-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Assorted concrete container types
gr
grThis package contains efficient general-purpose implementations of
grvarious immutable container types including sets, maps, sequences,
grtrees, and graphs.
gr
grFor a walkthrough of what this package provides with examples of
grcommon operations see the <a>containers introduction</a>.
gr
grThe declared cost of each operation is either worst-case or amortized,
grbut remains valid even if structures are shared.
@package containers
@version 0.5.11.0


module Utils.Containers.Internal.BitUtil
bitcount :: Int -> Word -> Int

-- | Return a word where only the highest bit is set.
highestBitMask :: Word -> Word
shiftLL :: Word -> Int -> Word
shiftRL :: Word -> Int -> Word
wordSize :: Int


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grAn extremely light-weight, fast, and limited representation of a
grstring of up to (2*WORDSIZE - 2) bits. In fact, there are two
grrepresentations, misleadingly named bit queue builder and bit queue.
grThe builder supports only <a>emptyQB</a>, creating an empty builder,
grand <a>snocQB</a>, enqueueing a bit. The bit queue builder is then
grturned into a bit queue using <a>buildQ</a>, after which bits can be
grremoved one by one using <a>unconsQ</a>. If the size limit is
grexceeded, further operations will silently produce nonsense.
module Utils.Containers.Internal.BitQueue
data BitQueue
data BitQueueB

-- | Create an empty bit queue builder. This is represented as a single
grguard bit in the most significant position.
emptyQB :: BitQueueB

-- | Enqueue a bit. This works by shifting the queue right one bit, then
grsetting the most significant bit as requested.
snocQB :: BitQueueB -> Bool -> BitQueueB

-- | Convert a bit queue builder to a bit queue. This shifts in a new guard
grbit on the left, and shifts right until the old guard bit falls off.
buildQ :: BitQueueB -> BitQueue

-- | Dequeue an element, or discover the queue is empty.
unconsQ :: BitQueue -> Maybe (Bool, BitQueue)

-- | Convert a bit queue to a list of bits by unconsing. This is used to
grtest that the queue functions properly.
toListQ :: BitQueue -> [Bool]
instance GHC.Show.Show Utils.Containers.Internal.BitQueue.BitQueue
instance GHC.Show.Show Utils.Containers.Internal.BitQueue.BitQueueB


-- | A strict pair
module Utils.Containers.Internal.StrictPair

-- | The same as a regular Haskell pair, but
gr
gr<pre>
gr(x :*: _|_) = (_|_ :*: y) = _|_
gr</pre>
data StrictPair a b
(:*:) :: !a -> !b -> StrictPair a b

-- | Convert a strict pair to a standard pair.
toPair :: StrictPair a b -> (a, b)


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grAn efficient implementation of sets.
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.Set (Set)
grimport qualified Data.Set as Set
gr</pre>
gr
grThe implementation of <a>Set</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
gr
grNote that the implementation is <i>left-biased</i> -- the elements of
gra first argument are always preferred to the second, for example in
gr<a>union</a> or <a>insert</a>. Of course, left-biasing can only be
grobserved when equality is an equivalence relation instead of
grstructural equality.
gr
gr<i>Warning</i>: The size of the set must not exceed
gr<tt>maxBound::Int</tt>. Violation of this condition is not detected
grand if the size limit is exceeded, the behavior of the set is
grcompletely undefined.
module Data.Set.Internal

-- | A set of values <tt>a</tt>.
data Set a
Bin :: {-# UNPACK #-} !Size -> !a -> !(Set a) -> !(Set a) -> Set a
Tip :: Set a
type Size = Int

-- | <i>O(m*log(n/m+1)), m &lt;= n</i>. See <a>difference</a>.
(\\) :: Ord a => Set a -> Set a -> Set a
infixl 9 \\

-- | <i>O(1)</i>. Is this the empty set?
null :: Set a -> Bool

-- | <i>O(1)</i>. The number of elements in the set.
size :: Set a -> Int

-- | <i>O(log n)</i>. Is the element in the set?
member :: Ord a => a -> Set a -> Bool

-- | <i>O(log n)</i>. Is the element not in the set?
notMember :: Ord a => a -> Set a -> Bool

-- | <i>O(log n)</i>. Find largest element smaller than the given one.
gr
gr<pre>
grlookupLT 3 (fromList [3, 5]) == Nothing
grlookupLT 5 (fromList [3, 5]) == Just 3
gr</pre>
lookupLT :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find smallest element greater than the given one.
gr
gr<pre>
grlookupGT 4 (fromList [3, 5]) == Just 5
grlookupGT 5 (fromList [3, 5]) == Nothing
gr</pre>
lookupGT :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find largest element smaller or equal to the given
grone.
gr
gr<pre>
grlookupLE 2 (fromList [3, 5]) == Nothing
grlookupLE 4 (fromList [3, 5]) == Just 3
grlookupLE 5 (fromList [3, 5]) == Just 5
gr</pre>
lookupLE :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find smallest element greater or equal to the given
grone.
gr
gr<pre>
grlookupGE 3 (fromList [3, 5]) == Just 3
grlookupGE 4 (fromList [3, 5]) == Just 5
grlookupGE 6 (fromList [3, 5]) == Nothing
gr</pre>
lookupGE :: Ord a => a -> Set a -> Maybe a

-- | <i>O(n+m)</i>. Is this a subset? <tt>(s1 <a>isSubsetOf</a> s2)</tt>
grtells whether <tt>s1</tt> is a subset of <tt>s2</tt>.
isSubsetOf :: Ord a => Set a -> Set a -> Bool

-- | <i>O(n+m)</i>. Is this a proper subset? (ie. a subset but not equal).
isProperSubsetOf :: Ord a => Set a -> Set a -> Bool

-- | <i>O(n+m)</i>. Check whether two sets are disjoint (i.e. their
grintersection is empty).
gr
gr<pre>
grdisjoint (fromList [2,4,6])   (fromList [1,3])     == True
grdisjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
grdisjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
grdisjoint (fromList [])        (fromList [])        == True
gr</pre>
disjoint :: Ord a => Set a -> Set a -> Bool

-- | <i>O(1)</i>. The empty set.
empty :: Set a

-- | <i>O(1)</i>. Create a singleton set.
singleton :: a -> Set a

-- | <i>O(log n)</i>. Insert an element in a set. If the set already
grcontains an element equal to the given value, it is replaced with the
grnew value.
insert :: Ord a => a -> Set a -> Set a

-- | <i>O(log n)</i>. Delete an element from a set.
delete :: Ord a => a -> Set a -> Set a

-- | Calculate the power set of a set: the set of all its subsets.
gr
gr<pre>
grt <a>member</a> powerSet s == t <a>isSubsetOf</a> s
gr</pre>
gr
grExample:
gr
gr<pre>
grpowerSet (fromList [1,2,3]) =
gr  fromList [[], [1], [2], [3], [1,2], [1,3], [2,3], [1,2,3]]
gr</pre>
powerSet :: Set a -> Set (Set a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The union of two sets, preferring
grthe first set when equal elements are encountered.
union :: Ord a => Set a -> Set a -> Set a

-- | The union of a list of sets: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
unions :: Ord a => [Set a] -> Set a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two sets.
difference :: Ord a => Set a -> Set a -> Set a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The intersection of two sets.
grElements of the result come from the first set, so for example
gr
gr<pre>
grimport qualified Data.Set as S
grdata AB = A | B deriving Show
grinstance Ord AB where compare _ _ = EQ
grinstance Eq AB where _ == _ = True
grmain = print (S.singleton A `S.intersection` S.singleton B,
gr              S.singleton B `S.intersection` S.singleton A)
gr</pre>
gr
grprints <tt>(fromList [A],fromList [B])</tt>.
intersection :: Ord a => Set a -> Set a -> Set a

-- | Calculate the Cartesian product of two sets.
gr
gr<pre>
grcartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)
gr</pre>
gr
grExample:
gr
gr<pre>
grcartesianProduct (fromList [1,2]) (fromList [<tt>a</tt>,<tt>b</tt>]) =
gr  fromList [(1,<tt>a</tt>), (1,<tt>b</tt>), (2,<tt>a</tt>), (2,<tt>b</tt>)]
gr</pre>
cartesianProduct :: Set a -> Set b -> Set (a, b)

-- | Calculate the disjoin union of two sets.
gr
gr<pre>
grdisjointUnion xs ys = map Left xs <a>union</a> map Right ys
gr</pre>
gr
grExample:
gr
gr<pre>
grdisjointUnion (fromList [1,2]) (fromList ["hi", "bye"]) =
gr  fromList [Left 1, Left 2, Right "hi", Right "bye"]
gr</pre>
disjointUnion :: Set a -> Set b -> Set (Either a b)

-- | <i>O(n)</i>. Filter all elements that satisfy the predicate.
filter :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Take while a predicate on the elements holds. The
gruser is responsible for ensuring that for all elements <tt>j</tt> and
gr<tt>k</tt> in the set, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See
grnote at <a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> p . <a>toList</a>
grtakeWhileAntitone p = <a>filter</a> p
gr</pre>
takeWhileAntitone :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Drop while a predicate on the elements holds. The
gruser is responsible for ensuring that for all elements <tt>j</tt> and
gr<tt>k</tt> in the set, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See
grnote at <a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> p . <a>toList</a>
grdropWhileAntitone p = <a>filter</a> (not . p)
gr</pre>
dropWhileAntitone :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Divide a set at the point where a predicate on the
grelements stops holding. The user is responsible for ensuring that for
grall elements <tt>j</tt> and <tt>k</tt> in the set, <tt>j &lt; k ==&gt;
grp j &gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the set at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first element and to fail
grafter the last element).
spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a)

-- | <i>O(n)</i>. Partition the set into two sets, one with all elements
grthat satisfy the predicate and one with all elements that don't
grsatisfy the predicate. See also <a>split</a>.
partition :: (a -> Bool) -> Set a -> (Set a, Set a)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> x set</tt>) is a
grpair <tt>(set1,set2)</tt> where <tt>set1</tt> comprises the elements
grof <tt>set</tt> less than <tt>x</tt> and <tt>set2</tt> comprises the
grelements of <tt>set</tt> greater than <tt>x</tt>.
split :: Ord a => a -> Set a -> (Set a, Set a)

-- | <i>O(log n)</i>. Performs a <a>split</a> but also returns whether the
grpivot element was found in the original set.
splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a)

-- | <i>O(1)</i>. Decompose a set into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a set in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst subset less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList [1..6]) ==
gr  [fromList [1,2,3],fromList [4],fromList [5,6]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubsets, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Set a -> [Set a]

-- | <i>O(log n)</i>. Lookup the <i>index</i> of an element, which is its
grzero-based index in the sorted sequence of elements. The index is a
grnumber from <i>0</i> up to, but not including, the <a>size</a> of the
grset.
gr
gr<pre>
grisJust   (lookupIndex 2 (fromList [5,3])) == False
grfromJust (lookupIndex 3 (fromList [5,3])) == 0
grfromJust (lookupIndex 5 (fromList [5,3])) == 1
grisJust   (lookupIndex 6 (fromList [5,3])) == False
gr</pre>
lookupIndex :: Ord a => a -> Set a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of an element, which is its
grzero-based index in the sorted sequence of elements. The index is a
grnumber from <i>0</i> up to, but not including, the <a>size</a> of the
grset. Calls <a>error</a> when the element is not a <a>member</a> of the
grset.
gr
gr<pre>
grfindIndex 2 (fromList [5,3])    Error: element is not in the set
grfindIndex 3 (fromList [5,3]) == 0
grfindIndex 5 (fromList [5,3]) == 1
grfindIndex 6 (fromList [5,3])    Error: element is not in the set
gr</pre>
findIndex :: Ord a => a -> Set a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sorted sequence of elements. If the
gr<i>index</i> is out of range (less than zero, greater or equal to
gr<a>size</a> of the set), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [5,3]) == 3
grelemAt 1 (fromList [5,3]) == 5
grelemAt 2 (fromList [5,3])    Error: index out of range
gr</pre>
elemAt :: Int -> Set a -> a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sorted sequence of elements. If the
gr<i>index</i> is out of range (less than zero, greater or equal to
gr<a>size</a> of the set), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0    (fromList [5,3]) == singleton 5
grdeleteAt 1    (fromList [5,3]) == singleton 3
grdeleteAt 2    (fromList [5,3])    Error: index out of range
grdeleteAt (-1) (fromList [5,3])    Error: index out of range
gr</pre>
deleteAt :: Int -> Set a -> Set a

-- | Take a given number of elements in order, beginning with the smallest
grones.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Set a -> Set a

-- | Drop a given number of elements in order, beginning with the smallest
grones.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Set a -> Set a

-- | <i>O(log n)</i>. Split a set at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Set a -> (Set a, Set a)

-- | <i>O(n*log n)</i>. <tt><a>map</a> f s</tt> is the set obtained by
grapplying <tt>f</tt> to each element of <tt>s</tt>.
gr
grIt's worth noting that the size of the result may be smaller if, for
grsome <tt>(x,y)</tt>, <tt>x /= y &amp;&amp; f x == f y</tt>
map :: Ord b => (a -> b) -> Set a -> Set b

-- | <i>O(n)</i>. The
gr
gr<tt><a>mapMonotonic</a> f s == <a>map</a> f s</tt>, but works only
grwhen <tt>f</tt> is strictly increasing. <i>The precondition is not
grchecked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapMonotonic f s == map f s
gr    where ls = toList s
gr</pre>
mapMonotonic :: (a -> b) -> Set a -> Set b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoAscList set = foldr (:) [] set
gr</pre>
foldr :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoDescList set = foldl (flip (:)) [] set
gr</pre>
foldl :: (a -> b -> a) -> a -> Set b -> a

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Set b -> a

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.
gr
gr<i>Please note that fold will be deprecated in the future and
grremoved.</i>
fold :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(log n)</i>. The minimal element of a set.
lookupMin :: Set a -> Maybe a

-- | <i>O(log n)</i>. The maximal element of a set.
lookupMax :: Set a -> Maybe a

-- | <i>O(log n)</i>. The minimal element of a set.
findMin :: Set a -> a

-- | <i>O(log n)</i>. The maximal element of a set.
findMax :: Set a -> a

-- | <i>O(log n)</i>. Delete the minimal element. Returns an empty set if
grthe set is empty.
deleteMin :: Set a -> Set a

-- | <i>O(log n)</i>. Delete the maximal element. Returns an empty set if
grthe set is empty.
deleteMax :: Set a -> Set a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin set = (findMin set, deleteMin set)
gr</pre>
deleteFindMin :: Set a -> (a, Set a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax set = (findMax set, deleteMax set)
gr</pre>
deleteFindMax :: Set a -> (a, Set a)

-- | <i>O(log n)</i>. Retrieves the maximal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
maxView :: Set a -> Maybe (a, Set a)

-- | <i>O(log n)</i>. Retrieves the minimal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
minView :: Set a -> Maybe (a, Set a)

-- | <i>O(n)</i>. An alias of <a>toAscList</a>. The elements of a set in
grascending order. Subject to list fusion.
elems :: Set a -> [a]

-- | <i>O(n)</i>. Convert the set to a list of elements. Subject to list
grfusion.
toList :: Set a -> [a]

-- | <i>O(n*log n)</i>. Create a set from a list of elements.
gr
grIf the elements are ordered, a linear-time implementation is used,
grwith the performance equal to <a>fromDistinctAscList</a>.
fromList :: Ord a => [a] -> Set a

-- | <i>O(n)</i>. Convert the set to an ascending list of elements. Subject
grto list fusion.
toAscList :: Set a -> [a]

-- | <i>O(n)</i>. Convert the set to a descending list of elements. Subject
grto list fusion.
toDescList :: Set a -> [a]

-- | <i>O(n)</i>. Build a set from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
fromAscList :: Eq a => [a] -> Set a

-- | <i>O(n)</i>. Build a set from an ascending list of distinct elements
grin linear time. <i>The precondition (input list is strictly ascending)
gris not checked.</i>
fromDistinctAscList :: [a] -> Set a

-- | <i>O(n)</i>. Build a set from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
fromDescList :: Eq a => [a] -> Set a

-- | <i>O(n)</i>. Build a set from a descending list of distinct elements
grin linear time. <i>The precondition (input list is strictly
grdescending) is not checked.</i>
fromDistinctDescList :: [a] -> Set a

-- | <i>O(n)</i>. Show the tree that implements the set. The tree is shown
grin a compressed, hanging format.
showTree :: Show a => Set a -> String

-- | <i>O(n)</i>. The expression (<tt>showTreeWith hang wide map</tt>)
grshows the tree that implements the set. If <tt>hang</tt> is
gr<tt>True</tt>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grSet&gt; putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
gr4
gr+--2
gr|  +--1
gr|  +--3
gr+--5
gr
grSet&gt; putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
gr4
gr|
gr+--2
gr|  |
gr|  +--1
gr|  |
gr|  +--3
gr|
gr+--5
gr
grSet&gt; putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
gr+--5
gr|
gr4
gr|
gr|  +--3
gr|  |
gr+--2
gr   |
gr   +--1
gr</pre>
showTreeWith :: Show a => Bool -> Bool -> Set a -> String

-- | <i>O(n)</i>. Test if the internal set structure is valid.
valid :: Ord a => Set a -> Bool
bin :: a -> Set a -> Set a -> Set a
balanced :: Set a -> Bool
link :: a -> Set a -> Set a -> Set a
merge :: Set a -> Set a -> Set a
instance GHC.Base.Semigroup (Data.Set.Internal.MergeSet a)
instance GHC.Base.Monoid (Data.Set.Internal.MergeSet a)
instance GHC.Classes.Ord a => GHC.Base.Monoid (Data.Set.Internal.Set a)
instance GHC.Classes.Ord a => GHC.Base.Semigroup (Data.Set.Internal.Set a)
instance Data.Foldable.Foldable Data.Set.Internal.Set
instance (Data.Data.Data a, GHC.Classes.Ord a) => Data.Data.Data (Data.Set.Internal.Set a)
instance GHC.Classes.Ord a => GHC.Exts.IsList (Data.Set.Internal.Set a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Set.Internal.Set a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Set.Internal.Set a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Set.Internal.Set a)
instance Data.Functor.Classes.Eq1 Data.Set.Internal.Set
instance Data.Functor.Classes.Ord1 Data.Set.Internal.Set
instance Data.Functor.Classes.Show1 Data.Set.Internal.Set
instance (GHC.Read.Read a, GHC.Classes.Ord a) => GHC.Read.Read (Data.Set.Internal.Set a)
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Set.Internal.Set a)


-- | <h1>Finite Sets</h1>
gr
grThe <tt><a>Set</a> e</tt> type represents a set of elements of type
gr<tt>e</tt>. Most operations require that <tt>e</tt> be an instance of
grthe <a>Ord</a> class. A <a>Set</a> is strict in its elements.
gr
grFor a walkthrough of the most commonly used functions see the <a>sets
grintroduction</a>.
gr
grNote that the implementation is generally <i>left-biased</i>.
grFunctions that take two sets as arguments and combine them, such as
gr<a>union</a> and <a>intersection</a>, prefer the entries in the first
grargument to those in the second. Of course, this bias can only be
grobserved when equality is an equivalence relation instead of
grstructural equality.
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.Set (Set)
grimport qualified Data.Set as Set
gr</pre>
gr
gr<h2>Warning</h2>
gr
grThe size of the set must not exceed <tt>maxBound::Int</tt>. Violation
grof this condition is not detected and if the size limit is exceeded,
grits behaviour is undefined.
gr
gr<h2>Implementation</h2>
gr
grThe implementation of <a>Set</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
module Data.Set

-- | A set of values <tt>a</tt>.
data Set a

-- | <i>O(m*log(n/m+1)), m &lt;= n</i>. See <a>difference</a>.
(\\) :: Ord a => Set a -> Set a -> Set a
infixl 9 \\

-- | <i>O(1)</i>. Is this the empty set?
null :: Set a -> Bool

-- | <i>O(1)</i>. The number of elements in the set.
size :: Set a -> Int

-- | <i>O(log n)</i>. Is the element in the set?
member :: Ord a => a -> Set a -> Bool

-- | <i>O(log n)</i>. Is the element not in the set?
notMember :: Ord a => a -> Set a -> Bool

-- | <i>O(log n)</i>. Find largest element smaller than the given one.
gr
gr<pre>
grlookupLT 3 (fromList [3, 5]) == Nothing
grlookupLT 5 (fromList [3, 5]) == Just 3
gr</pre>
lookupLT :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find smallest element greater than the given one.
gr
gr<pre>
grlookupGT 4 (fromList [3, 5]) == Just 5
grlookupGT 5 (fromList [3, 5]) == Nothing
gr</pre>
lookupGT :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find largest element smaller or equal to the given
grone.
gr
gr<pre>
grlookupLE 2 (fromList [3, 5]) == Nothing
grlookupLE 4 (fromList [3, 5]) == Just 3
grlookupLE 5 (fromList [3, 5]) == Just 5
gr</pre>
lookupLE :: Ord a => a -> Set a -> Maybe a

-- | <i>O(log n)</i>. Find smallest element greater or equal to the given
grone.
gr
gr<pre>
grlookupGE 3 (fromList [3, 5]) == Just 3
grlookupGE 4 (fromList [3, 5]) == Just 5
grlookupGE 6 (fromList [3, 5]) == Nothing
gr</pre>
lookupGE :: Ord a => a -> Set a -> Maybe a

-- | <i>O(n+m)</i>. Is this a subset? <tt>(s1 <a>isSubsetOf</a> s2)</tt>
grtells whether <tt>s1</tt> is a subset of <tt>s2</tt>.
isSubsetOf :: Ord a => Set a -> Set a -> Bool

-- | <i>O(n+m)</i>. Is this a proper subset? (ie. a subset but not equal).
isProperSubsetOf :: Ord a => Set a -> Set a -> Bool

-- | <i>O(n+m)</i>. Check whether two sets are disjoint (i.e. their
grintersection is empty).
gr
gr<pre>
grdisjoint (fromList [2,4,6])   (fromList [1,3])     == True
grdisjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
grdisjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
grdisjoint (fromList [])        (fromList [])        == True
gr</pre>
disjoint :: Ord a => Set a -> Set a -> Bool

-- | <i>O(1)</i>. The empty set.
empty :: Set a

-- | <i>O(1)</i>. Create a singleton set.
singleton :: a -> Set a

-- | <i>O(log n)</i>. Insert an element in a set. If the set already
grcontains an element equal to the given value, it is replaced with the
grnew value.
insert :: Ord a => a -> Set a -> Set a

-- | <i>O(log n)</i>. Delete an element from a set.
delete :: Ord a => a -> Set a -> Set a

-- | Calculate the power set of a set: the set of all its subsets.
gr
gr<pre>
grt <a>member</a> powerSet s == t <a>isSubsetOf</a> s
gr</pre>
gr
grExample:
gr
gr<pre>
grpowerSet (fromList [1,2,3]) =
gr  fromList [[], [1], [2], [3], [1,2], [1,3], [2,3], [1,2,3]]
gr</pre>
powerSet :: Set a -> Set (Set a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The union of two sets, preferring
grthe first set when equal elements are encountered.
union :: Ord a => Set a -> Set a -> Set a

-- | The union of a list of sets: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
unions :: Ord a => [Set a] -> Set a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two sets.
difference :: Ord a => Set a -> Set a -> Set a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The intersection of two sets.
grElements of the result come from the first set, so for example
gr
gr<pre>
grimport qualified Data.Set as S
grdata AB = A | B deriving Show
grinstance Ord AB where compare _ _ = EQ
grinstance Eq AB where _ == _ = True
grmain = print (S.singleton A `S.intersection` S.singleton B,
gr              S.singleton B `S.intersection` S.singleton A)
gr</pre>
gr
grprints <tt>(fromList [A],fromList [B])</tt>.
intersection :: Ord a => Set a -> Set a -> Set a

-- | Calculate the Cartesian product of two sets.
gr
gr<pre>
grcartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys)
gr</pre>
gr
grExample:
gr
gr<pre>
grcartesianProduct (fromList [1,2]) (fromList [<tt>a</tt>,<tt>b</tt>]) =
gr  fromList [(1,<tt>a</tt>), (1,<tt>b</tt>), (2,<tt>a</tt>), (2,<tt>b</tt>)]
gr</pre>
cartesianProduct :: Set a -> Set b -> Set (a, b)

-- | Calculate the disjoin union of two sets.
gr
gr<pre>
grdisjointUnion xs ys = map Left xs <a>union</a> map Right ys
gr</pre>
gr
grExample:
gr
gr<pre>
grdisjointUnion (fromList [1,2]) (fromList ["hi", "bye"]) =
gr  fromList [Left 1, Left 2, Right "hi", Right "bye"]
gr</pre>
disjointUnion :: Set a -> Set b -> Set (Either a b)

-- | <i>O(n)</i>. Filter all elements that satisfy the predicate.
filter :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Take while a predicate on the elements holds. The
gruser is responsible for ensuring that for all elements <tt>j</tt> and
gr<tt>k</tt> in the set, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See
grnote at <a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> p . <a>toList</a>
grtakeWhileAntitone p = <a>filter</a> p
gr</pre>
takeWhileAntitone :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Drop while a predicate on the elements holds. The
gruser is responsible for ensuring that for all elements <tt>j</tt> and
gr<tt>k</tt> in the set, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See
grnote at <a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> p . <a>toList</a>
grdropWhileAntitone p = <a>filter</a> (not . p)
gr</pre>
dropWhileAntitone :: (a -> Bool) -> Set a -> Set a

-- | <i>O(log n)</i>. Divide a set at the point where a predicate on the
grelements stops holding. The user is responsible for ensuring that for
grall elements <tt>j</tt> and <tt>k</tt> in the set, <tt>j &lt; k ==&gt;
grp j &gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the set at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first element and to fail
grafter the last element).
spanAntitone :: (a -> Bool) -> Set a -> (Set a, Set a)

-- | <i>O(n)</i>. Partition the set into two sets, one with all elements
grthat satisfy the predicate and one with all elements that don't
grsatisfy the predicate. See also <a>split</a>.
partition :: (a -> Bool) -> Set a -> (Set a, Set a)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> x set</tt>) is a
grpair <tt>(set1,set2)</tt> where <tt>set1</tt> comprises the elements
grof <tt>set</tt> less than <tt>x</tt> and <tt>set2</tt> comprises the
grelements of <tt>set</tt> greater than <tt>x</tt>.
split :: Ord a => a -> Set a -> (Set a, Set a)

-- | <i>O(log n)</i>. Performs a <a>split</a> but also returns whether the
grpivot element was found in the original set.
splitMember :: Ord a => a -> Set a -> (Set a, Bool, Set a)

-- | <i>O(1)</i>. Decompose a set into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a set in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst subset less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList [1..6]) ==
gr  [fromList [1,2,3],fromList [4],fromList [5,6]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubsets, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Set a -> [Set a]

