tip p2 bm2) =
    let minbit = bitmapOf p1
        ge_minbit = complement (minbit - 1)
        maxbit = bitmapOf (p1 .|. (m1 .|. (m1 - 1)))
        le_maxbit = maxbit .|. (maxbit - 1)
    -- TODO(wrengr): should we manually inline/unroll 'lookupPrefix'
    -- and 'restrictBM' here, in order to avoid redundant case analyses?
    in restrictBM (bm2 .&. ge_minbit .&. le_maxbit) (lookupPrefix p2 t1)
restrictKeys (Bin _ _ _ _) IntSet.Nil = Nil
restrictKeys t1@(Tip k1 _) t2
    | k1 `IntSet.member` t2 = t1
    | otherwise = Nil
restrictKeys Nil _ = Nil


-- | /O(min(n,W))/. Restrict to the sub-map with all keys matching
-- a key prefix.
lookupPrefix :: IntSetPrefix -> IntMap a -> IntMap a
lookupPrefix !kp t@(Bin p m l r)
    | m .&. IntSet.suffixBitMask /= 0 =
        if p .&. IntSet.prefixBitMask == kp then t else Nil
    | nomatch kp p m = Nil
    | zero kp m      = lookupPrefix kp l
    | otherwise      = lookupPrefix kp r
lookupPrefix kp t@(Tip kx _)
    | (kx .&. IntSet.prefixBitMask) == kp = t
    | otherwise = Nil
lookupPrefix _ Nil = Nil


restrictBM :: IntSetBitMap -> IntMap a -> IntMap a
restrictBM 0 _ = Nil
restrictBM bm (Bin p m l r) =
    let leftBits = bitmapOf (p .|. m) - 1
        bmL = bm .&. leftBits
        bmR = bm `xor` bmL -- = (bm .&. complement leftBits)
    in  bin p m (restrictBM bmL l) (restrictBM bmR r)
restrictBM bm t@(Tip k _)
    -- TODO(wrengr): need we manually inline 'IntSet.Member' here?
    | k `IntSet.member` IntSet.Tip (k .&. IntSet.prefixBitMask) bm = t
    | otherwise = Nil
restrictBM _ Nil = Nil


-- | /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_link p1 (g1 t1) p2 (g2 t2)
      where
        merge1 | nomatch p2 p1 m1  = maybe_link 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_link 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' _) = merge0 t2' k2' t1'
      where
        merge0 t2 k2 t1@(Bin p1 m1 l1 r1)
          | nomatch k2 p1 m1 = maybe_link p1 (g1 t1) k2 (g2 t2)
          | zero k2 m1 = bin' p1 m1 (merge0 t2 k2 l1) (g1 r1)
          | otherwise  = bin' p1 m1 (g1 l1) (merge0 t2 k2 r1)
        merge0 t2 k2 t1@(Tip k1 _)
          | k1 == k2 = f t1 t2
          | otherwise = maybe_link k1 (g1 t1) k2 (g2 t2)
        merge0 t2 _  Nil = g2 t2

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

    go t1'@(Tip k1' _) t2' = merge0 t1' k1' t2'
      where
        merge0 t1 k1 t2@(Bin p2 m2 l2 r2)
          | nomatch k1 p2 m2 = maybe_link k1 (g1 t1) p2 (g2 t2)
          | zero k1 m2 = bin' p2 m2 (merge0 t1 k1 l2) (g2 r2)
          | otherwise  = bin' p2 m2 (g2 l2) (merge0 t1 k1 r2)
        merge0 t1 k1 t2@(Tip k2 _)
          | k1 == k2 = f t1 t2
          | otherwise = maybe_link k1 (g1 t1) k2 (g2 t2)
        merge0 t1 _  Nil = g1 t1

    go Nil t2 = g2 t2

    maybe_link _ Nil _ t2 = t2
    maybe_link _ t1 _ Nil = t1
    maybe_link p1 t1 p2 t2 = link p1 t1 p2 t2
    {-# INLINE maybe_link #-}
{-# INLINE mergeWithKey' #-}


{--------------------------------------------------------------------
  mergeA
--------------------------------------------------------------------}

-- | A tactic for dealing with keys present in one map but not the
-- other in 'merge' or 'mergeA'.
--
-- A tactic of type @WhenMissing f k x z@ is an abstract representation
-- of a function of type @Key -> x -> f (Maybe z)@.
--
-- @since 0.5.9

data WhenMissing f x y = WhenMissing
  { missingSubtree :: IntMap x -> f (IntMap y)
  , missingKey :: Key -> x -> f (Maybe y)}