-- | <i>O(log n)</i>. Lookup the <i>index</i> of an element, which is its
grzero-based index in the sorted sequence of elements. The index is a
grnumber from <i>0</i> up to, but not including, the <a>size</a> of the
grset.
gr
gr<pre>
grisJust   (lookupIndex 2 (fromList [5,3])) == False
grfromJust (lookupIndex 3 (fromList [5,3])) == 0
grfromJust (lookupIndex 5 (fromList [5,3])) == 1
grisJust   (lookupIndex 6 (fromList [5,3])) == False
gr</pre>
lookupIndex :: Ord a => a -> Set a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of an element, which is its
grzero-based index in the sorted sequence of elements. The index is a
grnumber from <i>0</i> up to, but not including, the <a>size</a> of the
grset. Calls <a>error</a> when the element is not a <a>member</a> of the
grset.
gr
gr<pre>
grfindIndex 2 (fromList [5,3])    Error: element is not in the set
grfindIndex 3 (fromList [5,3]) == 0
grfindIndex 5 (fromList [5,3]) == 1
grfindIndex 6 (fromList [5,3])    Error: element is not in the set
gr</pre>
findIndex :: Ord a => a -> Set a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sorted sequence of elements. If the
gr<i>index</i> is out of range (less than zero, greater or equal to
gr<a>size</a> of the set), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [5,3]) == 3
grelemAt 1 (fromList [5,3]) == 5
grelemAt 2 (fromList [5,3])    Error: index out of range
gr</pre>
elemAt :: Int -> Set a -> a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sorted sequence of elements. If the
gr<i>index</i> is out of range (less than zero, greater or equal to
gr<a>size</a> of the set), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0    (fromList [5,3]) == singleton 5
grdeleteAt 1    (fromList [5,3]) == singleton 3
grdeleteAt 2    (fromList [5,3])    Error: index out of range
grdeleteAt (-1) (fromList [5,3])    Error: index out of range
gr</pre>
deleteAt :: Int -> Set a -> Set a

-- | Take a given number of elements in order, beginning with the smallest
grones.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Set a -> Set a

-- | Drop a given number of elements in order, beginning with the smallest
grones.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Set a -> Set a

-- | <i>O(log n)</i>. Split a set at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Set a -> (Set a, Set a)

-- | <i>O(n*log n)</i>. <tt><a>map</a> f s</tt> is the set obtained by
grapplying <tt>f</tt> to each element of <tt>s</tt>.
gr
grIt's worth noting that the size of the result may be smaller if, for
grsome <tt>(x,y)</tt>, <tt>x /= y &amp;&amp; f x == f y</tt>
map :: Ord b => (a -> b) -> Set a -> Set b

-- | <i>O(n)</i>. The
gr
gr<tt><a>mapMonotonic</a> f s == <a>map</a> f s</tt>, but works only
grwhen <tt>f</tt> is strictly increasing. <i>The precondition is not
grchecked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapMonotonic f s == map f s
gr    where ls = toList s
gr</pre>
mapMonotonic :: (a -> b) -> Set a -> Set b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoAscList set = foldr (:) [] set
gr</pre>
foldr :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoDescList set = foldl (flip (:)) [] set
gr</pre>
foldl :: (a -> b -> a) -> a -> Set b -> a

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Set b -> a

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.
gr
gr<i>Please note that fold will be deprecated in the future and
grremoved.</i>
fold :: (a -> b -> b) -> b -> Set a -> b

-- | <i>O(log n)</i>. The minimal element of a set.
lookupMin :: Set a -> Maybe a

-- | <i>O(log n)</i>. The maximal element of a set.
lookupMax :: Set a -> Maybe a

-- | <i>O(log n)</i>. The minimal element of a set.
findMin :: Set a -> a

-- | <i>O(log n)</i>. The maximal element of a set.
findMax :: Set a -> a

-- | <i>O(log n)</i>. Delete the minimal element. Returns an empty set if
grthe set is empty.
deleteMin :: Set a -> Set a

-- | <i>O(log n)</i>. Delete the maximal element. Returns an empty set if
grthe set is empty.
deleteMax :: Set a -> Set a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin set = (findMin set, deleteMin set)
gr</pre>
deleteFindMin :: Set a -> (a, Set a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax set = (findMax set, deleteMax set)
gr</pre>
deleteFindMax :: Set a -> (a, Set a)

-- | <i>O(log n)</i>. Retrieves the maximal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
maxView :: Set a -> Maybe (a, Set a)

-- | <i>O(log n)</i>. Retrieves the minimal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
minView :: Set a -> Maybe (a, Set a)

-- | <i>O(n)</i>. An alias of <a>toAscList</a>. The elements of a set in
grascending order. Subject to list fusion.
elems :: Set a -> [a]

-- | <i>O(n)</i>. Convert the set to a list of elements. Subject to list
grfusion.
toList :: Set a -> [a]

-- | <i>O(n*log n)</i>. Create a set from a list of elements.
gr
grIf the elements are ordered, a linear-time implementation is used,
grwith the performance equal to <a>fromDistinctAscList</a>.
fromList :: Ord a => [a] -> Set a

-- | <i>O(n)</i>. Convert the set to an ascending list of elements. Subject
grto list fusion.
toAscList :: Set a -> [a]

-- | <i>O(n)</i>. Convert the set to a descending list of elements. Subject
grto list fusion.
toDescList :: Set a -> [a]

-- | <i>O(n)</i>. Build a set from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
fromAscList :: Eq a => [a] -> Set a

-- | <i>O(n)</i>. Build a set from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
fromDescList :: Eq a => [a] -> Set a

-- | <i>O(n)</i>. Build a set from an ascending list of distinct elements
grin linear time. <i>The precondition (input list is strictly ascending)
gris not checked.</i>
fromDistinctAscList :: [a] -> Set a

-- | <i>O(n)</i>. Build a set from a descending list of distinct elements
grin linear time. <i>The precondition (input list is strictly
grdescending) is not checked.</i>
fromDistinctDescList :: [a] -> Set a

-- | <i>O(n)</i>. Show the tree that implements the set. The tree is shown
grin a compressed, hanging format.
showTree :: Show a => Set a -> String

-- | <i>O(n)</i>. The expression (<tt>showTreeWith hang wide map</tt>)
grshows the tree that implements the set. If <tt>hang</tt> is
gr<tt>True</tt>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grSet&gt; putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
gr4
gr+--2
gr|  +--1
gr|  +--3
gr+--5
gr
grSet&gt; putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
gr4
gr|
gr+--2
gr|  |
gr|  +--1
gr|  |
gr|  +--3
gr|
gr+--5
gr
grSet&gt; putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
gr+--5
gr|
gr4
gr|
gr|  +--3
gr|  |
gr+--2
gr   |
gr   +--1
gr</pre>
showTreeWith :: Show a => Bool -> Bool -> Set a -> String

-- | <i>O(n)</i>. Test if the internal set structure is valid.
valid :: Ord a => Set a -> Bool


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grGeneral purpose finite sequences. Apart from being finite and having
grstrict operations, sequences also differ from lists in supporting a
grwider variety of operations efficiently.
gr
grAn amortized running time is given for each operation, with
gr&lt;math&gt; referring to the length of the sequence and &lt;math&gt;
grbeing the integral index used by some operations. These bounds hold
greven in a persistent (shared) setting.
gr
grThe implementation uses 2-3 finger trees annotated with sizes, as
grdescribed in section 4.2 of
gr
gr<ul>
gr<li>Ralf Hinze and Ross Paterson, "Finger trees: a simple
grgeneral-purpose data structure", <i>Journal of Functional
grProgramming</i> 16:2 (2006) pp 197-217.
gr<a>http://staff.city.ac.uk/~ross/papers/FingerTree.html</a></li>
gr</ul>
gr
gr<i>Note</i>: Many of these operations have the same names as similar
groperations on lists in the <a>Prelude</a>. The ambiguity may be
grresolved using either qualification or the <tt>hiding</tt> clause.
gr
gr<i>Warning</i>: The size of a <a>Seq</a> must not exceed
gr<tt>maxBound::Int</tt>. Violation of this condition is not detected
grand if the size limit is exceeded, the behaviour of the sequence is
grundefined. This is unlikely to occur in most applications, but some
grcare may be required when using <a>&gt;&lt;</a>, <a>&lt;*&gt;</a>,
gr<a>*&gt;</a>, or <a>&gt;&gt;</a>, particularly repeatedly and
grparticularly in combination with <a>replicate</a> or
gr<a>fromFunction</a>.
module Data.Sequence.Internal
newtype Elem a
Elem :: a -> Elem a
[getElem] :: Elem a -> a
data FingerTree a
EmptyT :: FingerTree a
Single :: a -> FingerTree a
Deep :: {-# UNPACK #-} !Int -> !(Digit a) -> (FingerTree (Node a)) -> !(Digit a) -> FingerTree a
data Node a
Node2 :: {-# UNPACK #-} !Int -> a -> a -> Node a
Node3 :: {-# UNPACK #-} !Int -> a -> a -> a -> Node a
data Digit a
One :: a -> Digit a
Two :: a -> a -> Digit a
Three :: a -> a -> a -> Digit a
Four :: a -> a -> a -> a -> Digit a
class Sized a
size :: Sized a => a -> Int
class MaybeForce a

-- | General-purpose finite sequences.
newtype Seq a
Seq :: (FingerTree (Elem a)) -> Seq a
newtype State s a
State :: s -> (s, a) -> State s a
[runState] :: State s a -> s -> (s, a)
execState :: State s a -> s -> a
foldDigit :: (b -> b -> b) -> (a -> b) -> Digit a -> b
foldNode :: (b -> b -> b) -> (a -> b) -> Node a -> b
foldWithIndexDigit :: Sized a => (b -> b -> b) -> (Int -> a -> b) -> Int -> Digit a -> b
foldWithIndexNode :: Sized a => (m -> m -> m) -> (Int -> a -> m) -> Int -> Node a -> m

-- | &lt;math&gt;. The empty sequence.
empty :: Seq a

-- | &lt;math&gt;. A singleton sequence.
singleton :: a -> Seq a

-- | &lt;math&gt;. Add an element to the left end of a sequence. Mnemonic:
gra triangle with the single element at the pointy end.
(<|) :: a -> Seq a -> Seq a
infixr 5 <|

-- | &lt;math&gt;. Add an element to the right end of a sequence. Mnemonic:
gra triangle with the single element at the pointy end.
(|>) :: Seq a -> a -> Seq a
infixl 5 |>

-- | &lt;math&gt;. Concatenate two sequences.
(><) :: Seq a -> Seq a -> Seq a
infixr 5 ><

-- | &lt;math&gt;. Create a sequence from a finite list of elements. There
gris a function <a>toList</a> in the opposite direction for all
grinstances of the <a>Foldable</a> class, including <a>Seq</a>.
fromList :: [a] -> Seq a

-- | &lt;math&gt;. Convert a given sequence length and a function
grrepresenting that sequence into a sequence.
fromFunction :: Int -> (Int -> a) -> Seq a

-- | &lt;math&gt;. Create a sequence consisting of the elements of an
gr<a>Array</a>. Note that the resulting sequence elements may be
grevaluated lazily (as on GHC), so you must force the entire structure
grto be sure that the original array can be garbage-collected.
fromArray :: Ix i => Array i a -> Seq a

-- | &lt;math&gt;. <tt>replicate n x</tt> is a sequence consisting of
gr<tt>n</tt> copies of <tt>x</tt>.
replicate :: Int -> a -> Seq a

-- | <a>replicateA</a> is an <a>Applicative</a> version of
gr<a>replicate</a>, and makes &lt;math&gt; calls to <a>liftA2</a> and
gr<a>pure</a>.
gr
gr<pre>
grreplicateA n x = sequenceA (replicate n x)
gr</pre>
replicateA :: Applicative f => Int -> f a -> f (Seq a)

-- | <a>replicateM</a> is a sequence counterpart of <a>replicateM</a>.
gr
gr<pre>
grreplicateM n x = sequence (replicate n x)
gr</pre>
gr
grFor <tt>base &gt;= 4.8.0</tt> and <tt>containers &gt;= 0.5.11</tt>,
gr<a>replicateM</a> is a synonym for <a>replicateA</a>.
replicateM :: Applicative m => Int -> m a -> m (Seq a)

-- | <i>O(</i>log<i> k)</i>. <tt><a>cycleTaking</a> k xs</tt> forms a
grsequence of length <tt>k</tt> by repeatedly concatenating <tt>xs</tt>
grwith itself. <tt>xs</tt> may only be empty if <tt>k</tt> is 0.
gr
gr<pre>
grcycleTaking k = fromList . take k . cycle . toList
gr</pre>
cycleTaking :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Constructs a sequence by repeated application of a
grfunction to a seed value.
gr
gr<pre>
griterateN n f x = fromList (Prelude.take n (Prelude.iterate f x))
gr</pre>
iterateN :: Int -> (a -> a) -> a -> Seq a

-- | Builds a sequence from a seed value. Takes time linear in the number
grof generated elements. <i>WARNING:</i> If the number of generated
grelements is infinite, this method will not terminate.
unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a

-- | <tt><a>unfoldl</a> f x</tt> is equivalent to <tt><a>reverse</a>
gr(<a>unfoldr</a> (<a>fmap</a> swap . f) x)</tt>.
unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a

-- | &lt;math&gt;. Is this the empty sequence?
null :: Seq a -> Bool

-- | &lt;math&gt;. The number of elements in the sequence.
length :: Seq a -> Int

-- | View of the left end of a sequence.
data ViewL a

-- | empty sequence
EmptyL :: ViewL a

-- | leftmost element and the rest of the sequence
(:<) :: a -> Seq a -> ViewL a

-- | &lt;math&gt;. Analyse the left end of a sequence.
viewl :: Seq a -> ViewL a

-- | View of the right end of a sequence.
data ViewR a

-- | empty sequence
EmptyR :: ViewR a

-- | the sequence minus the rightmost element, and the rightmost element
(:>) :: Seq a -> a -> ViewR a

-- | &lt;math&gt;. Analyse the right end of a sequence.
viewr :: Seq a -> ViewR a

-- | <a>scanl</a> is similar to <a>foldl</a>, but returns a sequence of
grreduced values from the left:
gr
gr<pre>
grscanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...]
gr</pre>
scanl :: (a -> b -> a) -> a -> Seq b -> Seq a

-- | <a>scanl1</a> is a variant of <a>scanl</a> that has no starting value
grargument:
gr
gr<pre>
grscanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]
gr</pre>
scanl1 :: (a -> a -> a) -> Seq a -> Seq a

-- | <a>scanr</a> is the right-to-left dual of <a>scanl</a>.
scanr :: (a -> b -> b) -> b -> Seq a -> Seq b

-- | <a>scanr1</a> is a variant of <a>scanr</a> that has no starting value
grargument.
scanr1 :: (a -> a -> a) -> Seq a -> Seq a

-- | &lt;math&gt;. Returns a sequence of all suffixes of this sequence,
grlongest first. For example,
gr
gr<pre>
grtails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]
gr</pre>
gr
grEvaluating the &lt;math&gt;th suffix takes &lt;math&gt;, but
grevaluating every suffix in the sequence takes &lt;math&gt; due to
grsharing.
tails :: Seq a -> Seq (Seq a)

-- | &lt;math&gt;. Returns a sequence of all prefixes of this sequence,
grshortest first. For example,
gr
gr<pre>
grinits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]
gr</pre>
gr
grEvaluating the &lt;math&gt;th prefix takes &lt;math&gt;, but
grevaluating every prefix in the sequence takes &lt;math&gt; due to
grsharing.
inits :: Seq a -> Seq (Seq a)

-- | &lt;math&gt;. <tt>chunksOf c xs</tt> splits <tt>xs</tt> into chunks of
grsize <tt>c&gt;0</tt>. If <tt>c</tt> does not divide the length of
gr<tt>xs</tt> evenly, then the last element of the result will be short.
gr
grSide note: the given performance bound is missing some messy terms
grthat only really affect edge cases. Performance degrades smoothly from
gr&lt;math&gt; (for &lt;math&gt;) to &lt;math&gt; (for &lt;math&gt;).
grThe true bound is more like &lt;math&gt;
chunksOf :: Int -> Seq a -> Seq (Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the prefix length.
gr<a>takeWhileL</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns the longest prefix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt>.
takeWhileL :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the suffix length.
gr<a>takeWhileR</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns the longest suffix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt>.
gr
gr<tt><a>takeWhileR</a> p xs</tt> is equivalent to <tt><a>reverse</a>
gr(<a>takeWhileL</a> p (<a>reverse</a> xs))</tt>.
takeWhileR :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the prefix length.
gr<tt><a>dropWhileL</a> p xs</tt> returns the suffix remaining after
gr<tt><a>takeWhileL</a> p xs</tt>.
dropWhileL :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the suffix length.
gr<tt><a>dropWhileR</a> p xs</tt> returns the prefix remaining after
gr<tt><a>takeWhileR</a> p xs</tt>.
gr
gr<tt><a>dropWhileR</a> p xs</tt> is equivalent to <tt><a>reverse</a>
gr(<a>dropWhileL</a> p (<a>reverse</a> xs))</tt>.
dropWhileR :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the prefix length. <a>spanl</a>,
grapplied to a predicate <tt>p</tt> and a sequence <tt>xs</tt>, returns
gra pair whose first element is the longest prefix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt> and the second element
gris the remainder of the sequence.
spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the suffix length. <a>spanr</a>,
grapplied to a predicate <tt>p</tt> and a sequence <tt>xs</tt>, returns
gra pair whose <i>first</i> element is the longest <i>suffix</i>
gr(possibly empty) of <tt>xs</tt> of elements that satisfy <tt>p</tt>
grand the second element is the remainder of the sequence.
spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the breakpoint index.
gr<a>breakl</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns a pair whose first element is the longest prefix
gr(possibly empty) of <tt>xs</tt> of elements that <i>do not satisfy</i>
gr<tt>p</tt> and the second element is the remainder of the sequence.
gr
gr<tt><a>breakl</a> p</tt> is equivalent to <tt><a>spanl</a> (not .
grp)</tt>.
breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | <tt><a>breakr</a> p</tt> is equivalent to <tt><a>spanr</a> (not .
grp)</tt>.
breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt;. The <a>partition</a> function takes a predicate
gr<tt>p</tt> and a sequence <tt>xs</tt> and returns sequences of those
grelements which do and do not satisfy the predicate.
partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt;. The <a>filter</a> function takes a predicate <tt>p</tt>
grand a sequence <tt>xs</tt> and returns a sequence of those elements
grwhich satisfy the predicate.
filter :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt;. The element at the specified position, counting from 0.
grIf the specified position is negative or at least the length of the
grsequence, <a>lookup</a> returns <a>Nothing</a>.
gr
gr<pre>
gr0 &lt;= i &lt; length xs ==&gt; lookup i xs == Just (toList xs !! i)
gr</pre>
gr
gr<pre>
gri &lt; 0 || i &gt;= length xs ==&gt; lookup i xs = Nothing
gr</pre>
gr
grUnlike <a>index</a>, this can be used to retrieve an element without
grforcing it. For example, to insert the fifth element of a sequence
gr<tt>xs</tt> into a <a>Map</a> <tt>m</tt> at key <tt>k</tt>, you could
gruse
gr
gr<pre>
grcase lookup 5 xs of
gr  Nothing -&gt; m
gr  Just x -&gt; <a>insert</a> k x m
gr</pre>
lookup :: Int -> Seq a -> Maybe a

-- | &lt;math&gt;. A flipped, infix version of <a>lookup</a>.
(!?) :: Seq a -> Int -> Maybe a

-- | &lt;math&gt;. The element at the specified position, counting from 0.
grThe argument should thus be a non-negative integer less than the size
grof the sequence. If the position is out of range, <a>index</a> fails
grwith an error.
gr
gr<pre>
grxs `index` i = toList xs !! i
gr</pre>
gr
grCaution: <a>index</a> necessarily delays retrieving the requested
grelement until the result is forced. It can therefore lead to a space
grleak if the result is stored, unforced, in another structure. To
grretrieve an element immediately without forcing it, use <a>lookup</a>
gror '(!?)'.
index :: Seq a -> Int -> a

-- | &lt;math&gt;. Update the element at the specified position. If the
grposition is out of range, the original sequence is returned.
gr<a>adjust</a> can lead to poor performance and even memory leaks,
grbecause it does not force the new value before installing it in the
grsequence. <a>adjust'</a> should usually be preferred.
adjust :: (a -> a) -> Int -> Seq a -> Seq a

-- | &lt;math&gt;. Update the element at the specified position. If the
grposition is out of range, the original sequence is returned. The new
grvalue is forced before it is installed in the sequence.
gr
gr<pre>
gradjust' f i xs =
gr case xs !? i of
gr   Nothing -&gt; xs
gr   Just x -&gt; let !x' = f x
gr             in update i x' xs
gr</pre>
adjust' :: forall a. (a -> a) -> Int -> Seq a -> Seq a

-- | &lt;math&gt;. Replace the element at the specified position. If the
grposition is out of range, the original sequence is returned.
update :: Int -> a -> Seq a -> Seq a

-- | &lt;math&gt;. The first <tt>i</tt> elements of a sequence. If
gr<tt>i</tt> is negative, <tt><a>take</a> i s</tt> yields the empty
grsequence. If the sequence contains fewer than <tt>i</tt> elements, the
grwhole sequence is returned.
take :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Elements of a sequence after the first <tt>i</tt>. If
gr<tt>i</tt> is negative, <tt><a>drop</a> i s</tt> yields the whole
grsequence. If the sequence contains fewer than <tt>i</tt> elements, the
grempty sequence is returned.
drop :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. <tt><a>insertAt</a> i x xs</tt> inserts <tt>x</tt> into
gr<tt>xs</tt> at the index <tt>i</tt>, shifting the rest of the sequence
grover.
gr
gr<pre>
grinsertAt 2 x (fromList [a,b,c,d]) = fromList [a,b,x,c,d]
grinsertAt 4 x (fromList [a,b,c,d]) = insertAt 10 x (fromList [a,b,c,d])
gr                                  = fromList [a,b,c,d,x]
gr</pre>
gr
gr<pre>
grinsertAt i x xs = take i xs &gt;&lt; singleton x &gt;&lt; drop i xs
gr</pre>
insertAt :: Int -> a -> Seq a -> Seq a

-- | &lt;math&gt;. Delete the element of a sequence at a given index.
grReturn the original sequence if the index is out of range.
gr
gr<pre>
grdeleteAt 2 [a,b,c,d] = [a,b,d]
grdeleteAt 4 [a,b,c,d] = deleteAt (-1) [a,b,c,d] = [a,b,c,d]
gr</pre>
deleteAt :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Split a sequence at a given position. <tt><a>splitAt</a>
gri s = (<a>take</a> i s, <a>drop</a> i s)</tt>.
splitAt :: Int -> Seq a -> (Seq a, Seq a)

-- | <a>elemIndexL</a> finds the leftmost index of the specified element,
grif it is present, and otherwise <a>Nothing</a>.
elemIndexL :: Eq a => a -> Seq a -> Maybe Int

-- | <a>elemIndicesL</a> finds the indices of the specified element, from
grleft to right (i.e. in ascending order).
elemIndicesL :: Eq a => a -> Seq a -> [Int]

-- | <a>elemIndexR</a> finds the rightmost index of the specified element,
grif it is present, and otherwise <a>Nothing</a>.
elemIndexR :: Eq a => a -> Seq a -> Maybe Int

-- | <a>elemIndicesR</a> finds the indices of the specified element, from
grright to left (i.e. in descending order).
elemIndicesR :: Eq a => a -> Seq a -> [Int]

-- | <tt><a>findIndexL</a> p xs</tt> finds the index of the leftmost
grelement that satisfies <tt>p</tt>, if any exist.
findIndexL :: (a -> Bool) -> Seq a -> Maybe Int

-- | <tt><a>findIndicesL</a> p</tt> finds all indices of elements that
grsatisfy <tt>p</tt>, in ascending order.
findIndicesL :: (a -> Bool) -> Seq a -> [Int]

-- | <tt><a>findIndexR</a> p xs</tt> finds the index of the rightmost
grelement that satisfies <tt>p</tt>, if any exist.
findIndexR :: (a -> Bool) -> Seq a -> Maybe Int

-- | <tt><a>findIndicesR</a> p</tt> finds all indices of elements that
grsatisfy <tt>p</tt>, in descending order.
findIndicesR :: (a -> Bool) -> Seq a -> [Int]
foldMapWithIndex :: Monoid m => (Int -> a -> m) -> Seq a -> m

-- | <a>foldlWithIndex</a> is a version of <a>foldl</a> that also provides
graccess to the index of each element.
foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b

-- | <a>foldrWithIndex</a> is a version of <a>foldr</a> that also provides
graccess to the index of each element.
foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b

-- | A generalization of <a>fmap</a>, <a>mapWithIndex</a> takes a mapping
grfunction that also depends on the element's index, and applies it to
grevery element in the sequence.
mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b

-- | <a>traverseWithIndex</a> is a version of <a>traverse</a> that also
groffers access to the index of each element.
traverseWithIndex :: Applicative f => (Int -> a -> f b) -> Seq a -> f (Seq b)

-- | &lt;math&gt;. The reverse of a sequence.
reverse :: Seq a -> Seq a

-- | &lt;math&gt;. Intersperse an element between the elements of a
grsequence.
gr
gr<pre>
grintersperse a empty = empty
grintersperse a (singleton x) = singleton x
grintersperse a (fromList [x,y]) = fromList [x,a,y]
grintersperse a (fromList [x,y,z]) = fromList [x,a,y,a,z]
gr</pre>
intersperse :: a -> Seq a -> Seq a
liftA2Seq :: (a -> b -> c) -> Seq a -> Seq b -> Seq c

-- | &lt;math&gt;. <a>zip</a> takes two sequences and returns a sequence of
grcorresponding pairs. If one input is short, excess elements are
grdiscarded from the right end of the longer sequence.
zip :: Seq a -> Seq b -> Seq (a, b)

-- | &lt;math&gt;. <a>zipWith</a> generalizes <a>zip</a> by zipping with
grthe function given as the first argument, instead of a tupling
grfunction. For example, <tt>zipWith (+)</tt> is applied to two
grsequences to take the sequence of corresponding sums.
zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c

-- | &lt;math&gt;. <a>zip3</a> takes three sequences and returns a sequence
grof triples, analogous to <a>zip</a>.
zip3 :: Seq a -> Seq b -> Seq c -> Seq (a, b, c)

-- | &lt;math&gt;. <a>zipWith3</a> takes a function which combines three
grelements, as well as three sequences and returns a sequence of their
grpoint-wise combinations, analogous to <a>zipWith</a>.
zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d