-- | @since 0.5.9
instance (Applicative f, Monad f) => Functor (WhenMissing f x) where
  fmap = mapWhenMissing
  {-# INLINE fmap #-}


-- | @since 0.5.9
instance (Applicative f, Monad f) => Category.Category (WhenMissing f)
  where
    id = preserveMissing
    f . g =
      traverseMaybeMissing $ \ k x -> do
        y <- missingKey g k x
        case y of
          Nothing -> pure Nothing
          Just q  -> missingKey f k q
    {-# INLINE id #-}
    {-# INLINE (.) #-}


-- | Equivalent to @ReaderT k (ReaderT x (MaybeT f))@.
--
-- @since 0.5.9
instance (Applicative f, Monad f) => Applicative (WhenMissing f x) where
  pure x = mapMissing (\ _ _ -> x)
  f <*> g =
    traverseMaybeMissing $ \k x -> do
      res1 <- missingKey f k x
      case res1 of
        Nothing -> pure Nothing
        Just r  -> (pure $!) . fmap r =<< missingKey g k x
  {-# INLINE pure #-}
  {-# INLINE (<*>) #-}


-- | Equivalent to @ReaderT k (ReaderT x (MaybeT f))@.
--
-- @since 0.5.9
instance (Applicative f, Monad f) => Monad (WhenMissing f x) where
#if !MIN_VERSION_base(4,8,0)
  return = pure
#endif
  m >>= f =
    traverseMaybeMissing $ \k x -> do
      res1 <- missingKey m k x
      case res1 of
        Nothing -> pure Nothing
        Just r  -> missingKey (f r) k x
  {-# INLINE (>>=) #-}


-- | Map covariantly over a @'WhenMissing' f x@.
--
-- @since 0.5.9
mapWhenMissing
  :: (Applicative f, Monad f)
  => (a -> b)
  -> WhenMissing f x a
  -> WhenMissing f x b
mapWhenMissing f t = WhenMissing
  { missingSubtree = \m -> missingSubtree t m >>= \m' -> pure $! fmap f m'
  , missingKey     = \k x -> missingKey t k x >>= \q -> (pure $! fmap f q) }
{-# INLINE mapWhenMissing #-}


-- | Map covariantly over a @'WhenMissing' f x@, using only a
-- 'Functor f' constraint.
mapGentlyWhenMissing
  :: Functor f
  => (a -> b)
  -> WhenMissing f x a
  -> WhenMissing f x b
mapGentlyWhenMissing f t = WhenMissing
  { missingSubtree = \m -> fmap f <$> missingSubtree t m
  , missingKey     = \k x -> fmap f <$> missingKey t k x }
{-# INLINE mapGentlyWhenMissing #-}


-- | Map covariantly over a @'WhenMatched' f k x@, using only a
-- 'Functor f' constraint.
mapGentlyWhenMatched
  :: Functor f
  => (a -> b)
  -> WhenMatched f x y a
  -> WhenMatched f x y b
mapGentlyWhenMatched f t =
  zipWithMaybeAMatched $ \k x y -> fmap f <$> runWhenMatched t k x y
{-# INLINE mapGentlyWhenMatched #-}


-- | Map contravariantly over a @'WhenMissing' f _ x@.
--
-- @since 0.5.9
lmapWhenMissing :: (b -> a) -> WhenMissing f a x -> WhenMissing f b x
lmapWhenMissing f t = WhenMissing
  { missingSubtree = \m -> missingSubtree t (fmap f m)
  , missingKey     = \k x -> missingKey t k (f x) }
{-# INLINE lmapWhenMissing #-}


-- | Map contravariantly over a @'WhenMatched' f _ y z@.
--
-- @since 0.5.9
contramapFirstWhenMatched
  :: (b -> a)
  -> WhenMatched f a y z
  -> WhenMatched f b y z
contramapFirstWhenMatched f t =
  WhenMatched $ \k x y -> runWhenMatched t k (f x) y
{-# INLINE contramapFirstWhenMatched #-}


-- | Map contravariantly over a @'WhenMatched' f x _ z@.
--
-- @since 0.5.9
contramapSecondWhenMatched
  :: (b -> a)
  -> WhenMatched f x a z
  -> WhenMatched f x b z
contramapSecondWhenMatched f t =
  WhenMatched $ \k x y -> runWhenMatched t k x (f y)
{-# INLINE contramapSecondWhenMatched #-}


#if !MIN_VERSION_base(4,8,0)
newtype Identity a = Identity {runIdentity :: a}

instance Functor Identity where
    fmap f (Identity x) = Identity (f x)

instance Applicative Identity where
    pure = Identity
    Identity f <*> Identity x = Identity (f x)
#endif

-- | A tactic for dealing with keys present in one map but not the
-- other in 'merge'.
--
-- A tactic of type @SimpleWhenMissing x z@ is an abstract
-- representation of a function of type @Key -> x -> Maybe z@.
--
-- @since 0.5.9
type SimpleWhenMissing = WhenMissing Identity


-- | A tactic for dealing with keys present in both maps in 'merge'
-- or 'mergeA'.
--
-- A tactic of type @WhenMatched f x y z@ is an abstract representation
-- of a function of type @Key -> x -> y -> f (Maybe z)@.
--
-- @since 0.5.9
newtype WhenMatched f x y z = WhenMatched
  { matchedKey :: Key -> x -> y -> f (Maybe z) }


-- | Along with zipWithMaybeAMatched, witnesses the isomorphism
-- between @WhenMatched f x y z@ and @Key -> x -> y -> f (Maybe z)@.
--
-- @since 0.5.9
runWhenMatched :: WhenMatched f x y z -> Key -> x -> y -> f (Maybe z)
runWhenMatched = matchedKey
{-# INLINE runWhenMatched #-}


-- | Along with traverseMaybeMissing, witnesses the isomorphism
-- between @WhenMissing f x y@ and @Key -> x -> f (Maybe y)@.
--
-- @since 0.5.9
runWhenMissing :: WhenMissing f x y -> Key-> x -> f (Maybe y)
runWhenMissing = missingKey
{-# INLINE runWhenMissing #-}


-- | @since 0.5.9
instance Functor f => Functor (WhenMatched f x y) where
  fmap = mapWhenMatched
  {-# INLINE fmap #-}


-- | @since 0.5.9
instance (Monad f, Applicative f) => Category.Category (WhenMatched f x)
  where
    id = zipWithMatched (\_ _ y -> y)
    f . g =
      zipWithMaybeAMatched $ \k x y -> do
        res <- runWhenMatched g k x y
        case res of
          Nothing -> pure Nothing
          Just r  -> runWhenMatched f k x r
    {-# INLINE id #-}
    {-# INLINE (.) #-}


-- | Equivalent to @ReaderT Key (ReaderT x (ReaderT y (MaybeT f)))@
--
-- @since 0.5.9
instance (Monad f, Applicative f) => Applicative (WhenMatched f x y) where
  pure x = zipWithMatched (\_ _ _ -> x)
  fs <*> xs =
    zipWithMaybeAMatched $ \k x y -> do
      res <- runWhenMatched fs k x y
      case res of
        Nothing -> pure Nothing
        Just r  -> (pure $!) . fmap r =<< runWhenMatched xs k x y
  {-# INLINE pure #-}
  {-# INLINE (<*>) #-}


-- | Equivalent to @ReaderT Key (ReaderT x (ReaderT y (MaybeT f)))@
--
-- @since 0.5.9
instance (Monad f, Applicative f) => Monad (WhenMatched f x y) where
#if !MIN_VERSION_base(4,8,0)
  return = pure
#endif
  m >>= f =
    zipWithMaybeAMatched $ \k x y -> do
      res <- runWhenMatched m k x y
      case res of
        Nothing -> pure Nothing
        Just r  -> runWhenMatched (f r) k x y
  {-# INLINE (>>=) #-}


-- | Map covariantly over a @'WhenMatched' f x y@.
--
-- @since 0.5.9
mapWhenMatched
  :: Functor f
  => (a -> b)
  -> WhenMatched f x y a
  -> WhenMatched f x y b
mapWhenMatched f (WhenMatched g) =
  WhenMatched $ \k x y -> fmap (fmap f) (g k x y)
{-# INLINE mapWhenMatched #-}


-- | A tactic for dealing with keys present in both maps in 'merge'.
--
-- A tactic of type @SimpleWhenMatched x y z@ is an abstract
-- representation of a function of type @Key -> x -> y -> Maybe z@.
--
-- @since 0.5.9
type SimpleWhenMatched = WhenMatched Identity


-- | When a key is found in both maps, apply a function to the key
-- and values and use the result in the merged map.
--
-- > zipWithMatched
-- >   :: (Key -> x -> y -> z)
-- >   -> SimpleWhenMatched x y z
--
-- @since 0.5.9
zipWithMatched
  :: Applicative f
  => (Key -> x -> y -> z)
  -> WhenMatched f x y z
zipWithMatched f = WhenMatched $ \ k x y -> pure . Just $ f k x y
{-# INLINE zipWithMatched #-}