-- | &lt;math&gt;. <a>zip4</a> takes four sequences and returns a sequence
grof quadruples, analogous to <a>zip</a>.
zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d)

-- | &lt;math&gt;. <a>zipWith4</a> takes a function which combines four
grelements, as well as four sequences and returns a sequence of their
grpoint-wise combinations, analogous to <a>zipWith</a>.
zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e

-- | Unzip a sequence of pairs.
gr
gr<pre>
grunzip ps = ps `<tt>seq'</tt> (<a>fmap</a> <a>fst</a> ps) (<a>fmap</a> <a>snd</a> ps)
gr</pre>
gr
grExample:
gr
gr<pre>
grunzip $ fromList [(1,"a"), (2,"b"), (3,"c")] =
gr  (fromList [1,2,3], fromList ["a", "b", "c"])
gr</pre>
gr
grSee the note about efficiency at <a>unzipWith</a>.
unzip :: Seq (a, b) -> (Seq a, Seq b)

-- | &lt;math&gt;. Unzip a sequence using a function to divide elements.
gr
gr<pre>
grunzipWith f xs == <a>unzip</a> (<a>fmap</a> f xs)
gr</pre>
gr
grEfficiency note:
gr
gr<tt>unzipWith</tt> produces its two results in lockstep. If you
grcalculate <tt> unzipWith f xs </tt> and fully force <i>either</i> of
grthe results, then the entire structure of the <i>other</i> one will be
grbuilt as well. This behavior allows the garbage collector to collect
greach calculated pair component as soon as it dies, without having to
grwait for its mate to die. If you do not need this behavior, you may be
grbetter off simply calculating the sequence of pairs and using
gr<a>fmap</a> to extract each component sequence.
unzipWith :: (a -> (b, c)) -> Seq a -> (Seq b, Seq c)
instance GHC.Read.Read a => GHC.Read.Read (Data.Sequence.Internal.ViewR a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Sequence.Internal.ViewR a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Sequence.Internal.ViewR a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Sequence.Internal.ViewR a)
instance GHC.Read.Read a => GHC.Read.Read (Data.Sequence.Internal.ViewL a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Sequence.Internal.ViewL a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Sequence.Internal.ViewL a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Sequence.Internal.ViewL a)
instance Data.Data.Data a => Data.Data.Data (Data.Sequence.Internal.ViewL a)
instance GHC.Generics.Generic1 Data.Sequence.Internal.ViewL
instance GHC.Generics.Generic (Data.Sequence.Internal.ViewL a)
instance Data.Data.Data a => Data.Data.Data (Data.Sequence.Internal.ViewR a)
instance GHC.Generics.Generic1 Data.Sequence.Internal.ViewR
instance GHC.Generics.Generic (Data.Sequence.Internal.ViewR a)
instance Data.Sequence.Internal.UnzipWith Data.Sequence.Internal.Elem
instance Data.Sequence.Internal.UnzipWith Data.Sequence.Internal.Node
instance Data.Sequence.Internal.UnzipWith Data.Sequence.Internal.Digit
instance Data.Sequence.Internal.UnzipWith Data.Sequence.Internal.FingerTree
instance Data.Sequence.Internal.UnzipWith Data.Sequence.Internal.Seq
instance GHC.Base.Functor Data.Sequence.Internal.ViewR
instance Data.Foldable.Foldable Data.Sequence.Internal.ViewR
instance Data.Traversable.Traversable Data.Sequence.Internal.ViewR
instance Data.Data.Data a => Data.Data.Data (Data.Sequence.Internal.Seq a)
instance GHC.Base.Functor Data.Sequence.Internal.ViewL
instance Data.Foldable.Foldable Data.Sequence.Internal.ViewL
instance Data.Traversable.Traversable Data.Sequence.Internal.ViewL
instance GHC.Base.Functor Data.Sequence.Internal.Seq
instance Data.Foldable.Foldable Data.Sequence.Internal.Seq
instance Data.Traversable.Traversable Data.Sequence.Internal.Seq
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Sequence.Internal.Seq a)
instance GHC.Base.Monad Data.Sequence.Internal.Seq
instance Control.Monad.Fix.MonadFix Data.Sequence.Internal.Seq
instance GHC.Base.Applicative Data.Sequence.Internal.Seq
instance GHC.Base.MonadPlus Data.Sequence.Internal.Seq
instance GHC.Base.Alternative Data.Sequence.Internal.Seq
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Sequence.Internal.Seq a)
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.Sequence.Internal.Seq a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Sequence.Internal.Seq a)
instance Data.Functor.Classes.Show1 Data.Sequence.Internal.Seq
instance Data.Functor.Classes.Eq1 Data.Sequence.Internal.Seq
instance Data.Functor.Classes.Ord1 Data.Sequence.Internal.Seq
instance GHC.Read.Read a => GHC.Read.Read (Data.Sequence.Internal.Seq a)
instance Data.Functor.Classes.Read1 Data.Sequence.Internal.Seq
instance GHC.Base.Monoid (Data.Sequence.Internal.Seq a)
instance GHC.Base.Semigroup (Data.Sequence.Internal.Seq a)
instance GHC.Exts.IsList (Data.Sequence.Internal.Seq a)
instance (a ~ GHC.Types.Char) => Data.String.IsString (Data.Sequence.Internal.Seq a)
instance Control.Monad.Zip.MonadZip Data.Sequence.Internal.Seq
instance Data.Sequence.Internal.MaybeForce (Data.Sequence.Internal.Elem a)
instance Data.Sequence.Internal.Sized a => Data.Sequence.Internal.Sized (Data.Sequence.Internal.FingerTree a)
instance Data.Sequence.Internal.Sized (Data.Sequence.Internal.Elem a)
instance GHC.Base.Functor Data.Sequence.Internal.Elem
instance Data.Foldable.Foldable Data.Sequence.Internal.Elem
instance Data.Traversable.Traversable Data.Sequence.Internal.Elem
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Sequence.Internal.Elem a)
instance Data.Foldable.Foldable Data.Sequence.Internal.FingerTree
instance GHC.Base.Functor Data.Sequence.Internal.FingerTree
instance Data.Traversable.Traversable Data.Sequence.Internal.FingerTree
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Sequence.Internal.FingerTree a)
instance Data.Sequence.Internal.MaybeForce (Data.Sequence.Internal.Node a)
instance Data.Foldable.Foldable Data.Sequence.Internal.Node
instance GHC.Base.Functor Data.Sequence.Internal.Node
instance Data.Traversable.Traversable Data.Sequence.Internal.Node
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Sequence.Internal.Node a)
instance Data.Sequence.Internal.Sized (Data.Sequence.Internal.Node a)
instance Data.Foldable.Foldable Data.Sequence.Internal.Digit
instance GHC.Base.Functor Data.Sequence.Internal.Digit
instance Data.Traversable.Traversable Data.Sequence.Internal.Digit
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Sequence.Internal.Digit a)
instance Data.Sequence.Internal.Sized a => Data.Sequence.Internal.Sized (Data.Sequence.Internal.Digit a)
instance Data.Sequence.Internal.MaybeForce (Data.Sequence.Internal.ForceBox a)
instance Data.Sequence.Internal.Sized (Data.Sequence.Internal.ForceBox a)


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grThis module provides the various sorting implementations for
gr<a>Data.Sequence</a>. Further notes are available in the file
grsorting.md (in this directory).
module Data.Sequence.Internal.Sorting

-- | &lt;math&gt;. <a>sort</a> sorts the specified <a>Seq</a> by the
grnatural ordering of its elements. The sort is stable. If stability is
grnot required, <a>unstableSort</a> can be slightly faster.
sort :: Ord a => Seq a -> Seq a

-- | &lt;math&gt;. <a>sortBy</a> sorts the specified <a>Seq</a> according
grto the specified comparator. The sort is stable. If stability is not
grrequired, <a>unstableSortBy</a> can be slightly faster.
sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>sortOn</a> sorts the specified <a>Seq</a> by
grcomparing the results of a key function applied to each element.
gr<tt><a>sortOn</a> f</tt> is equivalent to <tt><a>sortBy</a>
gr(<a>compare</a> `<a>on</a>` f)</tt>, but has the performance advantage
grof only evaluating <tt>f</tt> once for each element in the input list.
grThis is called the decorate-sort-undecorate paradigm, or Schwartzian
grtransform.
gr
grAn example of using <a>sortOn</a> might be to sort a <a>Seq</a> of
grstrings according to their length:
gr
gr<pre>
grsortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]
gr</pre>
gr
grIf, instead, <a>sortBy</a> had been used, <a>length</a> would be
grevaluated on every comparison, giving &lt;math&gt; evaluations, rather
grthan &lt;math&gt;.
gr
grIf <tt>f</tt> is very cheap (for example a record selector, or
gr<a>fst</a>), <tt><a>sortBy</a> (<a>compare</a> `<a>on</a>` f)</tt>
grwill be faster than <tt><a>sortOn</a> f</tt>.
sortOn :: Ord b => (a -> b) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>unstableSort</a> sorts the specified <a>Seq</a> by
grthe natural ordering of its elements, but the sort is not stable. This
gralgorithm is frequently faster and uses less memory than <a>sort</a>.
unstableSort :: Ord a => Seq a -> Seq a

-- | &lt;math&gt;. A generalization of <a>unstableSort</a>,
gr<a>unstableSortBy</a> takes an arbitrary comparator and sorts the
grspecified sequence. The sort is not stable. This algorithm is
grfrequently faster and uses less memory than <a>sortBy</a>.
unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>unstableSortOn</a> sorts the specified <a>Seq</a> by
grcomparing the results of a key function applied to each element.
gr<tt><a>unstableSortOn</a> f</tt> is equivalent to
gr<tt><a>unstableSortBy</a> (<a>compare</a> `<a>on</a>` f)</tt>, but has
grthe performance advantage of only evaluating <tt>f</tt> once for each
grelement in the input list. This is called the decorate-sort-undecorate
grparadigm, or Schwartzian transform.
gr
grAn example of using <a>unstableSortOn</a> might be to sort a
gr<a>Seq</a> of strings according to their length:
gr
gr<pre>
grunstableSortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]
gr</pre>
gr
grIf, instead, <a>unstableSortBy</a> had been used, <a>length</a> would
grbe evaluated on every comparison, giving &lt;math&gt; evaluations,
grrather than &lt;math&gt;.
gr
grIf <tt>f</tt> is very cheap (for example a record selector, or
gr<a>fst</a>), <tt><a>unstableSortBy</a> (<a>compare</a> `<a>on</a>`
grf)</tt> will be faster than <tt><a>unstableSortOn</a> f</tt>.
unstableSortOn :: Ord b => (a -> b) -> Seq a -> Seq a

-- | A simple pairing heap.
data Queue e
Q :: !e -> (QList e) -> Queue e
data QList e
Nil :: QList e
QCons :: {-# UNPACK #-} !(Queue e) -> (QList e) -> QList e

-- | A pairing heap tagged with the original position of elements, to allow
grfor stable sorting.
data IndexedQueue e
IQ :: {-# UNPACK #-} !Int -> !e -> (IQList e) -> IndexedQueue e
data IQList e
IQNil :: IQList e
IQCons :: {-# UNPACK #-} !(IndexedQueue e) -> (IQList e) -> IQList e

-- | A pairing heap tagged with some key for sorting elements, for use in
gr<a>unstableSortOn</a>.
data TaggedQueue a b
TQ :: !a -> b -> (TQList a b) -> TaggedQueue a b
data TQList a b
TQNil :: TQList a b
TQCons :: {-# UNPACK #-} !(TaggedQueue a b) -> (TQList a b) -> TQList a b

-- | A pairing heap tagged with both a key and the original position of its
grelements, for use in <a>sortOn</a>.
data IndexedTaggedQueue e a
ITQ :: {-# UNPACK #-} !Int -> !e -> a -> (ITQList e a) -> IndexedTaggedQueue e a
data ITQList e a
ITQNil :: ITQList e a
ITQCons :: {-# UNPACK #-} !(IndexedTaggedQueue e a) -> (ITQList e a) -> ITQList e a

-- | <a>mergeQ</a> merges two <a>Queue</a>s.
mergeQ :: (a -> a -> Ordering) -> Queue a -> Queue a -> Queue a

-- | <a>mergeIQ</a> merges two <a>IndexedQueue</a>s, taking into account
grthe original position of the elements.
mergeIQ :: (a -> a -> Ordering) -> IndexedQueue a -> IndexedQueue a -> IndexedQueue a

-- | <a>mergeTQ</a> merges two <a>TaggedQueue</a>s, based on the tag value.
mergeTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> TaggedQueue a b -> TaggedQueue a b

-- | <a>mergeITQ</a> merges two <a>IndexedTaggedQueue</a>s, based on the
grtag value, taking into account the original position of the elements.
mergeITQ :: (a -> a -> Ordering) -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b -> IndexedTaggedQueue a b

-- | Pop the smallest element from the queue, using the supplied
grcomparator.
popMinQ :: (e -> e -> Ordering) -> Queue e -> (Queue e, e)

-- | Pop the smallest element from the queue, using the supplied
grcomparator, deferring to the item's original position when the
grcomparator returns <a>EQ</a>.
popMinIQ :: (e -> e -> Ordering) -> IndexedQueue e -> (IndexedQueue e, e)

-- | Pop the smallest element from the queue, using the supplied comparator
gron the tag.
popMinTQ :: (a -> a -> Ordering) -> TaggedQueue a b -> (TaggedQueue a b, b)

-- | Pop the smallest element from the queue, using the supplied comparator
gron the tag, deferring to the item's original position when the
grcomparator returns <a>EQ</a>.
popMinITQ :: (e -> e -> Ordering) -> IndexedTaggedQueue e b -> (IndexedTaggedQueue e b, b)
buildQ :: (b -> b -> Ordering) -> (a -> Queue b) -> FingerTree a -> Maybe (Queue b)
buildIQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedQueue b) -> Int -> FingerTree (Elem y) -> Maybe (IndexedQueue b)
buildTQ :: (b -> b -> Ordering) -> (a -> TaggedQueue b c) -> FingerTree a -> Maybe (TaggedQueue b c)
buildITQ :: (b -> b -> Ordering) -> (Int -> Elem y -> IndexedTaggedQueue b c) -> Int -> FingerTree (Elem y) -> Maybe (IndexedTaggedQueue b c)

-- | A <a>foldMap</a>-like function, specialized to the <a>Option</a>
grmonoid, which takes advantage of the internal structure of <a>Seq</a>
grto avoid wrapping in <a>Maybe</a> at certain points.
foldToMaybeTree :: (b -> b -> b) -> (a -> b) -> FingerTree a -> Maybe b

-- | A <tt>foldMapWithIndex</tt>-like function, specialized to the
gr<a>Option</a> monoid, which takes advantage of the internal structure
grof <a>Seq</a> to avoid wrapping in <a>Maybe</a> at certain points.
foldToMaybeWithIndexTree :: (b -> b -> b) -> (Int -> Elem y -> b) -> Int -> FingerTree (Elem y) -> Maybe b


-- | <h1>Finite sequences</h1>
gr
grThe <tt><a>Seq</a> a</tt> type represents a finite sequence of values
grof type <tt>a</tt>.
gr
grSequences generally behave very much like lists.
gr
gr<ul>
gr<li>The class instances for sequences are all based very closely on
grthose for lists.</li>
gr<li>Many functions in this module have the same names as functions in
grthe <a>Prelude</a> or in <a>Data.List</a>. In almost all cases, these
grfunctions behave analogously. For example, <a>filter</a> filters a
grsequence in exactly the same way that
gr<tt><a>Prelude</a>.<a>filter</a></tt> filters a list. The only major
grexception is the <a>lookup</a> function, which is based on the
grfunction by that name in <a>Data.IntMap</a> rather than the one in
gr<a>Prelude</a>.</li>
gr</ul>
gr
grThere are two major differences between sequences and lists:
gr
gr<ul>
gr<li>Sequences support a wider variety of efficient operations than do
grlists. Notably, they offer<ul><li>Constant-time access to both the
grfront and the rear with <a>&lt;|</a>, <a>|&gt;</a>, <a>viewl</a>,
gr<a>viewr</a>. For recent GHC versions, this can be done more
grconveniently using the bidirectional patterns <a>Empty</a>,
gr<a>:&lt;|</a>, and <a>:|&gt;</a>. See the detailed explanation in the
gr"Pattern synonyms" section.</li><li>Logarithmic-time concatenation
grwith <a>&gt;&lt;</a></li><li>Logarithmic-time splitting with
gr<a>splitAt</a>, <a>take</a> and <a>drop</a></li><li>Logarithmic-time
graccess to any element with <a>lookup</a>, <a>!?</a>, <a>index</a>,
gr<a>insertAt</a>, <a>deleteAt</a>, <a>adjust'</a>, and
gr<a>update</a></li></ul></li>
gr</ul>
gr
grNote that sequences are typically <i>slower</i> than lists when using
gronly operations for which they have the same big-(O) complexity:
grsequences make rather mediocre stacks!
gr
gr<ul>
gr<li>Whereas lists can be either finite or infinite, sequences are
gralways finite. As a result, a sequence is strict in its length.
grIgnoring efficiency, you can imagine that <a>Seq</a> is defined<pre>
grdata Seq a = Empty | a :&lt;| !(Seq a)</pre>This means that many
groperations on sequences are stricter than those on lists. For
grexample,<pre> (1 : undefined) !! 0 = 1</pre>but<pre> (1 :&lt;|
grundefined) <a>index</a> 0 = undefined</pre></li>
gr</ul>
gr
grSequences may also be compared to immutable <a>arrays</a> or
gr<a>vectors</a>. Like these structures, sequences support fast
grindexing, although not as fast. But editing an immutable array or
grvector, or combining it with another, generally requires copying the
grentire structure; sequences generally avoid that, copying only the
grportion that has changed.
gr
gr<h2>Detailed performance information</h2>
gr
grAn amortized running time is given for each operation, with <i>n</i>
grreferring to the length of the sequence and <i>i</i> being the
grintegral index used by some operations. These bounds hold even in a
grpersistent (shared) setting.
gr
grDespite sequences being structurally strict from a semantic
grstandpoint, they are in fact implemented using laziness internally. As
gra result, many operations can be performed <i>incrementally</i>,
grproducing their results as they are demanded. This greatly improves
grperformance in some cases. These functions include
gr
gr<ul>
gr<li>The <a>Functor</a> methods <a>fmap</a> and <a>&lt;$</a>, along
grwith <a>mapWithIndex</a></li>
gr<li>The <a>Applicative</a> methods <a>&lt;*&gt;</a>, <a>*&gt;</a>, and
gr<a>&lt;*</a></li>
gr<li>The zips: <a>zipWith</a>, <a>zip</a>, etc.</li>
gr<li><tt>heads</tt> and <a>tails</a></li>
gr<li><a>fromFunction</a>, <a>replicate</a>, <a>intersperse</a>, and
gr<a>cycleTaking</a></li>
gr<li><a>reverse</a></li>
gr<li><a>chunksOf</a></li>
gr</ul>
gr
grNote that the <a>Monad</a> method, <a>&gt;&gt;=</a>, is not
grparticularly lazy. It will take time proportional to the sum of the
grlogarithms of the individual result sequences to produce anything
grwhatsoever.
gr
grSeveral functions take special advantage of sharing to produce results
grusing much less time and memory than one might expect. These are
grdocumented individually for functions, but also include the methods
gr<a>&lt;$</a> and <a>*&gt;</a>, each of which take time and space
grproportional to the logarithm of the size of the result.
gr
gr<h2>Warning</h2>
gr
grThe size of a <a>Seq</a> must not exceed <tt>maxBound::Int</tt>.
grViolation of this condition is not detected and if the size limit is
grexceeded, the behaviour of the sequence is undefined. This is unlikely
grto occur in most applications, but some care may be required when
grusing <a>&gt;&lt;</a>, <a>&lt;*&gt;</a>, <a>*&gt;</a>, or
gr<a>&gt;&gt;</a>, particularly repeatedly and particularly in
grcombination with <a>replicate</a> or <a>fromFunction</a>.
gr
gr<h2>Implementation</h2>
gr
grThe implementation uses 2-3 finger trees annotated with sizes, as
grdescribed in section 4.2 of
gr
gr<ul>
gr<li>Ralf Hinze and Ross Paterson, <a>"Finger trees: a simple
grgeneral-purpose data structure"</a>, <i>Journal of Functional
grProgramming</i> 16:2 (2006) pp 197-217.</li>
gr</ul>
module Data.Sequence

-- | General-purpose finite sequences.
data Seq a

-- | &lt;math&gt;. The empty sequence.
empty :: Seq a

-- | &lt;math&gt;. A singleton sequence.
singleton :: a -> Seq a

-- | &lt;math&gt;. Add an element to the left end of a sequence. Mnemonic:
gra triangle with the single element at the pointy end.
(<|) :: a -> Seq a -> Seq a
infixr 5 <|

-- | &lt;math&gt;. Add an element to the right end of a sequence. Mnemonic:
gra triangle with the single element at the pointy end.
(|>) :: Seq a -> a -> Seq a
infixl 5 |>

-- | &lt;math&gt;. Concatenate two sequences.
(><) :: Seq a -> Seq a -> Seq a
infixr 5 ><

-- | &lt;math&gt;. Create a sequence from a finite list of elements. There
gris a function <a>toList</a> in the opposite direction for all
grinstances of the <a>Foldable</a> class, including <a>Seq</a>.
fromList :: [a] -> Seq a

-- | &lt;math&gt;. Convert a given sequence length and a function
grrepresenting that sequence into a sequence.
fromFunction :: Int -> (Int -> a) -> Seq a

-- | &lt;math&gt;. Create a sequence consisting of the elements of an
gr<a>Array</a>. Note that the resulting sequence elements may be
grevaluated lazily (as on GHC), so you must force the entire structure
grto be sure that the original array can be garbage-collected.
fromArray :: Ix i => Array i a -> Seq a

-- | &lt;math&gt;. <tt>replicate n x</tt> is a sequence consisting of
gr<tt>n</tt> copies of <tt>x</tt>.
replicate :: Int -> a -> Seq a

-- | <a>replicateA</a> is an <a>Applicative</a> version of
gr<a>replicate</a>, and makes &lt;math&gt; calls to <a>liftA2</a> and
gr<a>pure</a>.
gr
gr<pre>
grreplicateA n x = sequenceA (replicate n x)
gr</pre>
replicateA :: Applicative f => Int -> f a -> f (Seq a)

-- | <a>replicateM</a> is a sequence counterpart of <a>replicateM</a>.
gr
gr<pre>
grreplicateM n x = sequence (replicate n x)
gr</pre>
gr
grFor <tt>base &gt;= 4.8.0</tt> and <tt>containers &gt;= 0.5.11</tt>,
gr<a>replicateM</a> is a synonym for <a>replicateA</a>.
replicateM :: Applicative m => Int -> m a -> m (Seq a)

-- | <i>O(</i>log<i> k)</i>. <tt><a>cycleTaking</a> k xs</tt> forms a
grsequence of length <tt>k</tt> by repeatedly concatenating <tt>xs</tt>
grwith itself. <tt>xs</tt> may only be empty if <tt>k</tt> is 0.
gr
gr<pre>
grcycleTaking k = fromList . take k . cycle . toList
gr</pre>
cycleTaking :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Constructs a sequence by repeated application of a
grfunction to a seed value.
gr
gr<pre>
griterateN n f x = fromList (Prelude.take n (Prelude.iterate f x))
gr</pre>
iterateN :: Int -> (a -> a) -> a -> Seq a

-- | Builds a sequence from a seed value. Takes time linear in the number
grof generated elements. <i>WARNING:</i> If the number of generated
grelements is infinite, this method will not terminate.
unfoldr :: (b -> Maybe (a, b)) -> b -> Seq a

-- | <tt><a>unfoldl</a> f x</tt> is equivalent to <tt><a>reverse</a>
gr(<a>unfoldr</a> (<a>fmap</a> swap . f) x)</tt>.
unfoldl :: (b -> Maybe (b, a)) -> b -> Seq a

-- | &lt;math&gt;. Is this the empty sequence?
null :: Seq a -> Bool

-- | &lt;math&gt;. The number of elements in the sequence.
length :: Seq a -> Int

-- | View of the left end of a sequence.
data ViewL a

-- | empty sequence
EmptyL :: ViewL a

-- | leftmost element and the rest of the sequence
(:<) :: a -> Seq a -> ViewL a

-- | &lt;math&gt;. Analyse the left end of a sequence.
viewl :: Seq a -> ViewL a

-- | View of the right end of a sequence.
data ViewR a

-- | empty sequence
EmptyR :: ViewR a

-- | the sequence minus the rightmost element, and the rightmost element
(:>) :: Seq a -> a -> ViewR a

-- | &lt;math&gt;. Analyse the right end of a sequence.
viewr :: Seq a -> ViewR a

-- | <a>scanl</a> is similar to <a>foldl</a>, but returns a sequence of
grreduced values from the left:
gr
gr<pre>
grscanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...]
gr</pre>
scanl :: (a -> b -> a) -> a -> Seq b -> Seq a

-- | <a>scanl1</a> is a variant of <a>scanl</a> that has no starting value
grargument:
gr
gr<pre>
grscanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]
gr</pre>
scanl1 :: (a -> a -> a) -> Seq a -> Seq a

-- | <a>scanr</a> is the right-to-left dual of <a>scanl</a>.
scanr :: (a -> b -> b) -> b -> Seq a -> Seq b

-- | <a>scanr1</a> is a variant of <a>scanr</a> that has no starting value
grargument.
scanr1 :: (a -> a -> a) -> Seq a -> Seq a

-- | &lt;math&gt;. Returns a sequence of all suffixes of this sequence,
grlongest first. For example,
gr
gr<pre>
grtails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]
gr</pre>
gr
grEvaluating the &lt;math&gt;th suffix takes &lt;math&gt;, but
grevaluating every suffix in the sequence takes &lt;math&gt; due to
grsharing.
tails :: Seq a -> Seq (Seq a)

-- | &lt;math&gt;. Returns a sequence of all prefixes of this sequence,
grshortest first. For example,
gr
gr<pre>
grinits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]
gr</pre>
gr
grEvaluating the &lt;math&gt;th prefix takes &lt;math&gt;, but
grevaluating every prefix in the sequence takes &lt;math&gt; due to
grsharing.
inits :: Seq a -> Seq (Seq a)

-- | &lt;math&gt;. <tt>chunksOf c xs</tt> splits <tt>xs</tt> into chunks of
grsize <tt>c&gt;0</tt>. If <tt>c</tt> does not divide the length of
gr<tt>xs</tt> evenly, then the last element of the result will be short.
gr
grSide note: the given performance bound is missing some messy terms
grthat only really affect edge cases. Performance degrades smoothly from
gr&lt;math&gt; (for &lt;math&gt;) to &lt;math&gt; (for &lt;math&gt;).
grThe true bound is more like &lt;math&gt;
chunksOf :: Int -> Seq a -> Seq (Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the prefix length.
gr<a>takeWhileL</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns the longest prefix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt>.
takeWhileL :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the suffix length.
gr<a>takeWhileR</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns the longest suffix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt>.
gr
gr<tt><a>takeWhileR</a> p xs</tt> is equivalent to <tt><a>reverse</a>
gr(<a>takeWhileL</a> p (<a>reverse</a> xs))</tt>.
takeWhileR :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the prefix length.
gr<tt><a>dropWhileL</a> p xs</tt> returns the suffix remaining after
gr<tt><a>takeWhileL</a> p xs</tt>.
dropWhileL :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the suffix length.
gr<tt><a>dropWhileR</a> p xs</tt> returns the prefix remaining after
gr<tt><a>takeWhileR</a> p xs</tt>.
gr
gr<tt><a>dropWhileR</a> p xs</tt> is equivalent to <tt><a>reverse</a>
gr(<a>dropWhileL</a> p (<a>reverse</a> xs))</tt>.
dropWhileR :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt; where &lt;math&gt; is the prefix length. <a>spanl</a>,
grapplied to a predicate <tt>p</tt> and a sequence <tt>xs</tt>, returns
gra pair whose first element is the longest prefix (possibly empty) of
gr<tt>xs</tt> of elements that satisfy <tt>p</tt> and the second element
gris the remainder of the sequence.
spanl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the suffix length. <a>spanr</a>,
grapplied to a predicate <tt>p</tt> and a sequence <tt>xs</tt>, returns
gra pair whose <i>first</i> element is the longest <i>suffix</i>
gr(possibly empty) of <tt>xs</tt> of elements that satisfy <tt>p</tt>
grand the second element is the remainder of the sequence.
spanr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt; where &lt;math&gt; is the breakpoint index.
gr<a>breakl</a>, applied to a predicate <tt>p</tt> and a sequence
gr<tt>xs</tt>, returns a pair whose first element is the longest prefix
gr(possibly empty) of <tt>xs</tt> of elements that <i>do not satisfy</i>
gr<tt>p</tt> and the second element is the remainder of the sequence.
gr
gr<tt><a>breakl</a> p</tt> is equivalent to <tt><a>spanl</a> (not .
grp)</tt>.
breakl :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | <tt><a>breakr</a> p</tt> is equivalent to <tt><a>spanr</a> (not .
grp)</tt>.
breakr :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt;. The <a>partition</a> function takes a predicate
gr<tt>p</tt> and a sequence <tt>xs</tt> and returns sequences of those
grelements which do and do not satisfy the predicate.
partition :: (a -> Bool) -> Seq a -> (Seq a, Seq a)

-- | &lt;math&gt;. The <a>filter</a> function takes a predicate <tt>p</tt>
grand a sequence <tt>xs</tt> and returns a sequence of those elements
grwhich satisfy the predicate.
filter :: (a -> Bool) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>sort</a> sorts the specified <a>Seq</a> by the
grnatural ordering of its elements. The sort is stable. If stability is
grnot required, <a>unstableSort</a> can be slightly faster.
sort :: Ord a => Seq a -> Seq a

-- | &lt;math&gt;. <a>sortBy</a> sorts the specified <a>Seq</a> according
grto the specified comparator. The sort is stable. If stability is not
grrequired, <a>unstableSortBy</a> can be slightly faster.
sortBy :: (a -> a -> Ordering) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>sortOn</a> sorts the specified <a>Seq</a> by
grcomparing the results of a key function applied to each element.
gr<tt><a>sortOn</a> f</tt> is equivalent to <tt><a>sortBy</a>
gr(<a>compare</a> `<a>on</a>` f)</tt>, but has the performance advantage
grof only evaluating <tt>f</tt> once for each element in the input list.
grThis is called the decorate-sort-undecorate paradigm, or Schwartzian
grtransform.
gr
grAn example of using <a>sortOn</a> might be to sort a <a>Seq</a> of
grstrings according to their length:
gr
gr<pre>
grsortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]
gr</pre>
gr
grIf, instead, <a>sortBy</a> had been used, <a>length</a> would be
grevaluated on every comparison, giving &lt;math&gt; evaluations, rather
grthan &lt;math&gt;.
gr
grIf <tt>f</tt> is very cheap (for example a record selector, or
gr<a>fst</a>), <tt><a>sortBy</a> (<a>compare</a> `<a>on</a>` f)</tt>
grwill be faster than <tt><a>sortOn</a> f</tt>.
sortOn :: Ord b => (a -> b) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>unstableSort</a> sorts the specified <a>Seq</a> by
grthe natural ordering of its elements, but the sort is not stable. This
gralgorithm is frequently faster and uses less memory than <a>sort</a>.
unstableSort :: Ord a => Seq a -> Seq a

-- | &lt;math&gt;. A generalization of <a>unstableSort</a>,
gr<a>unstableSortBy</a> takes an arbitrary comparator and sorts the
grspecified sequence. The sort is not stable. This algorithm is
grfrequently faster and uses less memory than <a>sortBy</a>.
unstableSortBy :: (a -> a -> Ordering) -> Seq a -> Seq a

-- | &lt;math&gt;. <a>unstableSortOn</a> sorts the specified <a>Seq</a> by
grcomparing the results of a key function applied to each element.
gr<tt><a>unstableSortOn</a> f</tt> is equivalent to
gr<tt><a>unstableSortBy</a> (<a>compare</a> `<a>on</a>` f)</tt>, but has
grthe performance advantage of only evaluating <tt>f</tt> once for each
grelement in the input list. This is called the decorate-sort-undecorate
grparadigm, or Schwartzian transform.
gr
grAn example of using <a>unstableSortOn</a> might be to sort a
gr<a>Seq</a> of strings according to their length:
gr
gr<pre>
grunstableSortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"]
gr</pre>
gr
grIf, instead, <a>unstableSortBy</a> had been used, <a>length</a> would
grbe evaluated on every comparison, giving &lt;math&gt; evaluations,
grrather than &lt;math&gt;.
gr
grIf <tt>f</tt> is very cheap (for example a record selector, or
gr<a>fst</a>), <tt><a>unstableSortBy</a> (<a>compare</a> `<a>on</a>`
grf)</tt> will be faster than <tt><a>unstableSortOn</a> f</tt>.
unstableSortOn :: Ord b => (a -> b) -> Seq a -> Seq a

-- | &lt;math&gt;. The element at the specified position, counting from 0.
grIf the specified position is negative or at least the length of the
grsequence, <a>lookup</a> returns <a>Nothing</a>.
gr
gr<pre>
gr0 &lt;= i &lt; length xs ==&gt; lookup i xs == Just (toList xs !! i)
gr</pre>
gr
gr<pre>
gri &lt; 0 || i &gt;= length xs ==&gt; lookup i xs = Nothing
gr</pre>
gr
grUnlike <a>index</a>, this can be used to retrieve an element without
grforcing it. For example, to insert the fifth element of a sequence
gr<tt>xs</tt> into a <a>Map</a> <tt>m</tt> at key <tt>k</tt>, you could
gruse
gr
gr<pre>
grcase lookup 5 xs of
gr  Nothing -&gt; m
gr  Just x -&gt; <a>insert</a> k x m
gr</pre>
lookup :: Int -> Seq a -> Maybe a

-- | &lt;math&gt;. A flipped, infix version of <a>lookup</a>.
(!?) :: Seq a -> Int -> Maybe a

-- | &lt;math&gt;. The element at the specified position, counting from 0.
grThe argument should thus be a non-negative integer less than the size
grof the sequence. If the position is out of range, <a>index</a> fails
grwith an error.
gr
gr<pre>
grxs `index` i = toList xs !! i
gr</pre>
gr
grCaution: <a>index</a> necessarily delays retrieving the requested
grelement until the result is forced. It can therefore lead to a space
grleak if the result is stored, unforced, in another structure. To
grretrieve an element immediately without forcing it, use <a>lookup</a>
gror '(!?)'.
index :: Seq a -> Int -> a

-- | &lt;math&gt;. Update the element at the specified position. If the
grposition is out of range, the original sequence is returned.
gr<a>adjust</a> can lead to poor performance and even memory leaks,
grbecause it does not force the new value before installing it in the
grsequence. <a>adjust'</a> should usually be preferred.
adjust :: (a -> a) -> Int -> Seq a -> Seq a

-- | &lt;math&gt;. Update the element at the specified position. If the
grposition is out of range, the original sequence is returned. The new
grvalue is forced before it is installed in the sequence.
gr
gr<pre>
gradjust' f i xs =
gr case xs !? i of
gr   Nothing -&gt; xs
gr   Just x -&gt; let !x' = f x
gr             in update i x' xs
gr</pre>
adjust' :: forall a. (a -> a) -> Int -> Seq a -> Seq a

-- | &lt;math&gt;. Replace the element at the specified position. If the
grposition is out of range, the original sequence is returned.
update :: Int -> a -> Seq a -> Seq a

-- | &lt;math&gt;. The first <tt>i</tt> elements of a sequence. If
gr<tt>i</tt> is negative, <tt><a>take</a> i s</tt> yields the empty
grsequence. If the sequence contains fewer than <tt>i</tt> elements, the
grwhole sequence is returned.
take :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Elements of a sequence after the first <tt>i</tt>. If
gr<tt>i</tt> is negative, <tt><a>drop</a> i s</tt> yields the whole
grsequence. If the sequence contains fewer than <tt>i</tt> elements, the
grempty sequence is returned.
drop :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. <tt><a>insertAt</a> i x xs</tt> inserts <tt>x</tt> into
gr<tt>xs</tt> at the index <tt>i</tt>, shifting the rest of the sequence
grover.
gr
gr<pre>
grinsertAt 2 x (fromList [a,b,c,d]) = fromList [a,b,x,c,d]
grinsertAt 4 x (fromList [a,b,c,d]) = insertAt 10 x (fromList [a,b,c,d])
gr                                  = fromList [a,b,c,d,x]
gr</pre>
gr
gr<pre>
grinsertAt i x xs = take i xs &gt;&lt; singleton x &gt;&lt; drop i xs
gr</pre>
insertAt :: Int -> a -> Seq a -> Seq a

-- | &lt;math&gt;. Delete the element of a sequence at a given index.
grReturn the original sequence if the index is out of range.
gr
gr<pre>
grdeleteAt 2 [a,b,c,d] = [a,b,d]
grdeleteAt 4 [a,b,c,d] = deleteAt (-1) [a,b,c,d] = [a,b,c,d]
gr</pre>
deleteAt :: Int -> Seq a -> Seq a

-- | &lt;math&gt;. Split a sequence at a given position. <tt><a>splitAt</a>
gri s = (<a>take</a> i s, <a>drop</a> i s)</tt>.
splitAt :: Int -> Seq a -> (Seq a, Seq a)

-- | <a>elemIndexL</a> finds the leftmost index of the specified element,
grif it is present, and otherwise <a>Nothing</a>.
elemIndexL :: Eq a => a -> Seq a -> Maybe Int

-- | <a>elemIndicesL</a> finds the indices of the specified element, from
grleft to right (i.e. in ascending order).
elemIndicesL :: Eq a => a -> Seq a -> [Int]

-- | <a>elemIndexR</a> finds the rightmost index of the specified element,
grif it is present, and otherwise <a>Nothing</a>.
elemIndexR :: Eq a => a -> Seq a -> Maybe Int

-- | <a>elemIndicesR</a> finds the indices of the specified element, from
grright to left (i.e. in descending order).
elemIndicesR :: Eq a => a -> Seq a -> [Int]

-- | <tt><a>findIndexL</a> p xs</tt> finds the index of the leftmost
grelement that satisfies <tt>p</tt>, if any exist.
findIndexL :: (a -> Bool) -> Seq a -> Maybe Int

-- | <tt><a>findIndicesL</a> p</tt> finds all indices of elements that
grsatisfy <tt>p</tt>, in ascending order.
findIndicesL :: (a -> Bool) -> Seq a -> [Int]

-- | <tt><a>findIndexR</a> p xs</tt> finds the index of the rightmost
grelement that satisfies <tt>p</tt>, if any exist.
findIndexR :: (a -> Bool) -> Seq a -> Maybe Int

-- | <tt><a>findIndicesR</a> p</tt> finds all indices of elements that
grsatisfy <tt>p</tt>, in descending order.
findIndicesR :: (a -> Bool) -> Seq a -> [Int]
foldMapWithIndex :: Monoid m => (Int -> a -> m) -> Seq a -> m

-- | <a>foldlWithIndex</a> is a version of <a>foldl</a> that also provides
graccess to the index of each element.
foldlWithIndex :: (b -> Int -> a -> b) -> b -> Seq a -> b

-- | <a>foldrWithIndex</a> is a version of <a>foldr</a> that also provides
graccess to the index of each element.
foldrWithIndex :: (Int -> a -> b -> b) -> b -> Seq a -> b

-- | A generalization of <a>fmap</a>, <a>mapWithIndex</a> takes a mapping
grfunction that also depends on the element's index, and applies it to
grevery element in the sequence.
mapWithIndex :: (Int -> a -> b) -> Seq a -> Seq b

-- | <a>traverseWithIndex</a> is a version of <a>traverse</a> that also
groffers access to the index of each element.
traverseWithIndex :: Applicative f => (Int -> a -> f b) -> Seq a -> f (Seq b)

-- | &lt;math&gt;. The reverse of a sequence.
reverse :: Seq a -> Seq a

-- | &lt;math&gt;. Intersperse an element between the elements of a
grsequence.
gr
gr<pre>
grintersperse a empty = empty
grintersperse a (singleton x) = singleton x
grintersperse a (fromList [x,y]) = fromList [x,a,y]
grintersperse a (fromList [x,y,z]) = fromList [x,a,y,a,z]
gr</pre>
intersperse :: a -> Seq a -> Seq a

-- | &lt;math&gt;. <a>zip</a> takes two sequences and returns a sequence of
grcorresponding pairs. If one input is short, excess elements are
grdiscarded from the right end of the longer sequence.
zip :: Seq a -> Seq b -> Seq (a, b)

-- | &lt;math&gt;. <a>zipWith</a> generalizes <a>zip</a> by zipping with
grthe function given as the first argument, instead of a tupling
grfunction. For example, <tt>zipWith (+)</tt> is applied to two
grsequences to take the sequence of corresponding sums.
zipWith :: (a -> b -> c) -> Seq a -> Seq b -> Seq c

-- | &lt;math&gt;. <a>zip3</a> takes three sequences and returns a sequence
grof triples, analogous to <a>zip</a>.
zip3 :: Seq a -> Seq b -> Seq c -> Seq (a, b, c)

-- | &lt;math&gt;. <a>zipWith3</a> takes a function which combines three
grelements, as well as three sequences and returns a sequence of their
grpoint-wise combinations, analogous to <a>zipWith</a>.
zipWith3 :: (a -> b -> c -> d) -> Seq a -> Seq b -> Seq c -> Seq d

-- | &lt;math&gt;. <a>zip4</a> takes four sequences and returns a sequence
grof quadruples, analogous to <a>zip</a>.
zip4 :: Seq a -> Seq b -> Seq c -> Seq d -> Seq (a, b, c, d)

-- | &lt;math&gt;. <a>zipWith4</a> takes a function which combines four
grelements, as well as four sequences and returns a sequence of their
grpoint-wise combinations, analogous to <a>zipWith</a>.
zipWith4 :: (a -> b -> c -> d -> e) -> Seq a -> Seq b -> Seq c -> Seq d -> Seq e

-- | Unzip a sequence of pairs.
gr
gr<pre>
grunzip ps = ps `<tt>seq'</tt> (<a>fmap</a> <a>fst</a> ps) (<a>fmap</a> <a>snd</a> ps)
gr</pre>
gr
grExample:
gr
gr<pre>
grunzip $ fromList [(1,"a"), (2,"b"), (3,"c")] =
gr  (fromList [1,2,3], fromList ["a", "b", "c"])
gr</pre>
gr
grSee the note about efficiency at <a>unzipWith</a>.
unzip :: Seq (a, b) -> (Seq a, Seq b)

-- | &lt;math&gt;. Unzip a sequence using a function to divide elements.
gr
gr<pre>
grunzipWith f xs == <a>unzip</a> (<a>fmap</a> f xs)
gr</pre>
gr
grEfficiency note:
gr
gr<tt>unzipWith</tt> produces its two results in lockstep. If you
grcalculate <tt> unzipWith f xs </tt> and fully force <i>either</i> of
grthe results, then the entire structure of the <i>other</i> one will be
grbuilt as well. This behavior allows the garbage collector to collect
greach calculated pair component as soon as it dies, without having to
grwait for its mate to die. If you do not need this behavior, you may be
grbetter off simply calculating the sequence of pairs and using
gr<a>fmap</a> to extract each component sequence.
unzipWith :: (a -> (b, c)) -> Seq a -> (Seq b, Seq c)


-- | Multi-way trees (<i>aka</i> rose trees) and forests.
module Data.Tree

-- | Multi-way trees, also known as <i>rose trees</i>.
data Tree a
Node :: a -> Forest a -> Tree a

-- | label value
[rootLabel] :: Tree a -> a

-- | zero or more child trees
[subForest] :: Tree a -> Forest a
type Forest a = [Tree a]

-- | Neat 2-dimensional drawing of a tree.
drawTree :: Tree String -> String

-- | Neat 2-dimensional drawing of a forest.
drawForest :: Forest String -> String

-- | The elements of a tree in pre-order.
flatten :: Tree a -> [a]

-- | Lists of nodes at each level of the tree.
levels :: Tree a -> [[a]]

-- | Catamorphism on trees.
foldTree :: (a -> [b] -> b) -> Tree a -> b

-- | Build a tree from a seed value
unfoldTree :: (b -> (a, [b])) -> b -> Tree a

-- | Build a forest from a list of seed values
unfoldForest :: (b -> (a, [b])) -> [b] -> Forest a

-- | Monadic tree builder, in depth-first order
unfoldTreeM :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a)

-- | Monadic forest builder, in depth-first order
unfoldForestM :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a)

-- | Monadic tree builder, in breadth-first order, using an algorithm
gradapted from <i>Breadth-First Numbering: Lessons from a Small Exercise
grin Algorithm Design</i>, by Chris Okasaki, <i>ICFP'00</i>.
unfoldTreeM_BF :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a)

-- | Monadic forest builder, in breadth-first order, using an algorithm
gradapted from <i>Breadth-First Numbering: Lessons from a Small Exercise
grin Algorithm Design</i>, by Chris Okasaki, <i>ICFP'00</i>.
unfoldForestM_BF :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a)
instance GHC.Generics.Generic1 Data.Tree.Tree
instance GHC.Generics.Generic (Data.Tree.Tree a)
instance Data.Data.Data a => Data.Data.Data (Data.Tree.Tree a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Tree.Tree a)
instance GHC.Read.Read a => GHC.Read.Read (Data.Tree.Tree a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Tree.Tree a)
instance Data.Functor.Classes.Eq1 Data.Tree.Tree
instance Data.Functor.Classes.Ord1 Data.Tree.Tree
instance Data.Functor.Classes.Show1 Data.Tree.Tree
instance Data.Functor.Classes.Read1 Data.Tree.Tree
instance GHC.Base.Functor Data.Tree.Tree
instance GHC.Base.Applicative Data.Tree.Tree
instance GHC.Base.Monad Data.Tree.Tree
instance Control.Monad.Fix.MonadFix Data.Tree.Tree
instance Data.Traversable.Traversable Data.Tree.Tree
instance Data.Foldable.Foldable Data.Tree.Tree
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Tree.Tree a)
instance Control.Monad.Zip.MonadZip Data.Tree.Tree


-- | A version of the graph algorithms described in:
gr
gr<i>Structuring Depth-First Search Algorithms in Haskell</i>, by David
grKing and John Launchbury.
module Data.Graph

-- | The strongly connected components of a directed graph, topologically
grsorted.
stronglyConnComp :: Ord key => [(node, key, [key])] -> [SCC node]

-- | The strongly connected components of a directed graph, topologically
grsorted. The function is the same as <a>stronglyConnComp</a>, except
grthat all the information about each node retained. This interface is
grused when you expect to apply <a>SCC</a> to (some of) the result of
gr<a>SCC</a>, so you don't want to lose the dependency information.
stronglyConnCompR :: Ord key => [(node, key, [key])] -> [SCC (node, key, [key])]

-- | Strongly connected component.
data SCC vertex

-- | A single vertex that is not in any cycle.
AcyclicSCC :: vertex -> SCC vertex

-- | A maximal set of mutually reachable vertices.
CyclicSCC :: [vertex] -> SCC vertex

-- | The vertices of a strongly connected component.
flattenSCC :: SCC vertex -> [vertex]

-- | The vertices of a list of strongly connected components.
flattenSCCs :: [SCC a] -> [a]

-- | Adjacency list representation of a graph, mapping each vertex to its
grlist of successors.
type Graph = Table [Vertex]

-- | Table indexed by a contiguous set of vertices.
type Table a = Array Vertex a

-- | The bounds of a <a>Table</a>.
type Bounds = (Vertex, Vertex)

-- | An edge from the first vertex to the second.
type Edge = (Vertex, Vertex)

-- | Abstract representation of vertices.
type Vertex = Int

-- | Build a graph from a list of nodes uniquely identified by keys, with a
grlist of keys of nodes this node should have edges to. The out-list may
grcontain keys that don't correspond to nodes of the graph; they are
grignored.
graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)

-- | Identical to <a>graphFromEdges</a>, except that the return value does
grnot include the function which maps keys to vertices. This version of
gr<a>graphFromEdges</a> is for backwards compatibility.
graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))

-- | Build a graph from a list of edges.
buildG :: Bounds -> [Edge] -> Graph

-- | The graph obtained by reversing all edges.
transposeG :: Graph -> Graph

-- | All vertices of a graph.
vertices :: Graph -> [Vertex]

-- | All edges of a graph.
edges :: Graph -> [Edge]

-- | A table of the count of edges from each node.
outdegree :: Graph -> Table Int

-- | A table of the count of edges into each node.
indegree :: Graph -> Table Int

-- | A spanning forest of the part of the graph reachable from the listed
grvertices, obtained from a depth-first search of the graph starting at
greach of the listed vertices in order.
dfs :: Graph -> [Vertex] -> Forest Vertex

-- | A spanning forest of the graph, obtained from a depth-first search of
grthe graph starting from each vertex in an unspecified order.
dff :: Graph -> Forest Vertex

-- | A topological sort of the graph. The order is partially specified by
grthe condition that a vertex <i>i</i> precedes <i>j</i> whenever
gr<i>j</i> is reachable from <i>i</i> but not vice versa.
topSort :: Graph -> [Vertex]

-- | The connected components of a graph. Two vertices are connected if
grthere is a path between them, traversing edges in either direction.
components :: Graph -> Forest Vertex

-- | The strongly connected components of a graph.
scc :: Graph -> Forest Vertex

-- | The biconnected components of a graph. An undirected graph is
grbiconnected if the deletion of any vertex leaves it connected.
bcc :: Graph -> Forest [Vertex]

-- | A list of vertices reachable from a given vertex.
reachable :: Graph -> Vertex -> [Vertex]

-- | Is the second vertex reachable from the first?
path :: Graph -> Vertex -> Vertex -> Bool
type Forest a = [Tree a]

-- | Multi-way trees, also known as <i>rose trees</i>.
data Tree a
Node :: a -> Forest a -> Tree a
instance GHC.Read.Read vertex => GHC.Read.Read (Data.Graph.SCC vertex)
instance GHC.Show.Show vertex => GHC.Show.Show (Data.Graph.SCC vertex)
instance GHC.Classes.Eq vertex => GHC.Classes.Eq (Data.Graph.SCC vertex)
instance Data.Data.Data vertex => Data.Data.Data (Data.Graph.SCC vertex)
instance GHC.Generics.Generic1 Data.Graph.SCC
instance GHC.Generics.Generic (Data.Graph.SCC vertex)
instance GHC.Base.Monad (Data.Graph.SetM s)
instance GHC.Base.Functor (Data.Graph.SetM s)
instance GHC.Base.Applicative (Data.Graph.SetM s)
instance Data.Functor.Classes.Eq1 Data.Graph.SCC
instance Data.Functor.Classes.Show1 Data.Graph.SCC
instance Data.Functor.Classes.Read1 Data.Graph.SCC
instance Data.Foldable.Foldable Data.Graph.SCC
instance Data.Traversable.Traversable Data.Graph.SCC
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Graph.SCC a)
instance GHC.Base.Functor Data.Graph.SCC


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grAn efficient implementation of maps from keys to values
gr(dictionaries).
gr
grSince many function names (but not the type name) clash with
gr<a>Prelude</a> names, this module is usually imported
gr<tt>qualified</tt>, e.g.
gr
gr<pre>
grimport Data.Map (Map)
grimport qualified Data.Map as Map
gr</pre>
gr
grThe implementation of <a>Map</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
gr
grNote that the implementation is <i>left-biased</i> -- the elements of
gra first argument are always preferred to the second, for example in
gr<a>union</a> or <a>insert</a>.
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation <a>http://en.wikipedia.org/wiki/Big_O_notation</a>.
module Data.Map.Internal

-- | A Map from keys <tt>k</tt> to values <tt>a</tt>.
data Map k a
Bin :: {-# UNPACK #-} !Size -> !k -> a -> !(Map k a) -> !(Map k a) -> Map k a
Tip :: Map k a
type Size = Int

-- | <i>O(log n)</i>. Find the value at a key. Calls <a>error</a> when the
grelement can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: Ord k => Map k a -> k -> a
infixl 9 !