-- | When a key is found in both maps, apply a function to the key
-- and values to produce an action and use its result in the merged
-- map.
--
-- @since 0.5.9
zipWithAMatched
  :: Applicative f
  => (Key -> x -> y -> f z)
  -> WhenMatched f x y z
zipWithAMatched f = WhenMatched $ \ k x y -> Just <$> f k x y
{-# INLINE zipWithAMatched #-}


-- | When a key is found in both maps, apply a function to the key
-- and values and maybe use the result in the merged map.
--
-- > zipWithMaybeMatched
-- >   :: (Key -> x -> y -> Maybe z)
-- >   -> SimpleWhenMatched x y z
--
-- @since 0.5.9
zipWithMaybeMatched
  :: Applicative f
  => (Key -> x -> y -> Maybe z)
  -> WhenMatched f x y z
zipWithMaybeMatched f = WhenMatched $ \ k x y -> pure $ f k x y
{-# INLINE zipWithMaybeMatched #-}


-- | When a key is found in both maps, apply a function to the key
-- and values, perform the resulting action, and maybe use the
-- result in the merged map.
--
-- This is the fundamental 'WhenMatched' tactic.
--
-- @since 0.5.9
zipWithMaybeAMatched
  :: (Key -> x -> y -> f (Maybe z))
  -> WhenMatched f x y z
zipWithMaybeAMatched f = WhenMatched $ \ k x y -> f k x y
{-# INLINE zipWithMaybeAMatched #-}


-- | Drop all the entries whose keys are missing from the other
-- map.
--
-- > dropMissing :: SimpleWhenMissing x y
--
-- prop> dropMissing = mapMaybeMissing (\_ _ -> Nothing)
--
-- but @dropMissing@ is much faster.
--
-- @since 0.5.9
dropMissing :: Applicative f => WhenMissing f x y
dropMissing = WhenMissing
  { missingSubtree = const (pure Nil)
  , missingKey     = \_ _ -> pure Nothing }
{-# INLINE dropMissing #-}


-- | Preserve, unchanged, the entries whose keys are missing from
-- the other map.
--
-- > preserveMissing :: SimpleWhenMissing x x
--
-- prop> preserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -> Just x)
--
-- but @preserveMissing@ is much faster.
--
-- @since 0.5.9
preserveMissing :: Applicative f => WhenMissing f x x
preserveMissing = WhenMissing
  { missingSubtree = pure
  , missingKey     = \_ v -> pure (Just v) }
{-# INLINE preserveMissing #-}


-- | Map over the entries whose keys are missing from the other map.
--
-- > mapMissing :: (k -> x -> y) -> SimpleWhenMissing x y
--
-- prop> mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)
--
-- but @mapMissing@ is somewhat faster.
--
-- @since 0.5.9
mapMissing :: Applicative f => (Key -> x -> y) -> WhenMissing f x y
mapMissing f = WhenMissing
  { missingSubtree = \m -> pure $! mapWithKey f m
  , missingKey     = \k x -> pure $ Just (f k x) }
{-# INLINE mapMissing #-}


-- | Map over the entries whose keys are missing from the other
-- map, optionally removing some. This is the most powerful
-- 'SimpleWhenMissing' tactic, but others are usually more efficient.
--
-- > mapMaybeMissing :: (Key -> x -> Maybe y) -> SimpleWhenMissing x y
--
-- prop> mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))
--
-- but @mapMaybeMissing@ uses fewer unnecessary 'Applicative'
-- operations.
--
-- @since 0.5.9
mapMaybeMissing
  :: Applicative f => (Key -> x -> Maybe y) -> WhenMissing f x y
mapMaybeMissing f = WhenMissing
  { missingSubtree = \m -> pure $! mapMaybeWithKey f m
  , missingKey     = \k x -> pure $! f k x }
{-# INLINE mapMaybeMissing #-}


-- | Filter the entries whose keys are missing from the other map.
--
-- > filterMissing :: (k -> x -> Bool) -> SimpleWhenMissing x x
--
-- prop> filterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -> guard (f k x) *> Just x
--
-- but this should be a little faster.
--
-- @since 0.5.9
filterMissing
  :: Applicative f => (Key -> x -> Bool) -> WhenMissing f x x
filterMissing f = WhenMissing
  { missingSubtree = \m -> pure $! filterWithKey f m
  , missingKey     = \k x -> pure $! if f k x then Just x else Nothing }
{-# INLINE filterMissing #-}


-- | Filter the entries whose keys are missing from the other map
-- using some 'Applicative' action.
--
-- > filterAMissing f = Merge.Lazy.traverseMaybeMissing $
-- >   \k x -> (\b -> guard b *> Just x) <$> f k x
--
-- but this should be a little faster.
--
-- @since 0.5.9
filterAMissing
  :: Applicative f => (Key -> x -> f Bool) -> WhenMissing f x x
filterAMissing f = WhenMissing
  { missingSubtree = \m -> filterWithKeyA f m
  , missingKey     = \k x -> bool Nothing (Just x) <$> f k x }
{-# INLINE filterAMissing #-}


-- | /O(n)/. Filter keys and values using an 'Applicative' predicate.
filterWithKeyA
  :: Applicative f => (Key -> a -> f Bool) -> IntMap a -> f (IntMap a)
filterWithKeyA _ Nil           = pure Nil
filterWithKeyA f t@(Tip k x)   = (\b -> if b then t else Nil) <$> f k x
filterWithKeyA f (Bin p m l r) =
    liftA2 (bin p m) (filterWithKeyA f l) (filterWithKeyA f r)

-- | This wasn't in Data.Bool until 4.7.0, so we define it here
bool :: a -> a -> Bool -> a
bool f _ False = f
bool _ t True  = t


-- | Traverse over the entries whose keys are missing from the other
-- map.
--
-- @since 0.5.9
traverseMissing
  :: Applicative f => (Key -> x -> f y) -> WhenMissing f x y
traverseMissing f = WhenMissing
  { missingSubtree = traverseWithKey f
  , missingKey = \k x -> Just <$> f k x }
{-# INLINE traverseMissing #-}


-- | Traverse over the entries whose keys are missing from the other
-- map, optionally producing values to put in the result. This is
-- the most powerful 'WhenMissing' tactic, but others are usually
-- more efficient.
--
-- @since 0.5.9
traverseMaybeMissing
  :: Applicative f => (Key -> x -> f (Maybe y)) -> WhenMissing f x y
traverseMaybeMissing f = WhenMissing
  { missingSubtree = traverseMaybeWithKey f
  , missingKey = f }
{-# INLINE traverseMaybeMissing #-}


-- | /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)


-- | Merge two maps.
--
-- 'merge' takes two 'WhenMissing' tactics, a 'WhenMatched' tactic
-- and two maps. It uses the tactics to merge the maps. Its behavior
-- is best understood via its fundamental tactics, 'mapMaybeMissing'
-- and 'zipWithMaybeMatched'.
--
-- Consider
--
-- @
-- merge (mapMaybeMissing g1)
--              (mapMaybeMissing g2)
--              (zipWithMaybeMatched f)
--              m1 m2
-- @
--
-- Take, for example,
--
-- @
-- m1 = [(0, \'a\'), (1, \'b\'), (3, \'c\'), (4, \'d\')]
-- m2 = [(1, "one"), (2, "two"), (4, "three")]
-- @
--
-- 'merge' will first \"align\" these maps by key:
--
-- @
-- m1 = [(0, \'a\'), (1, \'b\'),               (3, \'c\'), (4, \'d\')]
-- m2 =           [(1, "one"), (2, "two"),           (4, "three")]
-- @
--
-- It will then pass the individual entries and pairs of entries
-- to @g1@, @g2@, or @f@ as appropriate:
--
-- @
-- maybes = [g1 0 \'a\', f 1 \'b\' "one", g2 2 "two", g1 3 \'c\', f 4 \'d\' "three"]
-- @
--
-- This produces a 'Maybe' for each key:
--
-- @
-- keys =     0        1          2           3        4
-- results = [Nothing, Just True, Just False, Nothing, Just True]
-- @
--
-- Finally, the @Just@ results are collected into a map:
--
-- @
-- return value = [(1, True), (2, False), (4, True)]
-- @
--
-- The other tactics below are optimizations or simplifications of
-- 'mapMaybeMissing' for special cases. Most importantly,
--
-- * 'dropMissing' drops all the keys.
-- * 'preserveMissing' leaves all the entries alone.
--
-- When 'merge' is given three arguments, it is inlined at the call
-- site. To prevent excessive inlining, you should typically use
-- 'merge' to define your custom combining functions.
--
--
-- Examples:
--
-- prop> unionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
-- prop> intersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
-- prop> differenceWith f = merge diffPreserve diffDrop f
-- prop> symmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -> Nothing)
-- prop> mapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
--
-- @since 0.5.9
merge
  :: SimpleWhenMissing a c -- ^ What to do with keys in @m1@ but not @m2@
  -> SimpleWhenMissing b c -- ^ What to do with keys in @m2@ but not @m1@
  -> SimpleWhenMatched a b c -- ^ What to do with keys in both @m1@ and @m2@
  -> IntMap a -- ^ Map @m1@
  -> IntMap b -- ^ Map @m2@
  -> IntMap c
merge g1 g2 f m1 m2 =
  runIdentity $ mergeA g1 g2 f m1 m2
{-# INLINE merge #-}