-- | <i>O(log n)</i>. Find the value at a key. Returns <a>Nothing</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 1 == Nothing
gr</pre>
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: Ord k => Map k a -> k -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.Map.null (empty)           == True
grData.Map.null (singleton 1 'a') == False
gr</pre>
null :: Map k a -> Bool

-- | <i>O(1)</i>. The number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: Map k a -> Int

-- | <i>O(log n)</i>. Is the key a member of the map? See also
gr<a>notMember</a>.
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Is the key not a member of the map? See also
gr<a>member</a>.
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Lookup the value at a key in the map.
gr
grThe function will return the corresponding value as <tt>(<a>Just</a>
grvalue)</tt>, or <a>Nothing</a> if the key isn't in the map.
gr
grAn example of using <tt>lookup</tt>:
gr
gr<pre>
grimport Prelude hiding (lookup)
grimport Data.Map
gr
gremployeeDept = fromList([("John","Sales"), ("Bob","IT")])
grdeptCountry = fromList([("IT","USA"), ("Sales","France")])
grcountryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])
gr
gremployeeCurrency :: String -&gt; Maybe String
gremployeeCurrency name = do
gr    dept &lt;- lookup name employeeDept
gr    country &lt;- lookup dept deptCountry
gr    lookup country countryCurrency
gr
grmain = do
gr    putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))
gr    putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
gr</pre>
gr
grThe output of this program:
gr
gr<pre>
grJohn's currency: Just "Euro"
grPete's currency: Nothing
gr</pre>
lookup :: Ord k => k -> Map k a -> Maybe a

-- | <i>O(log n)</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns default value
gr<tt>def</tt> when the key is not in the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: Ord k => a -> k -> Map k a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: Map k a

-- | <i>O(1)</i>. A map with a single element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: k -> a -> Map k a

-- | <i>O(log n)</i>. Insert a new key and value in the map. If the key is
gralready present in the map, the associated value is replaced with the
grsupplied value. <a>insert</a> is equivalent to <tt><a>insertWith</a>
gr<a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Ord k => k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining new value and old
grvalue. <tt><a>insertWith</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert the pair <tt>(key, f
grnew_value old_value)</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining key, new value and
grold value. <tt><a>insertWithKey</a> f key value mp</tt> will insert
grthe pair (key, value) into <tt>mp</tt> if key does not exist in the
grmap. If the key does exist, the function will insert the pair
gr<tt>(key,f key new_value old_value)</tt>. Note that the key passed to
grf is the same key passed to <a>insertWithKey</a>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Combines insert operation with old value retrieval.
grThe expression (<tt><a>insertLookupWithKey</a> f k x map</tt>) is a
grpair where the first element is equal to (<tt><a>lookup</a> k
grmap</tt>) and the second element equal to (<tt><a>insertWithKey</a> f
grk x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. Delete a key and its value from the map. When the key
gris not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Ord k => k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update a value at a specific key with the result of
grthe provided function. When the key is not a member of the map, the
groriginal map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Adjust a value at a specific key. When the key is not
gra member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>updateWithKey</a> f k
grmap</tt>) updates the value <tt>x</tt> at <tt>k</tt> (if it is in the
grmap). If (<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted.
grIf it is (<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the
grnew value <tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Lookup and update. See also <a>updateWithKey</a>. The
grfunction returns changed value, if it is updated. Returns the original
grkey value if the map entry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>alter</a> f k map</tt>) alters
grthe value <tt>x</tt> at <tt>k</tt>, or absence thereof. <a>alter</a>
grcan be used to insert, delete, or update a value in a <a>Map</a>. In
grshort : <tt><a>lookup</a> k (<a>alter</a> f k m) = f (<a>lookup</a> k
grm)</tt>.
gr
gr<pre>
grlet f _ = Nothing
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr
grlet f _ = Just "c"
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]
gr</pre>
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in a <a>Map</a>. In short: <tt><a>lookup</a> k &lt;$&gt;
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; Map Int String -&gt; IO (Map Int String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map. When used with
grtrivial functors like <a>Identity</a> and <a>Const</a>, it is often
grslightly slower than more specialized combinators like <a>lookup</a>
grand <a>insert</a>. However, when the functor is non-trivial and key
grcomparison is not particularly cheap, it is the fastest way.
gr
grNote on rewrite rules:
gr
grThis module includes GHC rewrite rules to optimize <a>alterF</a> for
grthe <a>Const</a> and <a>Identity</a> functors. In general, these rules
grimprove performance. The sole exception is that when using
gr<a>Identity</a>, deleting a key that is already absent takes longer
grthan it would without the rules. If you expect this to occur a very
grlarge fraction of the time, you might consider using a private copy of
grthe <a>Identity</a> type.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: (Functor f, Ord k) => (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression (<tt><a>union</a>
grt1 t2</tt>) takes the left-biased union of <tt>t1</tt> and
gr<tt>t2</tt>. It prefers <tt>t1</tt> when duplicate keys are
grencountered, i.e. (<tt><a>union</a> == <a>unionWith</a>
gr<a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: Ord k => Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | The union of a list of maps: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: Ord k => [Map k a] -> Map k a

-- | The union of a list of maps, with a combining operation:
gr(<tt><a>unionsWith</a> f == <a>foldl</a> (<a>unionWith</a> f)
gr<a>empty</a></tt>).
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two maps. Return
grelements of the first map not existing in the second map.
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the values
grof these keys. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection of two maps. Return
grdata in the first map for the keys existing in both maps.
gr(<tt><a>intersection</a> m1 m2 == <a>intersectionWith</a> <a>const</a>
grm1 m2</tt>).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMissing k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; Maybe z
gr</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMatched k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt;
grMaybe z </tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f k x y z</tt> and <tt>k -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f k x y</tt> and <tt>k -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f k x y -> k -> x -> f (Maybe y)

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: Ord k => SimpleWhenMissing k a c -> SimpleWhenMissing k b c -> SimpleWhenMatched k a b c -> Map k a -> Map k b -> Map k c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched :: (k -&gt; x -&gt; y -&gt; Maybe z)
gr                    -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (k -> x -> y -> Maybe z) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched :: (k -&gt; x -&gt; y -&gt; z)
gr               -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMatched :: Applicative f => (k -> x -> y -> z) -> WhenMatched f k x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (k -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (k -> x -> Maybe y) -> WhenMissing f k x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f k x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f k x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (k -> x -> y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (k -> x -> Bool) -> WhenMissing f k x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt> WhenMissing f k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; f (Maybe z)
gr</tt>.
data WhenMissing f k x y
WhenMissing :: Map k x -> f (Map k y) -> k -> x -> f (Maybe y) -> WhenMissing f k x y
[missingSubtree] :: WhenMissing f k x y -> Map k x -> f (Map k y)
[missingKey] :: WhenMissing f k x y -> k -> x -> f (Maybe y)

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt> WhenMatched f k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt; f
gr(Maybe z) </tt>.
newtype WhenMatched f k x y z
WhenMatched :: k -> x -> y -> f (Maybe z) -> WhenMatched f k x y z
[matchedKey] :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f, Ord k) => WhenMissing f k a c -> WhenMissing f k b c -> WhenMatched f k a b c -> Map k a -> Map k b -> f (Map k c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: (k -> x -> y -> f (Maybe z)) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (k -> x -> y -> f z) -> WhenMatched f k x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (k -> x -> f (Maybe y)) -> WhenMissing f k x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (k -> x -> f y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  k x -&gt; (b -&gt; guard b *&gt; Just x) <a>$</a> f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (k -> x -> f Bool) -> WhenMissing f k x x

-- | <i>O(n+m)</i>. An unsafe general combining function.
gr
grWARNING: This function can produce corrupt maps and its results may
grdepend on the internal structures of its inputs. Users should prefer
gr<a>merge</a> or <a>mergeA</a>.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>Map</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt>,
gr<tt><a>filterWithKey</a> f</tt>, or <tt><a>mapMaybeWithKey</a> f</tt>
grcould be used for any <tt>f</tt>.
mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f m == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (,) k <a>$</a> f k v)
gr(<a>toList</a> m)</tt> That is, behaves exactly like a regular
gr<a>traverse</a> except that the traversing function also has access to
grthe key associated with a value.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)

-- | <i>O(n)</i>. Traverse keys/values and collect the <a>Just</a> results.
traverseMaybeWithKey :: Applicative f => (k -> a -> f (Maybe b)) -> Map k a -> f (Map k b)

-- | <i>O(n)</i>. The function <a>mapAccum</a> threads an accumulating
grargument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <a>mapAccumWithKey</a> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <tt>mapAccumR</tt> threads an accumulating
grargument through the map in descending order of keys.
mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n*log n)</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained by
grapplying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n*log n)</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>. The value at the greater of the two
groriginal keys is used as the first argument to <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. <tt><a>mapKeysMonotonic</a> f s == <a>mapKeys</a> f
grs</tt>, but works only when <tt>f</tt> is strictly monotonic. That is,
grfor any values <tt>x</tt> and <tt>y</tt>, if <tt>x</tt> &lt;
gr<tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The precondition is
grnot checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
grvalid (mapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")])) == True
grvalid (mapKeysMonotonic (\ _ -&gt; 1)     (fromList [(5,"a"), (3,"b")])) == False
gr</pre>
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (k -> a -> m) -> Map k a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: Map k a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: Map k a -> [k]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Return all key/value pairs
grin the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. The set of all keys of the map.
gr
gr<pre>
grkeysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
grkeysSet empty == Data.Set.empty
gr</pre>
keysSet :: Map k a -> Set k

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.Set.empty == empty
gr</pre>
fromSet :: (k -> a) -> Set k -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: Map k a -> [(k, a)]

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs. See
gralso <a>fromAscList</a>. If the list contains more than one value for
grthe same key, the last value for the key is retained.
gr
grIf the keys of the list are ordered, linear-time implementation is
grused, with the performance equal to <a>fromDistinctAscList</a>.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: Ord k => [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWithKey</a>.
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ a1 ++ a2
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Build a map from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
grvalid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctAscList [(3,"b"), (5,"a")])          == True
grvalid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False
gr</pre>
fromDistinctAscList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
gr
gr<pre>
grfromDescList [(5,"a"), (3,"b")]          == fromList [(3, "b"), (5, "a")]
grfromDescList [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grfromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromDescListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctDescList [(5,"a"), (3,"b")])          == True
grvalid (fromDistinctDescList [(5,"a"), (5,"b"), (3,"b")]) == False
gr</pre>
fromDistinctDescList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Filter all values that satisfy the predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> Map k a -> Map k a

-- | <i>O(n)</i>. Filter all keys/values that satisfy the predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Take while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> (p . fst) . <a>toList</a>
grtakeWhileAntitone p = <a>filterWithKey</a> (k _ -&gt; p k)
gr</pre>
takeWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Drop while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> (p . fst) . <a>toList</a>
grdropWhileAntitone p = <a>filterWithKey</a> (k -&gt; not (p k))
gr</pre>
dropWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Divide a map at the point where a predicate on the
grkeys stops holding. The user is responsible for ensuring that for all
grkeys <tt>j</tt> and <tt>k</tt> in the map, <tt>j &lt; k ==&gt; p j
gr&gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the map at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first key and to fail after
grthe last key).
spanAntitone :: (k -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Restrict a <a>Map</a> to only
grthose keys found in a <a>Set</a>.
gr
gr<pre>
grm `<tt>restrictKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
grm `<tt>restrictKeys'</tt> s = m `<tt>intersect</tt> <a>fromSet</a> (const ()) s
gr</pre>
restrictKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Remove all keys in a <a>Set</a>
grfrom a <a>Map</a>.
gr
gr<pre>
grm `<tt>withoutKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
grm `<tt>withoutKeys'</tt> s = m `<tt>difference'</tt> <a>fromSet</a> (const ()) s
gr</pre>
withoutKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where the keys in <tt>map1</tt> are smaller
grthan <tt>k</tt> and the keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Ord k => k -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>splitLookup</a> k map</tt>)
grsplits a map just like <a>split</a> but also returns <tt><a>lookup</a>
grk map</tt>.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d')],fromList [(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Map k b -> [Map k b]

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. This function is defined as
gr(<tt><a>isSubmapOf</a> = <a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression
gr(<tt><a>isSubmapOfBy</a> f t1 t2</tt>) returns <a>True</a> if all keys
grin <tt>t1</tt> are in tree <tt>t2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
gr</pre>
isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). Defined as (<tt><a>isProperSubmapOf</a> =
gr<a>isProperSubmapOfBy</a> (==)</tt>).
isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). The expression (<tt><a>isProperSubmapOfBy</a> f
grm1 m2</tt>) returns <a>True</a> when <tt>m1</tt> and <tt>m2</tt> are
grnot equal, all keys in <tt>m1</tt> are in <tt>m2</tt>, and when
gr<tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(log n)</i>. Lookup the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
gr
gr<pre>
grisJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False
grfromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
grfromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
grisJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False
gr</pre>
lookupIndex :: Ord k => k -> Map k a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
grCalls <a>error</a> when the key is not a <a>member</a> of the map.
gr
gr<pre>
grfindIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
grfindIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
grfindIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
grfindIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
gr</pre>
findIndex :: Ord k => k -> Map k a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
grelemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
grelemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
elemAt :: Int -> Map k a -> (k, a)

-- | <i>O(log n)</i>. Update the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grupdateAt (\ _ _ -&gt; Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
grupdateAt (\ _ _ -&gt; Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
grupdateAt (\ _ _ -&gt; Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\ _ _ -&gt; Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grupdateAt (\_ _  -&gt; Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grupdateAt (\_ _  -&gt; Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grdeleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdeleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range
grdeleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range
gr</pre>
deleteAt :: Int -> Map k a -> Map k a

-- | Take a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Map k a -> Map k a

-- | Drop a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Split a map at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The minimal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b")
grfindMin empty = Nothing
gr</pre>
lookupMin :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The maximal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMax (fromList [(5,"a"), (3,"b")]) == Just (5,"a")
grlookupMax empty = Nothing
gr</pre>
lookupMax :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The minimal key of the map. Calls <a>error</a> if the
grmap is empty.
gr
gr<pre>
grfindMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
grfindMin empty                            Error: empty map has no minimal element
gr</pre>
findMin :: Map k a -> (k, a)
findMax :: Map k a -> (k, a)

-- | <i>O(log n)</i>. Delete the minimal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
grdeleteMin empty == empty
gr</pre>
deleteMin :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the maximal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
grdeleteMax empty == empty
gr</pre>
deleteMax :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
grdeleteFindMin                                            Error: can not return the minimal element of an empty map
gr</pre>
deleteFindMin :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
grdeleteFindMax empty                                      Error: can not return the maximal element of an empty map
gr</pre>
deleteFindMax :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Retrieves the value associated with minimal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grminView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
grminView empty == Nothing
gr</pre>
minView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the value associated with maximal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grmaxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
grmaxView empty == Nothing
gr</pre>
maxView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(log n)</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)
data AreWeStrict
Strict :: AreWeStrict
Lazy :: AreWeStrict
atKeyImpl :: (Functor f, Ord k) => AreWeStrict -> k -> (Maybe a -> f (Maybe a)) -> Map k a -> f (Map k a)
atKeyPlain :: Ord k => AreWeStrict -> k -> (Maybe a -> Maybe a) -> Map k a -> Map k a
bin :: k -> a -> Map k a -> Map k a -> Map k a
balance :: k -> a -> Map k a -> Map k a -> Map k a
balanceL :: k -> a -> Map k a -> Map k a -> Map k a
balanceR :: k -> a -> Map k a -> Map k a -> Map k a
delta :: Int
insertMax :: k -> a -> Map k a -> Map k a
link :: k -> a -> Map k a -> Map k a -> Map k a
link2 :: Map k a -> Map k a -> Map k a
glue :: Map k a -> Map k a -> Map k a
data MaybeS a
NothingS :: MaybeS a
JustS :: !a -> MaybeS a

-- | Identity functor and monad. (a non-strict monad)
newtype Identity a
Identity :: a -> Identity a
[runIdentity] :: Identity a -> a

-- | Map covariantly over a <tt><a>WhenMissing</a> f k x</tt>.
mapWhenMissing :: (Applicative f, Monad f) => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b

-- | Map contravariantly over a <tt><a>WhenMissing</a> f k _ x</tt>.
lmapWhenMissing :: (b -> a) -> WhenMissing f k a x -> WhenMissing f k b x

-- | Map contravariantly over a <tt><a>WhenMatched</a> f k _ y z</tt>.
contramapFirstWhenMatched :: (b -> a) -> WhenMatched f k a y z -> WhenMatched f k b y z

-- | Map contravariantly over a <tt><a>WhenMatched</a> f k x _ z</tt>.
contramapSecondWhenMatched :: (b -> a) -> WhenMatched f k x a z -> WhenMatched f k x b z

-- | Map covariantly over a <tt><a>WhenMissing</a> f k x</tt>, using only a
gr'Functor f' constraint.
mapGentlyWhenMissing :: Functor f => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x</tt>, using only a
gr'Functor f' constraint.
mapGentlyWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b
instance GHC.Base.Functor f => GHC.Base.Functor (Data.Map.Internal.WhenMatched f k x y)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => Control.Category.Category (Data.Map.Internal.WhenMatched f k x)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => GHC.Base.Applicative (Data.Map.Internal.WhenMatched f k x y)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => GHC.Base.Monad (Data.Map.Internal.WhenMatched f k x y)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Functor (Data.Map.Internal.WhenMissing f k x)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => Control.Category.Category (Data.Map.Internal.WhenMissing f k)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Applicative (Data.Map.Internal.WhenMissing f k x)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Monad (Data.Map.Internal.WhenMissing f k x)
instance GHC.Classes.Ord k => GHC.Base.Monoid (Data.Map.Internal.Map k v)
instance GHC.Classes.Ord k => GHC.Base.Semigroup (Data.Map.Internal.Map k v)
instance (Data.Data.Data k, Data.Data.Data a, GHC.Classes.Ord k) => Data.Data.Data (Data.Map.Internal.Map k a)
instance GHC.Classes.Ord k => GHC.Exts.IsList (Data.Map.Internal.Map k v)
instance (GHC.Classes.Eq k, GHC.Classes.Eq a) => GHC.Classes.Eq (Data.Map.Internal.Map k a)
instance (GHC.Classes.Ord k, GHC.Classes.Ord v) => GHC.Classes.Ord (Data.Map.Internal.Map k v)
instance Data.Functor.Classes.Eq2 Data.Map.Internal.Map
instance GHC.Classes.Eq k => Data.Functor.Classes.Eq1 (Data.Map.Internal.Map k)
instance Data.Functor.Classes.Ord2 Data.Map.Internal.Map
instance GHC.Classes.Ord k => Data.Functor.Classes.Ord1 (Data.Map.Internal.Map k)
instance Data.Functor.Classes.Show2 Data.Map.Internal.Map
instance GHC.Show.Show k => Data.Functor.Classes.Show1 (Data.Map.Internal.Map k)
instance (GHC.Classes.Ord k, GHC.Read.Read k) => Data.Functor.Classes.Read1 (Data.Map.Internal.Map k)
instance GHC.Base.Functor (Data.Map.Internal.Map k)
instance Data.Traversable.Traversable (Data.Map.Internal.Map k)
instance Data.Foldable.Foldable (Data.Map.Internal.Map k)
instance (Control.DeepSeq.NFData k, Control.DeepSeq.NFData a) => Control.DeepSeq.NFData (Data.Map.Internal.Map k a)
instance (GHC.Classes.Ord k, GHC.Read.Read k, GHC.Read.Read e) => GHC.Read.Read (Data.Map.Internal.Map k e)
instance (GHC.Show.Show k, GHC.Show.Show a) => GHC.Show.Show (Data.Map.Internal.Map k a)


-- | This module defines an API for writing functions that merge two maps.
grThe key functions are <a>merge</a> and <a>mergeA</a>. Each of these
grcan be used with several different "merge tactics".
gr
grThe <a>merge</a> and <a>mergeA</a> functions are shared by the lazy
grand strict modules. Only the choice of merge tactics determines
grstrictness. If you use <a>mapMissing</a> from
gr<a>Data.Map.Merge.Strict</a> then the results will be forced before
grthey are inserted. If you use <a>mapMissing</a> from this module then
grthey will not.
gr
gr<h2>Efficiency note</h2>
gr
grThe <tt>Category</tt>, <a>Applicative</a>, and <a>Monad</a> instances
grfor <a>WhenMissing</a> tactics are included because they are valid.
grHowever, they are inefficient in many cases and should usually be
gravoided. The instances for <a>WhenMatched</a> tactics should not pose
grany major efficiency problems.
module Data.Map.Merge.Lazy

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMissing k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; Maybe z
gr</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMatched k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt;
grMaybe z </tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: Ord k => SimpleWhenMissing k a c -> SimpleWhenMissing k b c -> SimpleWhenMatched k a b c -> Map k a -> Map k b -> Map k c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched :: (k -&gt; x -&gt; y -&gt; Maybe z)
gr                    -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (k -> x -> y -> Maybe z) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched :: (k -&gt; x -&gt; y -&gt; z)
gr               -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMatched :: Applicative f => (k -> x -> y -> z) -> WhenMatched f k x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (k -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (k -> x -> Maybe y) -> WhenMissing f k x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f k x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f k x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (k -> x -> y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (k -> x -> Bool) -> WhenMissing f k x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt> WhenMissing f k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; f (Maybe z)
gr</tt>.
data WhenMissing f k x y

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt> WhenMatched f k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt; f
gr(Maybe z) </tt>.
data WhenMatched f k x y z

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f, Ord k) => WhenMissing f k a c -> WhenMissing f k b c -> WhenMatched f k a b c -> Map k a -> Map k b -> f (Map k c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: (k -> x -> y -> f (Maybe z)) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (k -> x -> y -> f z) -> WhenMatched f k x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (k -> x -> f (Maybe y)) -> WhenMissing f k x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (k -> x -> f y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  k x -&gt; (b -&gt; guard b *&gt; Just x) <a>$</a> f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (k -> x -> f Bool) -> WhenMissing f k x x

-- | Map covariantly over a <tt><a>WhenMissing</a> f k x</tt>.
mapWhenMissing :: (Applicative f, Monad f) => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b

-- | Map contravariantly over a <tt><a>WhenMissing</a> f k _ x</tt>.
lmapWhenMissing :: (b -> a) -> WhenMissing f k a x -> WhenMissing f k b x

-- | Map contravariantly over a <tt><a>WhenMatched</a> f k _ y z</tt>.
contramapFirstWhenMatched :: (b -> a) -> WhenMatched f k a y z -> WhenMatched f k b y z

-- | Map contravariantly over a <tt><a>WhenMatched</a> f k x _ z</tt>.
contramapSecondWhenMatched :: (b -> a) -> WhenMatched f k x a z -> WhenMatched f k x b z

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f k x y z</tt> and <tt>k -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f k x y</tt> and <tt>k -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f k x y -> k -> x -> f (Maybe y)

module Data.Map.Internal.Debug

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format. See <a>showTreeWith</a>.
showTree :: (Show k, Show a) => Map k a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> showelem hang
grwide map</tt>) shows the tree that implements the map. Elements are
grshown using the <tt>showElem</tt> function. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grMap&gt; let t = fromDistinctAscList [(x,()) | x &lt;- [1..5]]
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True False t
gr(4,())
gr+--(2,())
gr|  +--(1,())
gr|  +--(3,())
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True True t
gr(4,())
gr|
gr+--(2,())
gr|  |
gr|  +--(1,())
gr|  |
gr|  +--(3,())
gr|
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) False True t
gr+--(5,())
gr|
gr(4,())
gr|
gr|  +--(3,())
gr|  |
gr+--(2,())
gr   |
gr   +--(1,())
gr</pre>
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String
showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS
showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS
showWide :: Bool -> [String] -> String -> String
showsBars :: [String] -> ShowS
node :: String
withBar :: [String] -> [String]
withEmpty :: [String] -> [String]

-- | <i>O(n)</i>. Test if the internal map structure is valid.
gr
gr<pre>
grvalid (fromAscList [(3,"b"), (5,"a")]) == True
grvalid (fromAscList [(5,"a"), (3,"b")]) == False
gr</pre>
valid :: Ord k => Map k a -> Bool

-- | Test if the keys are ordered correctly.
ordered :: Ord a => Map a b -> Bool

-- | Test if a map obeys the balance invariants.
balanced :: Map k a -> Bool

-- | Test if each node of a map reports its size correctly.
validsize :: Map a b -> Bool


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grAn efficient implementation of ordered maps from keys to values
gr(dictionaries).
gr
grAPI of this module is strict in both the keys and the values. If you
grneed value-lazy maps, use <a>Data.Map.Lazy</a> instead. The <a>Map</a>
grtype is shared between the lazy and strict modules, meaning that the
grsame <a>Map</a> value can be passed to functions in both modules
gr(although that is rarely needed).
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport qualified Data.Map.Strict as Map
gr</pre>
gr
grThe implementation of <a>Map</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
gr
grNote that the implementation is <i>left-biased</i> -- the elements of
gra first argument are always preferred to the second, for example in
gr<a>union</a> or <a>insert</a>.
gr
gr<i>Warning</i>: The size of the map must not exceed
gr<tt>maxBound::Int</tt>. Violation of this condition is not detected
grand if the size limit is exceeded, its behaviour is undefined.
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation (<a>http://en.wikipedia.org/wiki/Big_O_notation</a>).
gr
grBe aware that the <a>Functor</a>, <a>Traversable</a> and <tt>Data</tt>
grinstances are the same as for the <a>Data.Map.Lazy</a> module, so if
grthey are used on strict maps, the resulting maps will be lazy.
module Data.Map.Strict.Internal

-- | A Map from keys <tt>k</tt> to values <tt>a</tt>.
data Map k a
Bin :: {-# UNPACK #-} !Size -> !k -> a -> !(Map k a) -> !(Map k a) -> Map k a
Tip :: Map k a
type Size = Int

-- | <i>O(log n)</i>. Find the value at a key. Calls <a>error</a> when the
grelement can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: Ord k => Map k a -> k -> a
infixl 9 !

-- | <i>O(log n)</i>. Find the value at a key. Returns <a>Nothing</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 1 == Nothing
gr</pre>
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: Ord k => Map k a -> k -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.Map.null (empty)           == True
grData.Map.null (singleton 1 'a') == False
gr</pre>
null :: Map k a -> Bool

-- | <i>O(1)</i>. The number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: Map k a -> Int

-- | <i>O(log n)</i>. Is the key a member of the map? See also
gr<a>notMember</a>.
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Is the key not a member of the map? See also
gr<a>member</a>.
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Lookup the value at a key in the map.
gr
grThe function will return the corresponding value as <tt>(<a>Just</a>
grvalue)</tt>, or <a>Nothing</a> if the key isn't in the map.
gr
grAn example of using <tt>lookup</tt>:
gr
gr<pre>
grimport Prelude hiding (lookup)
grimport Data.Map
gr
gremployeeDept = fromList([("John","Sales"), ("Bob","IT")])
grdeptCountry = fromList([("IT","USA"), ("Sales","France")])
grcountryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])
gr
gremployeeCurrency :: String -&gt; Maybe String
gremployeeCurrency name = do
gr    dept &lt;- lookup name employeeDept
gr    country &lt;- lookup dept deptCountry
gr    lookup country countryCurrency
gr
grmain = do
gr    putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))
gr    putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
gr</pre>
gr
grThe output of this program:
gr
gr<pre>
grJohn's currency: Just "Euro"
grPete's currency: Nothing
gr</pre>
lookup :: Ord k => k -> Map k a -> Maybe a

-- | <i>O(log n)</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns default value
gr<tt>def</tt> when the key is not in the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: Ord k => a -> k -> Map k a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: Map k a

-- | <i>O(1)</i>. A map with a single element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: k -> a -> Map k a

-- | <i>O(log n)</i>. Insert a new key and value in the map. If the key is
gralready present in the map, the associated value is replaced with the
grsupplied value. <a>insert</a> is equivalent to <tt><a>insertWith</a>
gr<a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Ord k => k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining new value and old
grvalue. <tt><a>insertWith</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert the pair <tt>(key, f
grnew_value old_value)</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining key, new value and
grold value. <tt><a>insertWithKey</a> f key value mp</tt> will insert
grthe pair (key, value) into <tt>mp</tt> if key does not exist in the
grmap. If the key does exist, the function will insert the pair
gr<tt>(key,f key new_value old_value)</tt>. Note that the key passed to
grf is the same key passed to <a>insertWithKey</a>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Combines insert operation with old value retrieval.
grThe expression (<tt><a>insertLookupWithKey</a> f k x map</tt>) is a
grpair where the first element is equal to (<tt><a>lookup</a> k
grmap</tt>) and the second element equal to (<tt><a>insertWithKey</a> f
grk x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. Delete a key and its value from the map. When the key
gris not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Ord k => k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update a value at a specific key with the result of
grthe provided function. When the key is not a member of the map, the
groriginal map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Adjust a value at a specific key. When the key is not
gra member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>updateWithKey</a> f k
grmap</tt>) updates the value <tt>x</tt> at <tt>k</tt> (if it is in the
grmap). If (<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted.
grIf it is (<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the
grnew value <tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Lookup and update. See also <a>updateWithKey</a>. The
grfunction returns changed value, if it is updated. Returns the original
grkey value if the map entry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>alter</a> f k map</tt>) alters
grthe value <tt>x</tt> at <tt>k</tt>, or absence thereof. <a>alter</a>
grcan be used to insert, delete, or update a value in a <a>Map</a>. In
grshort : <tt><a>lookup</a> k (<a>alter</a> f k m) = f (<a>lookup</a> k
grm)</tt>.
gr
gr<pre>
grlet f _ = Nothing
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr
grlet f _ = Just "c"
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]
gr</pre>
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in a <a>Map</a>. In short: <tt><a>lookup</a> k &lt;$&gt;
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; Map Int String -&gt; IO (Map Int String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map. When used with
grtrivial functors like <a>Identity</a> and <a>Const</a>, it is often
grslightly slower than more specialized combinators like <a>lookup</a>
grand <a>insert</a>. However, when the functor is non-trivial and key
grcomparison is not particularly cheap, it is the fastest way.
gr
grNote on rewrite rules:
gr
grThis module includes GHC rewrite rules to optimize <a>alterF</a> for
grthe <a>Const</a> and <a>Identity</a> functors. In general, these rules
grimprove performance. The sole exception is that when using
gr<a>Identity</a>, deleting a key that is already absent takes longer
grthan it would without the rules. If you expect this to occur a very
grlarge fraction of the time, you might consider using a private copy of
grthe <a>Identity</a> type.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: (Functor f, Ord k) => (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression (<tt><a>union</a>
grt1 t2</tt>) takes the left-biased union of <tt>t1</tt> and
gr<tt>t2</tt>. It prefers <tt>t1</tt> when duplicate keys are
grencountered, i.e. (<tt><a>union</a> == <a>unionWith</a>
gr<a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: Ord k => Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | The union of a list of maps: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: Ord k => [Map k a] -> Map k a

-- | The union of a list of maps, with a combining operation:
gr(<tt><a>unionsWith</a> f == <a>foldl</a> (<a>unionWith</a> f)
gr<a>empty</a></tt>).
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two maps. Return
grelements of the first map not existing in the second map.
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the values
grof these keys. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection of two maps. Return
grdata in the first map for the keys existing in both maps.
gr(<tt><a>intersection</a> m1 m2 == <a>intersectionWith</a> <a>const</a>
grm1 m2</tt>).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMissing k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; Maybe z
gr</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMatched k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt;
grMaybe z </tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: Ord k => SimpleWhenMissing k a c -> SimpleWhenMissing k b c -> SimpleWhenMatched k a b c -> Map k a -> Map k b -> Map k c

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f k x y z</tt> and <tt>k -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f k x y</tt> and <tt>k -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f k x y -> k -> x -> f (Maybe y)

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched :: (k -&gt; x -&gt; y -&gt; Maybe z)
gr                    -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (k -> x -> y -> Maybe z) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched :: (k -&gt; x -&gt; y -&gt; z)
gr               -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMatched :: Applicative f => (k -> x -> y -> z) -> WhenMatched f k x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (k -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (k -> x -> Maybe y) -> WhenMissing f k x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f k x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f k x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (k -> x -> y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (k -> x -> Bool) -> WhenMissing f k x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt> WhenMissing f k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; f (Maybe z)
gr</tt>.
data WhenMissing f k x y
WhenMissing :: Map k x -> f (Map k y) -> k -> x -> f (Maybe y) -> WhenMissing f k x y
[missingSubtree] :: WhenMissing f k x y -> Map k x -> f (Map k y)
[missingKey] :: WhenMissing f k x y -> k -> x -> f (Maybe y)

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt> WhenMatched f k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt; f
gr(Maybe z) </tt>.
newtype WhenMatched f k x y z
WhenMatched :: k -> x -> y -> f (Maybe z) -> WhenMatched f k x y z
[matchedKey] :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f, Ord k) => WhenMissing f k a c -> WhenMissing f k b c -> WhenMatched f k a b c -> Map k a -> Map k b -> f (Map k c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: Applicative f => (k -> x -> y -> f (Maybe z)) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (k -> x -> y -> f z) -> WhenMatched f k x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (k -> x -> f (Maybe y)) -> WhenMissing f k x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (k -> x -> f y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  k x -&gt; (b -&gt; guard b *&gt; Just x) <a>$</a> f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (k -> x -> f Bool) -> WhenMissing f k x x

-- | Map covariantly over a <tt><a>WhenMissing</a> f k x</tt>.
mapWhenMissing :: Functor f => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b

-- | <i>O(n+m)</i>. An unsafe universal combining function.
gr
grWARNING: This function can produce corrupt maps and its results may
grdepend on the internal structures of its inputs. Users should prefer
gr<a>merge</a> or <a>mergeA</a>.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>Map</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt> or
gr<tt><a>filterWithKey</a> f</tt> could be used for any <tt>f</tt>.
mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f m == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (v' -&gt; v' <a>seq</a> (k,v'))
gr<a>$</a> f k v) (<a>toList</a> m)</tt> That is, it behaves much like a
grregular <a>traverse</a> except that the traversing function also has
graccess to the key associated with a value and the values are forced
grbefore they are installed in the result map.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)

-- | <i>O(n)</i>. Traverse keys/values and collect the <a>Just</a> results.
traverseMaybeWithKey :: Applicative f => (k -> a -> f (Maybe b)) -> Map k a -> f (Map k b)

-- | <i>O(n)</i>. The function <a>mapAccum</a> threads an accumulating
grargument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <a>mapAccumWithKey</a> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <tt>mapAccumR</tt> threads an accumulating
grargument through the map in descending order of keys.
mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n*log n)</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained by
grapplying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n*log n)</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>. The value at the greater of the two
groriginal keys is used as the first argument to <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. <tt><a>mapKeysMonotonic</a> f s == <a>mapKeys</a> f
grs</tt>, but works only when <tt>f</tt> is strictly monotonic. That is,
grfor any values <tt>x</tt> and <tt>y</tt>, if <tt>x</tt> &lt;
gr<tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The precondition is
grnot checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
grvalid (mapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")])) == True
grvalid (mapKeysMonotonic (\ _ -&gt; 1)     (fromList [(5,"a"), (3,"b")])) == False
gr</pre>
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (k -> a -> m) -> Map k a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: Map k a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: Map k a -> [k]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Return all key/value pairs
grin the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. The set of all keys of the map.
gr
gr<pre>
grkeysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
grkeysSet empty == Data.Set.empty
gr</pre>
keysSet :: Map k a -> Set k

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.Set.empty == empty
gr</pre>
fromSet :: (k -> a) -> Set k -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: Map k a -> [(k, a)]

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs. See
gralso <a>fromAscList</a>. If the list contains more than one value for
grthe same key, the last value for the key is retained.
gr
grIf the keys of the list are ordered, linear-time implementation is
grused, with the performance equal to <a>fromDistinctAscList</a>.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: Ord k => [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWithKey</a>.
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ a1 ++ a2
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Build a map from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
grvalid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctAscList [(3,"b"), (5,"a")])          == True
grvalid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False
gr</pre>
fromDistinctAscList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
gr
gr<pre>
grfromDescList [(5,"a"), (3,"b")]          == fromList [(3, "b"), (5, "a")]
grfromDescList [(5,"a"), (5,"b"), (3,"a")] == fromList [(3, "b"), (5, "b")]
grvalid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grfromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromDescListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctDescList [(5,"a"), (3,"b")])          == True
grvalid (fromDistinctDescList [(5,"a"), (3,"b"), (3,"a")]) == False
gr</pre>
fromDistinctDescList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Filter all values that satisfy the predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> Map k a -> Map k a

-- | <i>O(n)</i>. Filter all keys/values that satisfy the predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Restrict a <a>Map</a> to only
grthose keys found in a <a>Set</a>.
gr
gr<pre>
grm `<tt>restrictKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
grm `<tt>restrictKeys'</tt> s = m `<tt>intersect</tt> <a>fromSet</a> (const ()) s
gr</pre>
restrictKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Remove all keys in a <a>Set</a>
grfrom a <a>Map</a>.
gr
gr<pre>
grm `<tt>withoutKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
grm `<tt>withoutKeys'</tt> s = m `<tt>difference'</tt> <a>fromSet</a> (const ()) s
gr</pre>
withoutKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. Take while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> (p . fst) . <a>toList</a>
grtakeWhileAntitone p = <a>filterWithKey</a> (k _ -&gt; p k)
gr</pre>
takeWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Drop while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> (p . fst) . <a>toList</a>
grdropWhileAntitone p = <a>filterWithKey</a> (k -&gt; not (p k))
gr</pre>
dropWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Divide a map at the point where a predicate on the
grkeys stops holding. The user is responsible for ensuring that for all
grkeys <tt>j</tt> and <tt>k</tt> in the map, <tt>j &lt; k ==&gt; p j
gr&gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the map at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first key and to fail after
grthe last key).
spanAntitone :: (k -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where the keys in <tt>map1</tt> are smaller
grthan <tt>k</tt> and the keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Ord k => k -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>splitLookup</a> k map</tt>)
grsplits a map just like <a>split</a> but also returns <tt><a>lookup</a>
grk map</tt>.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d')],fromList [(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Map k b -> [Map k b]

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. This function is defined as
gr(<tt><a>isSubmapOf</a> = <a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression
gr(<tt><a>isSubmapOfBy</a> f t1 t2</tt>) returns <a>True</a> if all keys
grin <tt>t1</tt> are in tree <tt>t2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
gr</pre>
isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). Defined as (<tt><a>isProperSubmapOf</a> =
gr<a>isProperSubmapOfBy</a> (==)</tt>).
isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). The expression (<tt><a>isProperSubmapOfBy</a> f
grm1 m2</tt>) returns <a>True</a> when <tt>m1</tt> and <tt>m2</tt> are
grnot equal, all keys in <tt>m1</tt> are in <tt>m2</tt>, and when
gr<tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(log n)</i>. Lookup the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
gr
gr<pre>
grisJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False
grfromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
grfromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
grisJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False
gr</pre>
lookupIndex :: Ord k => k -> Map k a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
grCalls <a>error</a> when the key is not a <a>member</a> of the map.
gr
gr<pre>
grfindIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
grfindIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
grfindIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
grfindIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
gr</pre>
findIndex :: Ord k => k -> Map k a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
grelemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
grelemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
elemAt :: Int -> Map k a -> (k, a)

-- | <i>O(log n)</i>. Update the element at <i>index</i>. Calls
gr<a>error</a> when an invalid index is used.
gr
gr<pre>
grupdateAt (\ _ _ -&gt; Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
grupdateAt (\ _ _ -&gt; Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
grupdateAt (\ _ _ -&gt; Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\ _ _ -&gt; Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grupdateAt (\_ _  -&gt; Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grupdateAt (\_ _  -&gt; Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grdeleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdeleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range
grdeleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range
gr</pre>
deleteAt :: Int -> Map k a -> Map k a

-- | Take a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Map k a -> Map k a

-- | Drop a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Split a map at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The minimal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b")
grfindMin empty = Nothing
gr</pre>
lookupMin :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The maximal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMax (fromList [(5,"a"), (3,"b")]) == Just (5,"a")
grlookupMax empty = Nothing
gr</pre>
lookupMax :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The minimal key of the map. Calls <a>error</a> if the
grmap is empty.
gr
gr<pre>
grfindMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
grfindMin empty                            Error: empty map has no minimal element
gr</pre>
findMin :: Map k a -> (k, a)
findMax :: Map k a -> (k, a)

-- | <i>O(log n)</i>. Delete the minimal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
grdeleteMin empty == empty
gr</pre>
deleteMin :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the maximal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
grdeleteMax empty == empty
gr</pre>
deleteMax :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
grdeleteFindMin                                            Error: can not return the minimal element of an empty map
gr</pre>
deleteFindMin :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
grdeleteFindMax empty                                      Error: can not return the maximal element of an empty map
gr</pre>
deleteFindMax :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Retrieves the value associated with minimal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grminView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
grminView empty == Nothing
gr</pre>
minView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the value associated with maximal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grmaxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
grmaxView empty == Nothing
gr</pre>
maxView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(log n)</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format. See <a>showTreeWith</a>.

-- | <i>Deprecated: <a>showTree</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTree :: (Show k, Show a) => Map k a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> showelem hang
grwide map</tt>) shows the tree that implements the map. Elements are
grshown using the <tt>showElem</tt> function. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grMap&gt; let t = fromDistinctAscList [(x,()) | x &lt;- [1..5]]
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True False t
gr(4,())
gr+--(2,())
gr|  +--(1,())
gr|  +--(3,())
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True True t
gr(4,())
gr|
gr+--(2,())
gr|  |
gr|  +--(1,())
gr|  |
gr|  +--(3,())
gr|
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) False True t
gr+--(5,())
gr|
gr(4,())
gr|
gr|  +--(3,())
gr|  |
gr+--(2,())
gr   |
gr   +--(1,())
gr</pre>

-- | <i>Deprecated: <a>showTreeWith</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String

-- | <i>O(n)</i>. Test if the internal map structure is valid.
gr
gr<pre>
grvalid (fromAscList [(3,"b"), (5,"a")]) == True
grvalid (fromAscList [(5,"a"), (3,"b")]) == False
gr</pre>
valid :: Ord k => Map k a -> Bool


-- | <h1>Finite Maps (strict interface)</h1>
gr
grThe <tt><a>Map</a> k v</tt> type represents a finite map (sometimes
grcalled a dictionary) from keys of type <tt>k</tt> to values of type
gr<tt>v</tt>.
gr
grEach function in this module is careful to force values before
grinstalling them in a <a>Map</a>. This is usually more efficient when
grlaziness is not necessary. When laziness <i>is</i> required, use the
grfunctions in <a>Data.Map.Lazy</a>.
gr
grIn particular, the functions in this module obey the following law:
gr
gr<ul>
gr<li>If all values stored in all maps in the arguments are in WHNF,
grthen all values stored in all maps in the results will be in WHNF once
grthose maps are evaluated.</li>
gr</ul>
gr
grWhen deciding if this is the correct data structure to use, consider:
gr
gr<ul>
gr<li>If you are using <tt>Int</tt> keys, you will get much better
grperformance for most operations using <a>Data.IntMap.Strict</a>.</li>
gr<li>If you don't care about ordering, consider use
gr<tt>Data.HashMap.Strict</tt> from the <a>unordered-containers</a>
grpackage instead.</li>
gr</ul>
gr
grFor a walkthrough of the most commonly used functions see the <a>maps
grintroduction</a>.
gr
grThis module is intended to be imported qualified, to avoid name
grclashes with Prelude functions:
gr
gr<pre>
grimport qualified Data.Map.Strict as Map
gr</pre>
gr
grNote that the implementation is generally <i>left-biased</i>.
grFunctions that take two maps as arguments and combine them, such as
gr<a>union</a> and <a>intersection</a>, prefer the values in the first
grargument to those in the second.
gr
gr<h2>Detailed performance information</h2>
gr
grThe amortized running time is given for each operation, with <i>n</i>
grreferring to the number of entries in the map.
gr
grBenchmarks comparing <a>Data.Map.Strict</a> with other dictionary
grimplementations can be found at
gr<a>https://github.com/haskell-perf/dictionaries</a>.
gr
gr<h2>Warning</h2>
gr
grThe size of a <a>Map</a> must not exceed <tt>maxBound::Int</tt>.
grViolation of this condition is not detected and if the size limit is
grexceeded, its behaviour is undefined.
gr
grThe <a>Map</a> type is shared between the lazy and strict modules,
grmeaning that the same <a>Map</a> value can be passed to functions in
grboth modules. This means that the <tt>Functor</tt>,
gr<tt>Traversable</tt> and <tt>Data</tt> instances are the same as for
grthe <a>Data.Map.Lazy</a> module, so if they are used on strict maps,
grthe resulting maps may contain suspended values (thunks).
gr
gr<h2>Implementation</h2>
gr
grThe implementation of <a>Map</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
module Data.Map.Strict

-- | A Map from keys <tt>k</tt> to values <tt>a</tt>.
data Map k a

-- | <i>O(log n)</i>. Find the value at a key. Calls <a>error</a> when the
grelement can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: Ord k => Map k a -> k -> a
infixl 9 !

-- | <i>O(log n)</i>. Find the value at a key. Returns <a>Nothing</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 1 == Nothing
gr</pre>
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: Ord k => Map k a -> k -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.Map.null (empty)           == True
grData.Map.null (singleton 1 'a') == False
gr</pre>
null :: Map k a -> Bool

-- | <i>O(1)</i>. The number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: Map k a -> Int

-- | <i>O(log n)</i>. Is the key a member of the map? See also
gr<a>notMember</a>.
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Is the key not a member of the map? See also
gr<a>member</a>.
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Lookup the value at a key in the map.
gr
grThe function will return the corresponding value as <tt>(<a>Just</a>
grvalue)</tt>, or <a>Nothing</a> if the key isn't in the map.
gr
grAn example of using <tt>lookup</tt>:
gr
gr<pre>
grimport Prelude hiding (lookup)
grimport Data.Map
gr
gremployeeDept = fromList([("John","Sales"), ("Bob","IT")])
grdeptCountry = fromList([("IT","USA"), ("Sales","France")])
grcountryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])
gr
gremployeeCurrency :: String -&gt; Maybe String
gremployeeCurrency name = do
gr    dept &lt;- lookup name employeeDept
gr    country &lt;- lookup dept deptCountry
gr    lookup country countryCurrency
gr
grmain = do
gr    putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))
gr    putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
gr</pre>
gr
grThe output of this program:
gr
gr<pre>
grJohn's currency: Just "Euro"
grPete's currency: Nothing
gr</pre>
lookup :: Ord k => k -> Map k a -> Maybe a

-- | <i>O(log n)</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns default value
gr<tt>def</tt> when the key is not in the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: Ord k => a -> k -> Map k a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: Map k a

-- | <i>O(1)</i>. A map with a single element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: k -> a -> Map k a

-- | <i>O(log n)</i>. Insert a new key and value in the map. If the key is
gralready present in the map, the associated value is replaced with the
grsupplied value. <a>insert</a> is equivalent to <tt><a>insertWith</a>
gr<a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Ord k => k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining new value and old
grvalue. <tt><a>insertWith</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert the pair <tt>(key, f
grnew_value old_value)</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining key, new value and
grold value. <tt><a>insertWithKey</a> f key value mp</tt> will insert
grthe pair (key, value) into <tt>mp</tt> if key does not exist in the
grmap. If the key does exist, the function will insert the pair
gr<tt>(key,f key new_value old_value)</tt>. Note that the key passed to
grf is the same key passed to <a>insertWithKey</a>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Combines insert operation with old value retrieval.
grThe expression (<tt><a>insertLookupWithKey</a> f k x map</tt>) is a
grpair where the first element is equal to (<tt><a>lookup</a> k
grmap</tt>) and the second element equal to (<tt><a>insertWithKey</a> f
grk x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. Delete a key and its value from the map. When the key
gris not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Ord k => k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update a value at a specific key with the result of
grthe provided function. When the key is not a member of the map, the
groriginal map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Adjust a value at a specific key. When the key is not
gra member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>updateWithKey</a> f k
grmap</tt>) updates the value <tt>x</tt> at <tt>k</tt> (if it is in the
grmap). If (<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted.
grIf it is (<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the
grnew value <tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Lookup and update. See also <a>updateWithKey</a>. The
grfunction returns changed value, if it is updated. Returns the original
grkey value if the map entry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>alter</a> f k map</tt>) alters
grthe value <tt>x</tt> at <tt>k</tt>, or absence thereof. <a>alter</a>
grcan be used to insert, delete, or update a value in a <a>Map</a>. In
grshort : <tt><a>lookup</a> k (<a>alter</a> f k m) = f (<a>lookup</a> k
grm)</tt>.
gr
gr<pre>
grlet f _ = Nothing
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr
grlet f _ = Just "c"
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]
gr</pre>
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in a <a>Map</a>. In short: <tt><a>lookup</a> k &lt;$&gt;
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; Map Int String -&gt; IO (Map Int String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map. When used with
grtrivial functors like <a>Identity</a> and <a>Const</a>, it is often
grslightly slower than more specialized combinators like <a>lookup</a>
grand <a>insert</a>. However, when the functor is non-trivial and key
grcomparison is not particularly cheap, it is the fastest way.
gr
grNote on rewrite rules:
gr
grThis module includes GHC rewrite rules to optimize <a>alterF</a> for
grthe <a>Const</a> and <a>Identity</a> functors. In general, these rules
grimprove performance. The sole exception is that when using
gr<a>Identity</a>, deleting a key that is already absent takes longer
grthan it would without the rules. If you expect this to occur a very
grlarge fraction of the time, you might consider using a private copy of
grthe <a>Identity</a> type.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: (Functor f, Ord k) => (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression (<tt><a>union</a>
grt1 t2</tt>) takes the left-biased union of <tt>t1</tt> and
gr<tt>t2</tt>. It prefers <tt>t1</tt> when duplicate keys are
grencountered, i.e. (<tt><a>union</a> == <a>unionWith</a>
gr<a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: Ord k => Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | The union of a list of maps: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: Ord k => [Map k a] -> Map k a

-- | The union of a list of maps, with a combining operation:
gr(<tt><a>unionsWith</a> f == <a>foldl</a> (<a>unionWith</a> f)
gr<a>empty</a></tt>).
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two maps. Return
grelements of the first map not existing in the second map.
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the values
grof these keys. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection of two maps. Return
grdata in the first map for the keys existing in both maps.
gr(<tt><a>intersection</a> m1 m2 == <a>intersectionWith</a> <a>const</a>
grm1 m2</tt>).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n+m)</i>. An unsafe universal combining function.
gr
grWARNING: This function can produce corrupt maps and its results may
grdepend on the internal structures of its inputs. Users should prefer
gr<a>merge</a> or <a>mergeA</a>.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>Map</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt> or
gr<tt><a>filterWithKey</a> f</tt> could be used for any <tt>f</tt>.
mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f m == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (v' -&gt; v' <a>seq</a> (k,v'))
gr<a>$</a> f k v) (<a>toList</a> m)</tt> That is, it behaves much like a
grregular <a>traverse</a> except that the traversing function also has
graccess to the key associated with a value and the values are forced
grbefore they are installed in the result map.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)

-- | <i>O(n)</i>. Traverse keys/values and collect the <a>Just</a> results.
traverseMaybeWithKey :: Applicative f => (k -> a -> f (Maybe b)) -> Map k a -> f (Map k b)

-- | <i>O(n)</i>. The function <a>mapAccum</a> threads an accumulating
grargument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <a>mapAccumWithKey</a> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <tt>mapAccumR</tt> threads an accumulating
grargument through the map in descending order of keys.
mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n*log n)</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained by
grapplying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n*log n)</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>. The value at the greater of the two
groriginal keys is used as the first argument to <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. <tt><a>mapKeysMonotonic</a> f s == <a>mapKeys</a> f
grs</tt>, but works only when <tt>f</tt> is strictly monotonic. That is,
grfor any values <tt>x</tt> and <tt>y</tt>, if <tt>x</tt> &lt;
gr<tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The precondition is
grnot checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
grvalid (mapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")])) == True
grvalid (mapKeysMonotonic (\ _ -&gt; 1)     (fromList [(5,"a"), (3,"b")])) == False
gr</pre>
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (k -> a -> m) -> Map k a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: Map k a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: Map k a -> [k]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Return all key/value pairs
grin the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. The set of all keys of the map.
gr
gr<pre>
grkeysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
grkeysSet empty == Data.Set.empty
gr</pre>
keysSet :: Map k a -> Set k

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.Set.empty == empty
gr</pre>
fromSet :: (k -> a) -> Set k -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: Map k a -> [(k, a)]

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs. See
gralso <a>fromAscList</a>. If the list contains more than one value for
grthe same key, the last value for the key is retained.
gr
grIf the keys of the list are ordered, linear-time implementation is
grused, with the performance equal to <a>fromDistinctAscList</a>.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: Ord k => [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWithKey</a>.
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ a1 ++ a2
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Build a map from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
grvalid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctAscList [(3,"b"), (5,"a")])          == True
grvalid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False
gr</pre>
fromDistinctAscList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
gr
gr<pre>
grfromDescList [(5,"a"), (3,"b")]          == fromList [(3, "b"), (5, "a")]
grfromDescList [(5,"a"), (5,"b"), (3,"a")] == fromList [(3, "b"), (5, "b")]
grvalid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grfromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromDescListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctDescList [(5,"a"), (3,"b")])          == True
grvalid (fromDistinctDescList [(5,"a"), (3,"b"), (3,"a")]) == False
gr</pre>
fromDistinctDescList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Filter all values that satisfy the predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> Map k a -> Map k a

-- | <i>O(n)</i>. Filter all keys/values that satisfy the predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Restrict a <a>Map</a> to only
grthose keys found in a <a>Set</a>.
gr
gr<pre>
grm `<tt>restrictKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
grm `<tt>restrictKeys'</tt> s = m `<tt>intersect</tt> <a>fromSet</a> (const ()) s
gr</pre>
restrictKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Remove all keys in a <a>Set</a>
grfrom a <a>Map</a>.
gr
gr<pre>
grm `<tt>withoutKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
grm `<tt>withoutKeys'</tt> s = m `<tt>difference'</tt> <a>fromSet</a> (const ()) s
gr</pre>
withoutKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. Take while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> (p . fst) . <a>toList</a>
grtakeWhileAntitone p = <a>filterWithKey</a> (k _ -&gt; p k)
gr</pre>
takeWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Drop while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> (p . fst) . <a>toList</a>
grdropWhileAntitone p = <a>filterWithKey</a> (k -&gt; not (p k))
gr</pre>
dropWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Divide a map at the point where a predicate on the
grkeys stops holding. The user is responsible for ensuring that for all
grkeys <tt>j</tt> and <tt>k</tt> in the map, <tt>j &lt; k ==&gt; p j
gr&gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the map at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first key and to fail after
grthe last key).
spanAntitone :: (k -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where the keys in <tt>map1</tt> are smaller
grthan <tt>k</tt> and the keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Ord k => k -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>splitLookup</a> k map</tt>)
grsplits a map just like <a>split</a> but also returns <tt><a>lookup</a>
grk map</tt>.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d')],fromList [(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Map k b -> [Map k b]

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. This function is defined as
gr(<tt><a>isSubmapOf</a> = <a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression
gr(<tt><a>isSubmapOfBy</a> f t1 t2</tt>) returns <a>True</a> if all keys
grin <tt>t1</tt> are in tree <tt>t2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
gr</pre>
isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). Defined as (<tt><a>isProperSubmapOf</a> =
gr<a>isProperSubmapOfBy</a> (==)</tt>).
isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). The expression (<tt><a>isProperSubmapOfBy</a> f
grm1 m2</tt>) returns <a>True</a> when <tt>m1</tt> and <tt>m2</tt> are
grnot equal, all keys in <tt>m1</tt> are in <tt>m2</tt>, and when
gr<tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(log n)</i>. Lookup the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
gr
gr<pre>
grisJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False
grfromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
grfromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
grisJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False
gr</pre>
lookupIndex :: Ord k => k -> Map k a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
grCalls <a>error</a> when the key is not a <a>member</a> of the map.
gr
gr<pre>
grfindIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
grfindIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
grfindIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
grfindIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
gr</pre>
findIndex :: Ord k => k -> Map k a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
grelemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
grelemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
elemAt :: Int -> Map k a -> (k, a)

-- | <i>O(log n)</i>. Update the element at <i>index</i>. Calls
gr<a>error</a> when an invalid index is used.
gr
gr<pre>
grupdateAt (\ _ _ -&gt; Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
grupdateAt (\ _ _ -&gt; Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
grupdateAt (\ _ _ -&gt; Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\ _ _ -&gt; Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grupdateAt (\_ _  -&gt; Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grupdateAt (\_ _  -&gt; Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grdeleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdeleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range
grdeleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range
gr</pre>
deleteAt :: Int -> Map k a -> Map k a

-- | Take a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Map k a -> Map k a

-- | Drop a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Split a map at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The minimal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b")
grfindMin empty = Nothing
gr</pre>
lookupMin :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The maximal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMax (fromList [(5,"a"), (3,"b")]) == Just (5,"a")
grlookupMax empty = Nothing
gr</pre>
lookupMax :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The minimal key of the map. Calls <a>error</a> if the
grmap is empty.
gr
gr<pre>
grfindMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
grfindMin empty                            Error: empty map has no minimal element
gr</pre>
findMin :: Map k a -> (k, a)
findMax :: Map k a -> (k, a)

-- | <i>O(log n)</i>. Delete the minimal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
grdeleteMin empty == empty
gr</pre>
deleteMin :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the maximal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
grdeleteMax empty == empty
gr</pre>
deleteMax :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
grdeleteFindMin                                            Error: can not return the minimal element of an empty map
gr</pre>
deleteFindMin :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
grdeleteFindMax empty                                      Error: can not return the maximal element of an empty map
gr</pre>
deleteFindMax :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Retrieves the value associated with minimal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grminView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
grminView empty == Nothing
gr</pre>
minView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the value associated with maximal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grmaxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
grmaxView empty == Nothing
gr</pre>
maxView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(log n)</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format. See <a>showTreeWith</a>.