-- | An applicative version of 'merge'.
--
-- 'mergeA' takes two 'WhenMissing' tactics, a 'WhenMatched'
-- tactic and two maps. It uses the tactics to merge the maps.
-- Its behavior is best understood via its fundamental tactics,
-- 'traverseMaybeMissing' and 'zipWithMaybeAMatched'.
--
-- Consider
--
-- @
-- mergeA (traverseMaybeMissing g1)
--               (traverseMaybeMissing g2)
--               (zipWithMaybeAMatched f)
--               m1 m2
-- @
--
-- Take, for example,
--
-- @
-- m1 = [(0, \'a\'), (1, \'b\'), (3,\'c\'), (4, \'d\')]
-- m2 = [(1, "one"), (2, "two"), (4, "three")]
-- @
--
-- 'mergeA' will first \"align\" these maps by key:
--
-- @
-- m1 = [(0, \'a\'), (1, \'b\'),               (3, \'c\'), (4, \'d\')]
-- m2 =           [(1, "one"), (2, "two"),           (4, "three")]
-- @
--
-- It will then pass the individual entries and pairs of entries
-- to @g1@, @g2@, or @f@ as appropriate:
--
-- @
-- actions = [g1 0 \'a\', f 1 \'b\' "one", g2 2 "two", g1 3 \'c\', f 4 \'d\' "three"]
-- @
--
-- Next, it will perform the actions in the @actions@ list in order from
-- left to right.
--
-- @
-- keys =     0        1          2           3        4
-- results = [Nothing, Just True, Just False, Nothing, Just True]
-- @
--
-- Finally, the @Just@ results are collected into a map:
--
-- @
-- return value = [(1, True), (2, False), (4, True)]
-- @
--
-- The other tactics below are optimizations or simplifications of
-- 'traverseMaybeMissing' for special cases. Most importantly,
--
-- * 'dropMissing' drops all the keys.
-- * 'preserveMissing' leaves all the entries alone.
-- * 'mapMaybeMissing' does not use the 'Applicative' context.
--
-- When 'mergeA' is given three arguments, it is inlined at the call
-- site. To prevent excessive inlining, you should generally only use
-- 'mergeA' to define custom combining functions.
--
-- @since 0.5.9
mergeA
  :: (Applicative f)
  => WhenMissing f a c -- ^ What to do with keys in @m1@ but not @m2@
  -> WhenMissing f b c -- ^ What to do with keys in @m2@ but not @m1@
  -> WhenMatched f a b c -- ^ What to do with keys in both @m1@ and @m2@
  -> IntMap a -- ^ Map @m1@
  -> IntMap b -- ^ Map @m2@
  -> f (IntMap c)
mergeA
    WhenMissing{missingSubtree = g1t, missingKey = g1k}
    WhenMissing{missingSubtree = g2t, missingKey = g2k}
    WhenMatched{matchedKey = f}
    = go
  where
    go t1  Nil = g1t t1
    go Nil t2  = g2t t2

    -- This case is already covered below.
    -- go (Tip k1 x1) (Tip k2 x2) = mergeTips k1 x1 k2 x2

    go (Tip k1 x1) t2' = merge2 t2'
      where
        merge2 t2@(Bin p2 m2 l2 r2)
          | nomatch k1 p2 m2 = linkA k1 (subsingletonBy g1k k1 x1) p2 (g2t t2)
          | zero k1 m2       = liftA2 (bin p2 m2) (merge2 l2) (g2t r2)
          | otherwise        = liftA2 (bin p2 m2) (g2t l2) (merge2 r2)
        merge2 (Tip k2 x2)   = mergeTips k1 x1 k2 x2
        merge2 Nil           = subsingletonBy g1k k1 x1

    go t1' (Tip k2 x2) = merge1 t1'
      where
        merge1 t1@(Bin p1 m1 l1 r1)
          | nomatch k2 p1 m1 = linkA p1 (g1t t1) k2 (subsingletonBy g2k k2 x2)
          | zero k2 m1       = liftA2 (bin p1 m1) (merge1 l1) (g1t r1)
          | otherwise        = liftA2 (bin p1 m1) (g1t l1) (merge1 r1)
        merge1 (Tip k1 x1)   = mergeTips k1 x1 k2 x2
        merge1 Nil           = subsingletonBy g2k k2 x2