-- | <i>Deprecated: <a>showTree</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTree :: (Show k, Show a) => Map k a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> showelem hang
grwide map</tt>) shows the tree that implements the map. Elements are
grshown using the <tt>showElem</tt> function. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grMap&gt; let t = fromDistinctAscList [(x,()) | x &lt;- [1..5]]
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True False t
gr(4,())
gr+--(2,())
gr|  +--(1,())
gr|  +--(3,())
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True True t
gr(4,())
gr|
gr+--(2,())
gr|  |
gr|  +--(1,())
gr|  |
gr|  +--(3,())
gr|
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) False True t
gr+--(5,())
gr|
gr(4,())
gr|
gr|  +--(3,())
gr|  |
gr+--(2,())
gr   |
gr   +--(1,())
gr</pre>

-- | <i>Deprecated: <a>showTreeWith</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String

-- | <i>O(n)</i>. Test if the internal map structure is valid.
gr
gr<pre>
grvalid (fromAscList [(3,"b"), (5,"a")]) == True
grvalid (fromAscList [(5,"a"), (3,"b")]) == False
gr</pre>
valid :: Ord k => Map k a -> Bool


-- | This module defines an API for writing functions that merge two maps.
grThe key functions are <a>merge</a> and <a>mergeA</a>. Each of these
grcan be used with several different "merge tactics".
gr
grThe <a>merge</a> and <a>mergeA</a> functions are shared by the lazy
grand strict modules. Only the choice of merge tactics determines
grstrictness. If you use <a>mapMissing</a> from this module then the
grresults will be forced before they are inserted. If you use
gr<a>mapMissing</a> from <a>Data.Map.Merge.Lazy</a> then they will not.
gr
gr<h2>Efficiency note</h2>
gr
grThe <tt>Category</tt>, <a>Applicative</a>, and <a>Monad</a> instances
grfor <a>WhenMissing</a> tactics are included because they are valid.
grHowever, they are inefficient in many cases and should usually be
gravoided. The instances for <a>WhenMatched</a> tactics should not pose
grany major efficiency problems.
module Data.Map.Merge.Strict

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMissing k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; Maybe z
gr</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt> SimpleWhenMatched k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt;
grMaybe z </tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: Ord k => SimpleWhenMissing k a c -> SimpleWhenMissing k b c -> SimpleWhenMatched k a b c -> Map k a -> Map k b -> Map k c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched :: (k -&gt; x -&gt; y -&gt; Maybe z)
gr                    -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (k -> x -> y -> Maybe z) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched :: (k -&gt; x -&gt; y -&gt; z)
gr               -&gt; SimpleWhenMatched k x y z
gr</pre>
zipWithMatched :: Applicative f => (k -> x -> y -> z) -> WhenMatched f k x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (k -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (k -> x -> Maybe y) -> WhenMissing f k x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f k x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f k x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing k x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (k -> x -> y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing k x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (k -> x -> Bool) -> WhenMissing f k x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt> WhenMissing f k x z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; f (Maybe z)
gr</tt>.
data WhenMissing f k x y

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt> WhenMatched f k x y z </tt> is an abstract
grrepresentation of a function of type <tt> k -&gt; x -&gt; y -&gt; f
gr(Maybe z) </tt>.
data WhenMatched f k x y z

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f, Ord k) => WhenMissing f k a c -> WhenMissing f k b c -> WhenMatched f k a b c -> Map k a -> Map k b -> f (Map k c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: Applicative f => (k -> x -> y -> f (Maybe z)) -> WhenMatched f k x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (k -> x -> y -> f z) -> WhenMatched f k x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (k -> x -> f (Maybe y)) -> WhenMissing f k x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (k -> x -> f y) -> WhenMissing f k x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  k x -&gt; (b -&gt; guard b *&gt; Just x) <a>$</a> f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (k -> x -> f Bool) -> WhenMissing f k x x

-- | Map covariantly over a <tt><a>WhenMissing</a> f k x</tt>.
mapWhenMissing :: Functor f => (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f k x y z</tt> and <tt>k -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f k x y z -> k -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f k x y</tt> and <tt>k -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f k x y -> k -> x -> f (Maybe y)


-- | <h1>Finite Maps (lazy interface)</h1>
gr
grThe <tt><a>Map</a> k v</tt> type represents a finite map (sometimes
grcalled a dictionary) from keys of type <tt>k</tt> to values of type
gr<tt>v</tt>. A <a>Map</a> is strict in its keys but lazy in its values.
gr
grThe functions in <a>Data.Map.Strict</a> are careful to force values
grbefore installing them in a <a>Map</a>. This is usually more efficient
grin cases where laziness is not essential. The functions in this module
grdo not do so.
gr
grWhen deciding if this is the correct data structure to use, consider:
gr
gr<ul>
gr<li>If you are using <tt>Int</tt> keys, you will get much better
grperformance for most operations using <a>Data.IntMap.Lazy</a>.</li>
gr<li>If you don't care about ordering, consider using
gr<tt>Data.HashMap.Lazy</tt> from the <a>unordered-containers</a>
grpackage instead.</li>
gr</ul>
gr
grFor a walkthrough of the most commonly used functions see the <a>maps
grintroduction</a>.
gr
grThis module is intended to be imported qualified, to avoid name
grclashes with Prelude functions:
gr
gr<pre>
grimport qualified Data.Map.Lazy as Map
gr</pre>
gr
grNote that the implementation is generally <i>left-biased</i>.
grFunctions that take two maps as arguments and combine them, such as
gr<a>union</a> and <a>intersection</a>, prefer the values in the first
grargument to those in the second.
gr
gr<h2>Detailed performance information</h2>
gr
grThe amortized running time is given for each operation, with <i>n</i>
grreferring to the number of entries in the map.
gr
grBenchmarks comparing <a>Data.Map.Lazy</a> with other dictionary
grimplementations can be found at
gr<a>https://github.com/haskell-perf/dictionaries</a>.
gr
gr<h2>Warning</h2>
gr
grThe size of a <a>Map</a> must not exceed <tt>maxBound::Int</tt>.
grViolation of this condition is not detected and if the size limit is
grexceeded, its behaviour is undefined.
gr
gr<h2>Implementation</h2>
gr
grThe implementation of <a>Map</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
module Data.Map.Lazy

-- | A Map from keys <tt>k</tt> to values <tt>a</tt>.
data Map k a

-- | <i>O(log n)</i>. Find the value at a key. Calls <a>error</a> when the
grelement can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: Ord k => Map k a -> k -> a
infixl 9 !

-- | <i>O(log n)</i>. Find the value at a key. Returns <a>Nothing</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 1 == Nothing
gr</pre>
gr
gr<pre>
grfromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: Ord k => Map k a -> k -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: Ord k => Map k a -> Map k b -> Map k a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.Map.null (empty)           == True
grData.Map.null (singleton 1 'a') == False
gr</pre>
null :: Map k a -> Bool

-- | <i>O(1)</i>. The number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: Map k a -> Int

-- | <i>O(log n)</i>. Is the key a member of the map? See also
gr<a>notMember</a>.
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Is the key not a member of the map? See also
gr<a>member</a>.
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Ord k => k -> Map k a -> Bool

-- | <i>O(log n)</i>. Lookup the value at a key in the map.
gr
grThe function will return the corresponding value as <tt>(<a>Just</a>
grvalue)</tt>, or <a>Nothing</a> if the key isn't in the map.
gr
grAn example of using <tt>lookup</tt>:
gr
gr<pre>
grimport Prelude hiding (lookup)
grimport Data.Map
gr
gremployeeDept = fromList([("John","Sales"), ("Bob","IT")])
grdeptCountry = fromList([("IT","USA"), ("Sales","France")])
grcountryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])
gr
gremployeeCurrency :: String -&gt; Maybe String
gremployeeCurrency name = do
gr    dept &lt;- lookup name employeeDept
gr    country &lt;- lookup dept deptCountry
gr    lookup country countryCurrency
gr
grmain = do
gr    putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))
gr    putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))
gr</pre>
gr
grThe output of this program:
gr
gr<pre>
grJohn's currency: Just "Euro"
grPete's currency: Nothing
gr</pre>
lookup :: Ord k => k -> Map k a -> Maybe a

-- | <i>O(log n)</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns default value
gr<tt>def</tt> when the key is not in the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: Ord k => a -> k -> Map k a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: Map k a

-- | <i>O(1)</i>. A map with a single element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: k -> a -> Map k a

-- | <i>O(log n)</i>. Insert a new key and value in the map. If the key is
gralready present in the map, the associated value is replaced with the
grsupplied value. <a>insert</a> is equivalent to <tt><a>insertWith</a>
gr<a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Ord k => k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining new value and old
grvalue. <tt><a>insertWith</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert the pair <tt>(key, f
grnew_value old_value)</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Insert with a function, combining key, new value and
grold value. <tt><a>insertWithKey</a> f key value mp</tt> will insert
grthe pair (key, value) into <tt>mp</tt> if key does not exist in the
grmap. If the key does exist, the function will insert the pair
gr<tt>(key,f key new_value old_value)</tt>. Note that the key passed to
grf is the same key passed to <a>insertWithKey</a>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Combines insert operation with old value retrieval.
grThe expression (<tt><a>insertLookupWithKey</a> f k x map</tt>) is a
grpair where the first element is equal to (<tt><a>lookup</a> k
grmap</tt>) and the second element equal to (<tt><a>insertWithKey</a> f
grk x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. Delete a key and its value from the map. When the key
gris not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Ord k => k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update a value at a specific key with the result of
grthe provided function. When the key is not a member of the map, the
groriginal map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Adjust a value at a specific key. When the key is not
gra member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>updateWithKey</a> f k
grmap</tt>) updates the value <tt>x</tt> at <tt>k</tt> (if it is in the
grmap). If (<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted.
grIf it is (<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the
grnew value <tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. Lookup and update. See also <a>updateWithKey</a>. The
grfunction returns changed value, if it is updated. Returns the original
grkey value if the map entry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>alter</a> f k map</tt>) alters
grthe value <tt>x</tt> at <tt>k</tt>, or absence thereof. <a>alter</a>
grcan be used to insert, delete, or update a value in a <a>Map</a>. In
grshort : <tt><a>lookup</a> k (<a>alter</a> f k m) = f (<a>lookup</a> k
grm)</tt>.
gr
gr<pre>
grlet f _ = Nothing
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr
grlet f _ = Just "c"
gralter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]
gralter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]
gr</pre>
alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in a <a>Map</a>. In short: <tt><a>lookup</a> k &lt;$&gt;
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; Map Int String -&gt; IO (Map Int String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map. When used with
grtrivial functors like <a>Identity</a> and <a>Const</a>, it is often
grslightly slower than more specialized combinators like <a>lookup</a>
grand <a>insert</a>. However, when the functor is non-trivial and key
grcomparison is not particularly cheap, it is the fastest way.
gr
grNote on rewrite rules:
gr
grThis module includes GHC rewrite rules to optimize <a>alterF</a> for
grthe <a>Const</a> and <a>Identity</a> functors. In general, these rules
grimprove performance. The sole exception is that when using
gr<a>Identity</a>, deleting a key that is already absent takes longer
grthan it would without the rules. If you expect this to occur a very
grlarge fraction of the time, you might consider using a private copy of
grthe <a>Identity</a> type.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: (Functor f, Ord k) => (Maybe a -> f (Maybe a)) -> k -> Map k a -> f (Map k a)

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression (<tt><a>union</a>
grt1 t2</tt>) takes the left-biased union of <tt>t1</tt> and
gr<tt>t2</tt>. It prefers <tt>t1</tt> when duplicate keys are
grencountered, i.e. (<tt><a>union</a> == <a>unionWith</a>
gr<a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: Ord k => Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a

-- | The union of a list of maps: (<tt><a>unions</a> == <a>foldl</a>
gr<a>union</a> <a>empty</a></tt>).
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: Ord k => [Map k a] -> Map k a

-- | The union of a list of maps, with a combining operation:
gr(<tt><a>unionsWith</a> f == <a>foldl</a> (<a>unionWith</a> f)
gr<a>empty</a></tt>).
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: Ord k => (a -> a -> a) -> [Map k a] -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Difference of two maps. Return
grelements of the first map not existing in the second map.
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the values
grof these keys. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection of two maps. Return
grdata in the first map for the keys existing in both maps.
gr(<tt><a>intersection</a> m1 m2 == <a>intersectionWith</a> <a>const</a>
grm1 m2</tt>).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: Ord k => Map k a -> Map k b -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Intersection with a combining
grfunction.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n+m)</i>. An unsafe general combining function.
gr
grWARNING: This function can produce corrupt maps and its results may
grdepend on the internal structures of its inputs. Users should prefer
gr<a>merge</a> or <a>mergeA</a>.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>Map</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt>,
gr<tt><a>filterWithKey</a> f</tt>, or <tt><a>mapMaybeWithKey</a> f</tt>
grcould be used for any <tt>f</tt>.
mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (Map k a -> Map k c) -> (Map k b -> Map k c) -> Map k a -> Map k b -> Map k c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (k -> a -> b) -> Map k a -> Map k b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f m == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (,) k <a>$</a> f k v)
gr(<a>toList</a> m)</tt> That is, behaves exactly like a regular
gr<a>traverse</a> except that the traversing function also has access to
grthe key associated with a value.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)

-- | <i>O(n)</i>. Traverse keys/values and collect the <a>Just</a> results.
traverseMaybeWithKey :: Applicative f => (k -> a -> f (Maybe b)) -> Map k a -> f (Map k b)

-- | <i>O(n)</i>. The function <a>mapAccum</a> threads an accumulating
grargument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <a>mapAccumWithKey</a> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n)</i>. The function <tt>mapAccumR</tt> threads an accumulating
grargument through the map in descending order of keys.
mapAccumRWithKey :: (a -> k -> b -> (a, c)) -> a -> Map k b -> (a, Map k c)

-- | <i>O(n*log n)</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained by
grapplying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: Ord k2 => (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n*log n)</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>. The value at the greater of the two
groriginal keys is used as the first argument to <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. <tt><a>mapKeysMonotonic</a> f s == <a>mapKeys</a> f
grs</tt>, but works only when <tt>f</tt> is strictly monotonic. That is,
grfor any values <tt>x</tt> and <tt>y</tt>, if <tt>x</tt> &lt;
gr<tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The precondition is
grnot checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
grvalid (mapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")])) == True
grvalid (mapKeysMonotonic (\ _ -&gt; 1)     (fromList [(5,"a"), (3,"b")])) == False
gr</pre>
mapKeysMonotonic :: (k1 -> k2) -> Map k1 a -> Map k2 a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (k -> a -> m) -> Map k a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: Map k a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: Map k a -> [k]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Return all key/value pairs
grin the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. The set of all keys of the map.
gr
gr<pre>
grkeysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]
grkeysSet empty == Data.Set.empty
gr</pre>
keysSet :: Map k a -> Set k

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.Set.empty == empty
gr</pre>
fromSet :: (k -> a) -> Set k -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: Map k a -> [(k, a)]

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs. See
gralso <a>fromAscList</a>. If the list contains more than one value for
grthe same key, the last value for the key is retained.
gr
grIf the keys of the list are ordered, linear-time implementation is
grused, with the performance equal to <a>fromDistinctAscList</a>.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: Ord k => [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: Ord k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n*log n)</i>. Build a map from a list of key/value pairs with a
grcombining function. See also <a>fromAscListWithKey</a>.
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ a1 ++ a2
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: Map k a -> [(k, a)]

-- | <i>O(n)</i>. Build a map from an ascending list in linear time. <i>The
grprecondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True
grvalid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromAscListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grascending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True
grvalid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from an ascending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctAscList [(3,"b"), (5,"a")])          == True
grvalid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False
gr</pre>
fromDistinctAscList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time. <i>The
grprecondition (input list is descending) is not checked.</i>
gr
gr<pre>
grfromDescList [(5,"a"), (3,"b")]          == fromList [(3, "b"), (5, "a")]
grfromDescList [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "b")]
grvalid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescList :: Eq k => [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grfromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")]
grvalid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False
gr</pre>
fromDescListWith :: Eq k => (a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list in linear time with a
grcombining function for equal keys. <i>The precondition (input list is
grdescending) is not checked.</i>
gr
gr<pre>
grlet f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2
grfromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]
grvalid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True
grvalid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False
gr</pre>
fromDescListWithKey :: Eq k => (k -> a -> a -> a) -> [(k, a)] -> Map k a

-- | <i>O(n)</i>. Build a map from a descending list of distinct elements
grin linear time. <i>The precondition is not checked.</i>
gr
gr<pre>
grfromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")]
grvalid (fromDistinctDescList [(5,"a"), (3,"b")])          == True
grvalid (fromDistinctDescList [(5,"a"), (5,"b"), (3,"b")]) == False
gr</pre>
fromDistinctDescList :: [(k, a)] -> Map k a

-- | <i>O(n)</i>. Filter all values that satisfy the predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> Map k a -> Map k a

-- | <i>O(n)</i>. Filter all keys/values that satisfy the predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Restrict a <a>Map</a> to only
grthose keys found in a <a>Set</a>.
gr
gr<pre>
grm `<tt>restrictKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
grm `<tt>restrictKeys'</tt> s = m `<tt>intersect</tt> <a>fromSet</a> (const ()) s
gr</pre>
restrictKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Remove all keys in a <a>Set</a>
grfrom a <a>Map</a>.
gr
gr<pre>
grm `<tt>withoutKeys'</tt> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
grm `<tt>withoutKeys'</tt> s = m `<tt>difference'</tt> <a>fromSet</a> (const ()) s
gr</pre>
withoutKeys :: Ord k => Map k a -> Set k -> Map k a

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Partition the map according to a predicate. The first map
grcontains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. Take while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grtakeWhileAntitone p = <a>fromDistinctAscList</a> . <a>takeWhile</a> (p . fst) . <a>toList</a>
grtakeWhileAntitone p = <a>filterWithKey</a> (k _ -&gt; p k)
gr</pre>
takeWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Drop while a predicate on the keys holds. The user is
grresponsible for ensuring that for all keys <tt>j</tt> and <tt>k</tt>
grin the map, <tt>j &lt; k ==&gt; p j &gt;= p k</tt>. See note at
gr<a>spanAntitone</a>.
gr
gr<pre>
grdropWhileAntitone p = <a>fromDistinctAscList</a> . <a>dropWhile</a> (p . fst) . <a>toList</a>
grdropWhileAntitone p = <a>filterWithKey</a> (k -&gt; not (p k))
gr</pre>
dropWhileAntitone :: (k -> Bool) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Divide a map at the point where a predicate on the
grkeys stops holding. The user is responsible for ensuring that for all
grkeys <tt>j</tt> and <tt>k</tt> in the map, <tt>j &lt; k ==&gt; p j
gr&gt;= p k</tt>.
gr
gr<pre>
grspanAntitone p xs = (<a>takeWhileAntitone</a> p xs, <a>dropWhileAntitone</a> p xs)
grspanAntitone p xs = partition p xs
gr</pre>
gr
grNote: if <tt>p</tt> is not actually antitone, then
gr<tt>spanAntitone</tt> will split the map at some <i>unspecified</i>
grpoint where the predicate switches from holding to not holding (where
grthe predicate is seen to hold before the first key and to fail after
grthe last key).
spanAntitone :: (k -> Bool) -> Map k a -> (Map k a, Map k a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)

-- | <i>O(log n)</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where the keys in <tt>map1</tt> are smaller
grthan <tt>k</tt> and the keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Ord k => k -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The expression (<tt><a>splitLookup</a> k map</tt>)
grsplits a map just like <a>split</a> but also returns <tt><a>lookup</a>
grk map</tt>.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Ord k => k -> Map k a -> (Map k a, Maybe a, Map k a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d')],fromList [(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than three
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: Map k b -> [Map k b]

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. This function is defined as
gr(<tt><a>isSubmapOf</a> = <a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. The expression
gr(<tt><a>isSubmapOfBy</a> f t1 t2</tt>) returns <a>True</a> if all keys
grin <tt>t1</tt> are in tree <tt>t2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (&lt;)  (fromList [('a',1)]) (fromList [('a',1),('b',2)])
grisSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])
gr</pre>
isSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). Defined as (<tt><a>isProperSubmapOf</a> =
gr<a>isProperSubmapOfBy</a> (==)</tt>).
isProperSubmapOf :: (Ord k, Eq a) => Map k a -> Map k a -> Bool

-- | <i>O(m*log(n/m + 1)), m &lt;= n</i>. Is this a proper submap? (ie. a
grsubmap but not equal). The expression (<tt><a>isProperSubmapOfBy</a> f
grm1 m2</tt>) returns <a>True</a> when <tt>m1</tt> and <tt>m2</tt> are
grnot equal, all keys in <tt>m1</tt> are in <tt>m2</tt>, and when
gr<tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool

-- | <i>O(log n)</i>. Lookup the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
gr
gr<pre>
grisJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")]))   == False
grfromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0
grfromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1
grisJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")]))   == False
gr</pre>
lookupIndex :: Ord k => k -> Map k a -> Maybe Int

-- | <i>O(log n)</i>. Return the <i>index</i> of a key, which is its
grzero-based index in the sequence sorted by keys. The index is a number
grfrom <i>0</i> up to, but not including, the <a>size</a> of the map.
grCalls <a>error</a> when the key is not a <a>member</a> of the map.
gr
gr<pre>
grfindIndex 2 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
grfindIndex 3 (fromList [(5,"a"), (3,"b")]) == 0
grfindIndex 5 (fromList [(5,"a"), (3,"b")]) == 1
grfindIndex 6 (fromList [(5,"a"), (3,"b")])    Error: element is not in the map
gr</pre>
findIndex :: Ord k => k -> Map k a -> Int

-- | <i>O(log n)</i>. Retrieve an element by its <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grelemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")
grelemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")
grelemAt 2 (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
elemAt :: Int -> Map k a -> (k, a)

-- | <i>O(log n)</i>. Update the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grupdateAt (\ _ _ -&gt; Just "x") 0    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]
grupdateAt (\ _ _ -&gt; Just "x") 1    (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]
grupdateAt (\ _ _ -&gt; Just "x") 2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\ _ _ -&gt; Just "x") (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  0    (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grupdateAt (\_ _  -&gt; Nothing)  1    (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grupdateAt (\_ _  -&gt; Nothing)  2    (fromList [(5,"a"), (3,"b")])    Error: index out of range
grupdateAt (\_ _  -&gt; Nothing)  (-1) (fromList [(5,"a"), (3,"b")])    Error: index out of range
gr</pre>
updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the element at <i>index</i>, i.e. by its
grzero-based index in the sequence sorted by keys. If the <i>index</i>
gris out of range (less than zero, greater or equal to <a>size</a> of
grthe map), <a>error</a> is called.
gr
gr<pre>
grdeleteAt 0  (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
grdeleteAt 1  (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdeleteAt 2 (fromList [(5,"a"), (3,"b")])     Error: index out of range
grdeleteAt (-1) (fromList [(5,"a"), (3,"b")])  Error: index out of range
gr</pre>
deleteAt :: Int -> Map k a -> Map k a

-- | Take a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grtake n = <a>fromDistinctAscList</a> . <a>take</a> n . <a>toAscList</a>
gr</pre>
take :: Int -> Map k a -> Map k a

-- | Drop a given number of entries in key order, beginning with the
grsmallest keys.
gr
gr<pre>
grdrop n = <a>fromDistinctAscList</a> . <a>drop</a> n . <a>toAscList</a>
gr</pre>
drop :: Int -> Map k a -> Map k a

-- | <i>O(log n)</i>. Split a map at a particular index.
gr
gr<pre>
grsplitAt !n !xs = (<a>take</a> n xs, <a>drop</a> n xs)
gr</pre>
splitAt :: Int -> Map k a -> (Map k a, Map k a)

-- | <i>O(log n)</i>. The minimal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b")
grfindMin empty = Nothing
gr</pre>
lookupMin :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The maximal key of the map. Returns <a>Nothing</a> if
grthe map is empty.
gr
gr<pre>
grlookupMax (fromList [(5,"a"), (3,"b")]) == Just (5,"a")
grlookupMax empty = Nothing
gr</pre>
lookupMax :: Map k a -> Maybe (k, a)

-- | <i>O(log n)</i>. The minimal key of the map. Calls <a>error</a> if the
grmap is empty.
gr
gr<pre>
grfindMin (fromList [(5,"a"), (3,"b")]) == (3,"b")
grfindMin empty                            Error: empty map has no minimal element
gr</pre>
findMin :: Map k a -> (k, a)
findMax :: Map k a -> (k, a)

-- | <i>O(log n)</i>. Delete the minimal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]
grdeleteMin empty == empty
gr</pre>
deleteMin :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete the maximal key. Returns an empty map if the
grmap is empty.
gr
gr<pre>
grdeleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]
grdeleteMax empty == empty
gr</pre>
deleteMax :: Map k a -> Map k a

-- | <i>O(log n)</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])
grdeleteFindMin                                            Error: can not return the minimal element of an empty map
gr</pre>
deleteFindMin :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])
grdeleteFindMax empty                                      Error: can not return the maximal element of an empty map
gr</pre>
deleteFindMax :: Map k a -> ((k, a), Map k a)

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a

-- | <i>O(log n)</i>. Retrieves the value associated with minimal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grminView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")
grminView empty == Nothing
gr</pre>
minView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the value associated with maximal key of
grthe map, and the map stripped of that element, or <a>Nothing</a> if
grpassed an empty map.
gr
gr<pre>
grmaxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")
grmaxView empty == Nothing
gr</pre>
maxView :: Map k a -> Maybe (a, Map k a)

-- | <i>O(log n)</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(log n)</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: Map k a -> Maybe ((k, a), Map k a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format. See <a>showTreeWith</a>.