    go t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
      | shorter m1 m2  = merge1
      | shorter m2 m1  = merge2
      | p1 == p2       = liftA2 (bin p1 m1)   (go  l1 l2) (go r1 r2)
      | otherwise      = liftA2 (link_ p1 p2) (g1t t1)    (g2t   t2)
      where
        merge1 | nomatch p2 p1 m1  = liftA2 (link_ p1 p2) (g1t t1)    (g2t t2)
               | zero p2 m1        = liftA2 (bin p1 m1)   (go  l1 t2) (g1t r1)
               | otherwise         = liftA2 (bin p1 m1)   (g1t l1)    (go  r1 t2)
        merge2 | nomatch p1 p2 m2  = liftA2 (link_ p1 p2) (g1t t1)    (g2t    t2)
               | zero p1 m2        = liftA2 (bin p2 m2)   (go  t1 l2) (g2t    r2)
               | otherwise         = liftA2 (bin p2 m2)   (g2t    l2) (go  t1 r2)

    subsingletonBy gk k x = maybe Nil (Tip k) <$> gk k x
    {-# INLINE subsingletonBy #-}

    mergeTips k1 x1 k2 x2
      | k1 == k2  = maybe Nil (Tip k1) <$> f k1 x1 x2
      | k1 <  k2  = liftA2 (subdoubleton k1 k2) (g1k k1 x1) (g2k k2 x2)
        {-
        = link_ k1 k2 <$> subsingletonBy g1k k1 x1 <*> subsingletonBy g2k k2 x2
        -}
      | otherwise = liftA2 (subdoubleton k2 k1) (g2k k2 x2) (g1k k1 x1)
    {-# INLINE mergeTips #-}

    subdoubleton _ _   Nothing Nothing     = Nil
    subdoubleton _ k2  Nothing (Just y2)   = Tip k2 y2
    subdoubleton k1 _  (Just y1) Nothing   = Tip k1 y1
    subdoubleton k1 k2 (Just y1) (Just y2) = link k1 (Tip k1 y1) k2 (Tip k2 y2)
    {-# INLINE subdoubleton #-}

    link_ _  _  Nil t2  = t2
    link_ _  _  t1  Nil = t1
    link_ p1 p2 t1  t2  = link p1 t1 p2 t2
    {-# INLINE link_ #-}

    -- | A variant of 'link_' which makes sure to execute side-effects
    -- in the right order.
    linkA
        :: Applicative f
        => Prefix -> f (IntMap a)
        -> Prefix -> f (IntMap a)
        -> f (IntMap a)
    linkA p1 t1 p2 t2
      | zero p1 m = liftA2 (bin p m) t1 t2
      | otherwise = liftA2 (bin p m) t2 t1
      where
        m = branchMask p1 p2
        p = mask p1 m
    {-# INLINE linkA #-}
{-# INLINE mergeA #-}


{--------------------------------------------------------------------
  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 -> 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(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 -> 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"


data View a = View {-# UNPACK #-} !Key a !(IntMap a)

-- | /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
  _ -> Just $ case maxViewWithKeySure t of
                View k v t' -> ((k, v), t')
{-# INLINE maxViewWithKey #-}

maxViewWithKeySure :: IntMap a -> View a
maxViewWithKeySure t =
  case t of
    Nil -> error "maxViewWithKeySure Nil"
    Bin p m l r | m < 0 ->
      case go l of View k a l' -> View k a (binCheckLeft p m l' r)
    _ -> go t
  where
    go (Bin p m l r) =
        case go r of View k a r' -> View k a (binCheckRight p m l r')
    go (Tip k y) = View k y Nil
    go Nil = error "maxViewWithKey_go Nil"
-- See note on NOINLINE at minViewWithKeySure
{-# NOINLINE maxViewWithKeySure #-}

-- | /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
    _ -> Just $ case minViewWithKeySure t of
                  View k v t' -> ((k, v), t')
-- We inline this to give GHC the best possible chance of
-- getting rid of the Maybe, pair, and Int constructors, as
-- well as a thunk under the Just. That is, we really want to
-- be certain this inlines!
{-# INLINE minViewWithKey #-}

minViewWithKeySure :: IntMap a -> View a
minViewWithKeySure t =
  case t of
    Nil -> error "minViewWithKeySure Nil"
    Bin p m l r | m < 0 ->
      case go r of
        View k a r' -> View k a (binCheckRight p m l r')
    _ -> go t
  where
    go (Bin p m l r) =
        case go l of View k a l' -> View k a (binCheckLeft p m l' r)
    go (Tip k y) = View k y Nil
    go Nil = error "minViewWithKey_go Nil"
-- There's never anything significant to be gained by inlining
-- this. Sufficiently recent GHC versions will inline the wrapper
-- anyway, which should be good enough.
{-# NOINLINE minViewWithKeySure #-}

-- | /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)

-- | /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 = fmap (\((_, x), t') -> (x, t')) (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 = fmap (\((_, x), t') -> (x, t')) (minViewWithKey t)

-- | /O(min(n,W))/. Delete and find the maximal element.
-- This function throws an error if the map is empty. Use 'maxViewWithKey'
-- if the map may be empty.
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.
-- This function throws an error if the map is empty. Use 'minViewWithKey'
-- if the map may be empty.
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. Returns 'Nothing' if the map is empty.
lookupMin :: IntMap a -> Maybe (Key, a)
lookupMin Nil = Nothing
lookupMin (Tip k v) = Just (k,v)
lookupMin (Bin _ m l r)
  | m < 0     = go r
  | otherwise = go l
    where go (Tip k v)      = Just (k,v)
          go (Bin _ _ l' _) = go l'
          go Nil            = Nothing

-- | /O(min(n,W))/. The minimal key of the map. Calls 'error' if the map is empty.
-- Use 'minViewWithKey' if the map may be empty.
findMin :: IntMap a -> (Key, a)
findMin t
  | Just r <- lookupMin t = r
  | otherwise = error "findMin: empty map has no minimal element"

-- | /O(min(n,W))/. The maximal key of the map. Returns 'Nothing' if the map is empty.
lookupMax :: IntMap a -> Maybe (Key, a)
lookupMax Nil = Nothing
lookupMax (Tip k v) = Just (k,v)
lookupMax (Bin _ m l r)
  | m < 0     = go l
  | otherwise = go r
    where go (Tip k v)      = Just (k,v)
          go (Bin _ _ _ r') = go r'
          go Nil            = Nothing

-- | /O(min(n,W))/. The maximal key of the map. Calls 'error' if the map is empty.
-- Use 'maxViewWithKey' if the map may be empty.
findMax :: IntMap a -> (Key, a)
findMax t
  | Just r <- lookupMax t = r
  | otherwise = error "findMax: empty map has no maximal element"

-- | /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 = 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 (f . g) xs
 #-}
#endif
#if __GLASGOW_HASKELL__ >= 709
-- Safe coercions were introduced in 7.8, but did not play well with RULES yet.
{-# RULES
"map/coerce" map coerce = coerce
 #-}
#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__
{-# 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/map" forall f g xs . mapWithKey f (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
 #-}
#endif

-- | /O(n)/.
-- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@
-- That is, behaves exactly like a regular 'traverse' except that the traversing
-- function also has access to the key associated with a value.
--
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)
traverseWithKey f = go
  where
    go Nil = pure Nil
    go (Tip k v) = Tip k <$> f k v
    go (Bin p m l r)
      | m < 0     = liftA2 (Bin p m) (go r) (go l)
      | otherwise = liftA2 (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 = go
    where
    go Nil           = Nil
    go t@(Tip k x)   = if predicate k x then t else Nil
    go (Bin p m l r) = bin p m (go l) (go r)

-- | /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', gt)
        else
          case go k r of
            (lt :*: gt) ->
              let !gt' = union gt l
              in (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)


data SplitLookup a = SplitLookup !(IntMap a) !(Maybe a) !(IntMap a)

mapLT :: (IntMap a -> IntMap a) -> SplitLookup a -> SplitLookup a
mapLT f (SplitLookup lt fnd gt) = SplitLookup (f lt) fnd gt
{-# INLINE mapLT #-}

mapGT :: (IntMap a -> IntMap a) -> SplitLookup a -> SplitLookup a
mapGT f (SplitLookup lt fnd gt) = SplitLookup lt fnd (f gt)
{-# INLINE mapGT #-}

-- | /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
    case t of
      Bin _ m l r
        | m < 0 ->
          if k >= 0 -- handle negative numbers.
          then mapLT (union r) (go k l)
          else mapGT (`union` l) (go k r)
      _ -> go k t
  of SplitLookup lt fnd gt -> (lt, fnd, gt)
  where
    go k' t'@(Bin p m l r)
      | nomatch k' p m =
          if k' > p
          then SplitLookup t' Nothing Nil
          else SplitLookup Nil Nothing t'
      | zero k' m = mapGT (`union` r) (go k' l)
      | otherwise = mapLT (union l) (go k' r)
    go k' t'@(Tip ky y)
      | k' > ky   = SplitLookup t'  Nothing  Nil
      | k' < ky   = SplitLookup Nil Nothing  t'
      | otherwise = SplitLookup Nil (Just y) Nil
    go _ Nil      = SplitLookup 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
    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
    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
    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
<