-- | <i>Deprecated: <a>showTree</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTree :: (Show k, Show a) => Map k a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> showelem hang
grwide map</tt>) shows the tree that implements the map. Elements are
grshown using the <tt>showElem</tt> function. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
gr
gr<pre>
grMap&gt; let t = fromDistinctAscList [(x,()) | x &lt;- [1..5]]
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True False t
gr(4,())
gr+--(2,())
gr|  +--(1,())
gr|  +--(3,())
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) True True t
gr(4,())
gr|
gr+--(2,())
gr|  |
gr|  +--(1,())
gr|  |
gr|  +--(3,())
gr|
gr+--(5,())
gr
grMap&gt; putStrLn $ showTreeWith (\k x -&gt; show (k,x)) False True t
gr+--(5,())
gr|
gr(4,())
gr|
gr|  +--(3,())
gr|  |
gr+--(2,())
gr   |
gr   +--(1,())
gr</pre>

-- | <i>Deprecated: <a>showTreeWith</a> is now in
gr<a>Data.Map.Internal.Debug</a></i>
showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String

-- | <i>O(n)</i>. Test if the internal map structure is valid.
gr
gr<pre>
grvalid (fromAscList [(3,"b"), (5,"a")]) == True
grvalid (fromAscList [(5,"a"), (3,"b")]) == False
gr</pre>
valid :: Ord k => Map k a -> Bool


-- | <i>Note:</i> You should use <a>Data.Map.Strict</a> instead of this
grmodule if:
gr
gr<ul>
gr<li>You will eventually need all the values stored.</li>
gr<li>The stored values don't represent large virtual data structures to
grbe lazily computed.</li>
gr</ul>
gr
grAn efficient implementation of ordered maps from keys to values
gr(dictionaries).
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport qualified Data.Map as Map
gr</pre>
gr
grThe implementation of <a>Map</a> is based on <i>size balanced</i>
grbinary trees (or trees of <i>bounded balance</i>) as described by:
gr
gr<ul>
gr<li>Stephen Adams, "<i>Efficient sets: a balancing act</i>", Journal
grof Functional Programming 3(4):553-562, October 1993,
gr<a>http://www.swiss.ai.mit.edu/~adams/BB/</a>.</li>
gr<li>J. Nievergelt and E.M. Reingold, "<i>Binary search trees of
grbounded balance</i>", SIAM journal of computing 2(1), March 1973.</li>
gr</ul>
gr
grBounds for <a>union</a>, <a>intersection</a>, and <a>difference</a>
grare as given by
gr
gr<ul>
gr<li>Guy Blelloch, Daniel Ferizovic, and Yihan Sun, "<i>Just Join for
grParallel Ordered Sets</i>",
gr<a>https://arxiv.org/abs/1602.02120v3</a>.</li>
gr</ul>
gr
grNote that the implementation is <i>left-biased</i> -- the elements of
gra first argument are always preferred to the second, for example in
gr<a>union</a> or <a>insert</a>.
gr
gr<i>Warning</i>: The size of the map must not exceed
gr<tt>maxBound::Int</tt>. Violation of this condition is not detected
grand if the size limit is exceeded, its behaviour is undefined.
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation (<a>http://en.wikipedia.org/wiki/Big_O_notation</a>).
module Data.Map

-- | <i>O(log n)</i>. Same as <a>insertWith</a>, but the value being
grinserted to the map is evaluated to WHNF beforehand.
gr
grFor example, to update a counter:
gr
gr<pre>
grinsertWith' (+) k 1 m
gr</pre>

-- | <i>Deprecated: As of version 0.5, replaced by <a>insertWith</a>.</i>
insertWith' :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Same as <a>insertWithKey</a>, but the value being
grinserted to the map is evaluated to WHNF beforehand.

-- | <i>Deprecated: As of version 0.5, replaced by
gr<a>insertWithKey</a>.</i>
insertWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a

-- | <i>O(log n)</i>. Same as <a>insertLookupWithKey</a>, but the value
grbeing inserted to the map is evaluated to WHNF beforehand.

-- | <i>Deprecated: As of version 0.5, replaced by
gr<a>insertLookupWithKey</a>.</i>
insertLookupWithKey' :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.

-- | <i>Deprecated: As of version 0.5, replaced by <a>foldr</a>.</i>
fold :: (a -> b -> b) -> b -> Map k a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldrWithKey</a> and is present for compatibility only.

-- | <i>Deprecated: As of version 0.4, replaced by <a>foldrWithKey</a>.</i>
foldWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grAn efficient implementation of integer sets.
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.IntSet (IntSet)
grimport qualified Data.IntSet as IntSet
gr</pre>
gr
grThe implementation is based on <i>big-endian patricia trees</i>. This
grdata structure performs especially well on binary operations like
gr<a>union</a> and <a>intersection</a>. However, my benchmarks show that
grit is also (much) faster on insertions and deletions when compared to
gra generic size-balanced set implementation (see <a>Data.Set</a>).
gr
gr<ul>
gr<li>Chris Okasaki and Andy Gill, "<i>Fast Mergeable Integer Maps</i>",
grWorkshop on ML, September 1998, pages 77-86,
gr<a>http://citeseer.ist.psu.edu/okasaki98fast.html</a></li>
gr<li>D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve
grInformation Coded In Alphanumeric/", Journal of the ACM, 15(4),
grOctober 1968, pages 514-534.</li>
gr</ul>
gr
grAdditionally, this implementation places bitmaps in the leaves of the
grtree. Their size is the natural size of a machine word (32 or 64 bits)
grand greatly reduce memory footprint and execution times for dense
grsets, e.g. sets where it is likely that many values lie close to each
grother. The asymptotics are not affected by this optimization.
gr
grMany operations have a worst-case complexity of <i>O(min(n,W))</i>.
grThis means that the operation can become linear in the number of
grelements with a maximum of <i>W</i> -- the number of bits in an
gr<a>Int</a> (32 or 64).
module Data.IntSet.Internal

-- | A set of integers.
data IntSet
Bin :: {-# UNPACK #-} !Prefix -> {-# UNPACK #-} !Mask -> !IntSet -> !IntSet -> IntSet
Tip :: {-# UNPACK #-} !Prefix -> {-# UNPACK #-} !BitMap -> IntSet
Nil :: IntSet
type Key = Int
type Prefix = Int
type Mask = Int
type BitMap = Word

-- | <i>O(n+m)</i>. See <a>difference</a>.
(\\) :: IntSet -> IntSet -> IntSet
infixl 9 \\

-- | <i>O(1)</i>. Is the set empty?
null :: IntSet -> Bool

-- | <i>O(n)</i>. Cardinality of the set.
size :: IntSet -> Int

-- | <i>O(min(n,W))</i>. Is the value a member of the set?
member :: Key -> IntSet -> Bool

-- | <i>O(min(n,W))</i>. Is the element not in the set?
notMember :: Key -> IntSet -> Bool

-- | <i>O(log n)</i>. Find largest element smaller than the given one.
gr
gr<pre>
grlookupLT 3 (fromList [3, 5]) == Nothing
grlookupLT 5 (fromList [3, 5]) == Just 3
gr</pre>
lookupLT :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find smallest element greater than the given one.
gr
gr<pre>
grlookupGT 4 (fromList [3, 5]) == Just 5
grlookupGT 5 (fromList [3, 5]) == Nothing
gr</pre>
lookupGT :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find largest element smaller or equal to the given
grone.
gr
gr<pre>
grlookupLE 2 (fromList [3, 5]) == Nothing
grlookupLE 4 (fromList [3, 5]) == Just 3
grlookupLE 5 (fromList [3, 5]) == Just 5
gr</pre>
lookupLE :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find smallest element greater or equal to the given
grone.
gr
gr<pre>
grlookupGE 3 (fromList [3, 5]) == Just 3
grlookupGE 4 (fromList [3, 5]) == Just 5
grlookupGE 6 (fromList [3, 5]) == Nothing
gr</pre>
lookupGE :: Key -> IntSet -> Maybe Key

-- | <i>O(n+m)</i>. Is this a subset? <tt>(s1 <a>isSubsetOf</a> s2)</tt>
grtells whether <tt>s1</tt> is a subset of <tt>s2</tt>.
isSubsetOf :: IntSet -> IntSet -> Bool

-- | <i>O(n+m)</i>. Is this a proper subset? (ie. a subset but not equal).
isProperSubsetOf :: IntSet -> IntSet -> Bool

-- | <i>O(n+m)</i>. Check whether two sets are disjoint (i.e. their
grintersection is empty).
gr
gr<pre>
grdisjoint (fromList [2,4,6])   (fromList [1,3])     == True
grdisjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
grdisjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
grdisjoint (fromList [])        (fromList [])        == True
gr</pre>
disjoint :: IntSet -> IntSet -> Bool

-- | <i>O(1)</i>. The empty set.
empty :: IntSet

-- | <i>O(1)</i>. A set of one element.
singleton :: Key -> IntSet

-- | <i>O(min(n,W))</i>. Add a value to the set. There is no left- or right
grbias for IntSets.
insert :: Key -> IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete a value in the set. Returns the original
grset when the value was not present.
delete :: Key -> IntSet -> IntSet

-- | <i>O(n+m)</i>. The union of two sets.
union :: IntSet -> IntSet -> IntSet

-- | The union of a list of sets.
unions :: [IntSet] -> IntSet

-- | <i>O(n+m)</i>. Difference between two sets.
difference :: IntSet -> IntSet -> IntSet

-- | <i>O(n+m)</i>. The intersection of two sets.
intersection :: IntSet -> IntSet -> IntSet

-- | <i>O(n)</i>. Filter all elements that satisfy some predicate.
filter :: (Key -> Bool) -> IntSet -> IntSet

-- | <i>O(n)</i>. partition the set according to some predicate.
partition :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>split</a> x set</tt>) is a
grpair <tt>(set1,set2)</tt> where <tt>set1</tt> comprises the elements
grof <tt>set</tt> less than <tt>x</tt> and <tt>set2</tt> comprises the
grelements of <tt>set</tt> greater than <tt>x</tt>.
gr
gr<pre>
grsplit 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])
gr</pre>
split :: Key -> IntSet -> (IntSet, IntSet)

-- | <i>O(min(n,W))</i>. Performs a <a>split</a> but also returns whether
grthe pivot element was found in the original set.
splitMember :: Key -> IntSet -> (IntSet, Bool, IntSet)

-- | <i>O(1)</i>. Decompose a set into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a set in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]]
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than two
grsubsets, but you should not depend on this behaviour because it can
grchange in the future without notice. Also, the current version does
grnot continue splitting all the way to individual singleton sets -- it
grstops at some point.
splitRoot :: IntSet -> [IntSet]

-- | <i>O(n*min(n,W))</i>. <tt><a>map</a> f s</tt> is the set obtained by
grapplying <tt>f</tt> to each element of <tt>s</tt>.
gr
grIt's worth noting that the size of the result may be smaller if, for
grsome <tt>(x,y)</tt>, <tt>x /= y &amp;&amp; f x == f y</tt>
map :: (Key -> Key) -> IntSet -> IntSet

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoAscList set = foldr (:) [] set
gr</pre>
foldr :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoDescList set = foldl (flip (:)) [] set
gr</pre>
foldl :: (a -> Key -> a) -> a -> IntSet -> a

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> Key -> a) -> a -> IntSet -> a

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.
gr
gr<i>Please note that fold will be deprecated in the future and
grremoved.</i>
fold :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(min(n,W))</i>. The minimal element of the set.
findMin :: IntSet -> Key

-- | <i>O(min(n,W))</i>. The maximal element of a set.
findMax :: IntSet -> Key

-- | <i>O(min(n,W))</i>. Delete the minimal element. Returns an empty set
grif the set is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Set</a> – versions prior to 0.5 threw an error if the <a>IntSet</a>
grwas already empty.
deleteMin :: IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete the maximal element. Returns an empty set
grif the set is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Set</a> – versions prior to 0.5 threw an error if the <a>IntSet</a>
grwas already empty.
deleteMax :: IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin set = (findMin set, deleteMin set)
gr</pre>
deleteFindMin :: IntSet -> (Key, IntSet)

-- | <i>O(min(n,W))</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax set = (findMax set, deleteMax set)
gr</pre>
deleteFindMax :: IntSet -> (Key, IntSet)

-- | <i>O(min(n,W))</i>. Retrieves the maximal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
maxView :: IntSet -> Maybe (Key, IntSet)

-- | <i>O(min(n,W))</i>. Retrieves the minimal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
minView :: IntSet -> Maybe (Key, IntSet)

-- | <i>O(n)</i>. An alias of <a>toAscList</a>. The elements of a set in
grascending order. Subject to list fusion.
elems :: IntSet -> [Key]

-- | <i>O(n)</i>. Convert the set to a list of elements. Subject to list
grfusion.
toList :: IntSet -> [Key]

-- | <i>O(n*min(n,W))</i>. Create a set from a list of integers.
fromList :: [Key] -> IntSet

-- | <i>O(n)</i>. Convert the set to an ascending list of elements. Subject
grto list fusion.
toAscList :: IntSet -> [Key]

-- | <i>O(n)</i>. Convert the set to a descending list of elements. Subject
grto list fusion.
toDescList :: IntSet -> [Key]

-- | <i>O(n)</i>. Build a set from an ascending list of elements. <i>The
grprecondition (input list is ascending) is not checked.</i>
fromAscList :: [Key] -> IntSet

-- | <i>O(n)</i>. Build a set from an ascending list of distinct elements.
gr<i>The precondition (input list is strictly ascending) is not
grchecked.</i>
fromDistinctAscList :: [Key] -> IntSet

-- | <i>O(n)</i>. Show the tree that implements the set. The tree is shown
grin a compressed, hanging format.
showTree :: IntSet -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the set. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
showTreeWith :: Bool -> Bool -> IntSet -> String
match :: Int -> Prefix -> Mask -> Bool
suffixBitMask :: Int
prefixBitMask :: Int
bitmapOf :: Int -> BitMap
zero :: Int -> Mask -> Bool
instance GHC.Exts.IsList Data.IntSet.Internal.IntSet
instance GHC.Base.Monoid Data.IntSet.Internal.IntSet
instance GHC.Base.Semigroup Data.IntSet.Internal.IntSet
instance Data.Data.Data Data.IntSet.Internal.IntSet
instance GHC.Classes.Eq Data.IntSet.Internal.IntSet
instance GHC.Classes.Ord Data.IntSet.Internal.IntSet
instance GHC.Show.Show Data.IntSet.Internal.IntSet
instance GHC.Read.Read Data.IntSet.Internal.IntSet
instance Control.DeepSeq.NFData Data.IntSet.Internal.IntSet


-- | An efficient implementation of integer sets.
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.IntSet (IntSet)
grimport qualified Data.IntSet as IntSet
gr</pre>
gr
grThe implementation is based on <i>big-endian patricia trees</i>. This
grdata structure performs especially well on binary operations like
gr<a>union</a> and <a>intersection</a>. However, my benchmarks show that
grit is also (much) faster on insertions and deletions when compared to
gra generic size-balanced set implementation (see <a>Data.Set</a>).
gr
gr<ul>
gr<li>Chris Okasaki and Andy Gill, "<i>Fast Mergeable Integer Maps</i>",
grWorkshop on ML, September 1998, pages 77-86,
gr<a>http://citeseer.ist.psu.edu/okasaki98fast.html</a></li>
gr<li>D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve
grInformation Coded In Alphanumeric/", Journal of the ACM, 15(4),
grOctober 1968, pages 514-534.</li>
gr</ul>
gr
grAdditionally, this implementation places bitmaps in the leaves of the
grtree. Their size is the natural size of a machine word (32 or 64 bits)
grand greatly reduce memory footprint and execution times for dense
grsets, e.g. sets where it is likely that many values lie close to each
grother. The asymptotics are not affected by this optimization.
gr
grMany operations have a worst-case complexity of <i>O(min(n,W))</i>.
grThis means that the operation can become linear in the number of
grelements with a maximum of <i>W</i> -- the number of bits in an
gr<a>Int</a> (32 or 64).
module Data.IntSet

-- | A set of integers.
data IntSet
type Key = Int

-- | <i>O(n+m)</i>. See <a>difference</a>.
(\\) :: IntSet -> IntSet -> IntSet
infixl 9 \\

-- | <i>O(1)</i>. Is the set empty?
null :: IntSet -> Bool

-- | <i>O(n)</i>. Cardinality of the set.
size :: IntSet -> Int

-- | <i>O(min(n,W))</i>. Is the value a member of the set?
member :: Key -> IntSet -> Bool

-- | <i>O(min(n,W))</i>. Is the element not in the set?
notMember :: Key -> IntSet -> Bool

-- | <i>O(log n)</i>. Find largest element smaller than the given one.
gr
gr<pre>
grlookupLT 3 (fromList [3, 5]) == Nothing
grlookupLT 5 (fromList [3, 5]) == Just 3
gr</pre>
lookupLT :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find smallest element greater than the given one.
gr
gr<pre>
grlookupGT 4 (fromList [3, 5]) == Just 5
grlookupGT 5 (fromList [3, 5]) == Nothing
gr</pre>
lookupGT :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find largest element smaller or equal to the given
grone.
gr
gr<pre>
grlookupLE 2 (fromList [3, 5]) == Nothing
grlookupLE 4 (fromList [3, 5]) == Just 3
grlookupLE 5 (fromList [3, 5]) == Just 5
gr</pre>
lookupLE :: Key -> IntSet -> Maybe Key

-- | <i>O(log n)</i>. Find smallest element greater or equal to the given
grone.
gr
gr<pre>
grlookupGE 3 (fromList [3, 5]) == Just 3
grlookupGE 4 (fromList [3, 5]) == Just 5
grlookupGE 6 (fromList [3, 5]) == Nothing
gr</pre>
lookupGE :: Key -> IntSet -> Maybe Key

-- | <i>O(n+m)</i>. Is this a subset? <tt>(s1 <a>isSubsetOf</a> s2)</tt>
grtells whether <tt>s1</tt> is a subset of <tt>s2</tt>.
isSubsetOf :: IntSet -> IntSet -> Bool

-- | <i>O(n+m)</i>. Is this a proper subset? (ie. a subset but not equal).
isProperSubsetOf :: IntSet -> IntSet -> Bool

-- | <i>O(n+m)</i>. Check whether two sets are disjoint (i.e. their
grintersection is empty).
gr
gr<pre>
grdisjoint (fromList [2,4,6])   (fromList [1,3])     == True
grdisjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False
grdisjoint (fromList [1,2])     (fromList [1,2,3,4]) == False
grdisjoint (fromList [])        (fromList [])        == True
gr</pre>
disjoint :: IntSet -> IntSet -> Bool

-- | <i>O(1)</i>. The empty set.
empty :: IntSet

-- | <i>O(1)</i>. A set of one element.
singleton :: Key -> IntSet

-- | <i>O(min(n,W))</i>. Add a value to the set. There is no left- or right
grbias for IntSets.
insert :: Key -> IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete a value in the set. Returns the original
grset when the value was not present.
delete :: Key -> IntSet -> IntSet

-- | <i>O(n+m)</i>. The union of two sets.
union :: IntSet -> IntSet -> IntSet

-- | The union of a list of sets.
unions :: [IntSet] -> IntSet

-- | <i>O(n+m)</i>. Difference between two sets.
difference :: IntSet -> IntSet -> IntSet

-- | <i>O(n+m)</i>. The intersection of two sets.
intersection :: IntSet -> IntSet -> IntSet

-- | <i>O(n)</i>. Filter all elements that satisfy some predicate.
filter :: (Key -> Bool) -> IntSet -> IntSet

-- | <i>O(n)</i>. partition the set according to some predicate.
partition :: (Key -> Bool) -> IntSet -> (IntSet, IntSet)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>split</a> x set</tt>) is a
grpair <tt>(set1,set2)</tt> where <tt>set1</tt> comprises the elements
grof <tt>set</tt> less than <tt>x</tt> and <tt>set2</tt> comprises the
grelements of <tt>set</tt> greater than <tt>x</tt>.
gr
gr<pre>
grsplit 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5])
gr</pre>
split :: Key -> IntSet -> (IntSet, IntSet)

-- | <i>O(min(n,W))</i>. Performs a <a>split</a> but also returns whether
grthe pivot element was found in the original set.
splitMember :: Key -> IntSet -> (IntSet, Bool, IntSet)

-- | <i>O(1)</i>. Decompose a set into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a set in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]]
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than two
grsubsets, but you should not depend on this behaviour because it can
grchange in the future without notice. Also, the current version does
grnot continue splitting all the way to individual singleton sets -- it
grstops at some point.
splitRoot :: IntSet -> [IntSet]

-- | <i>O(n*min(n,W))</i>. <tt><a>map</a> f s</tt> is the set obtained by
grapplying <tt>f</tt> to each element of <tt>s</tt>.
gr
grIt's worth noting that the size of the result may be smaller if, for
grsome <tt>(x,y)</tt>, <tt>x /= y &amp;&amp; f x == f y</tt>
map :: (Key -> Key) -> IntSet -> IntSet

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoAscList set = foldr (:) [] set
gr</pre>
foldr :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(n)</i>. Fold the elements in the set using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grtoDescList set = foldl (flip (:)) [] set
gr</pre>
foldl :: (a -> Key -> a) -> a -> IntSet -> a

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> Key -> a) -> a -> IntSet -> a

-- | <i>O(n)</i>. Fold the elements in the set using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.
gr
gr<i>Please note that fold will be deprecated in the future and
grremoved.</i>
fold :: (Key -> b -> b) -> b -> IntSet -> b

-- | <i>O(min(n,W))</i>. The minimal element of the set.
findMin :: IntSet -> Key

-- | <i>O(min(n,W))</i>. The maximal element of a set.
findMax :: IntSet -> Key

-- | <i>O(min(n,W))</i>. Delete the minimal element. Returns an empty set
grif the set is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Set</a> – versions prior to 0.5 threw an error if the <a>IntSet</a>
grwas already empty.
deleteMin :: IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete the maximal element. Returns an empty set
grif the set is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Set</a> – versions prior to 0.5 threw an error if the <a>IntSet</a>
grwas already empty.
deleteMax :: IntSet -> IntSet

-- | <i>O(min(n,W))</i>. Delete and find the minimal element.
gr
gr<pre>
grdeleteFindMin set = (findMin set, deleteMin set)
gr</pre>
deleteFindMin :: IntSet -> (Key, IntSet)

-- | <i>O(min(n,W))</i>. Delete and find the maximal element.
gr
gr<pre>
grdeleteFindMax set = (findMax set, deleteMax set)
gr</pre>
deleteFindMax :: IntSet -> (Key, IntSet)

-- | <i>O(min(n,W))</i>. Retrieves the maximal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
maxView :: IntSet -> Maybe (Key, IntSet)

-- | <i>O(min(n,W))</i>. Retrieves the minimal key of the set, and the set
grstripped of that element, or <a>Nothing</a> if passed an empty set.
minView :: IntSet -> Maybe (Key, IntSet)

-- | <i>O(n)</i>. An alias of <a>toAscList</a>. The elements of a set in
grascending order. Subject to list fusion.
elems :: IntSet -> [Key]

-- | <i>O(n)</i>. Convert the set to a list of elements. Subject to list
grfusion.
toList :: IntSet -> [Key]

-- | <i>O(n*min(n,W))</i>. Create a set from a list of integers.
fromList :: [Key] -> IntSet

-- | <i>O(n)</i>. Convert the set to an ascending list of elements. Subject
grto list fusion.
toAscList :: IntSet -> [Key]

-- | <i>O(n)</i>. Convert the set to a descending list of elements. Subject
grto list fusion.
toDescList :: IntSet -> [Key]

-- | <i>O(n)</i>. Build a set from an ascending list of elements. <i>The
grprecondition (input list is ascending) is not checked.</i>
fromAscList :: [Key] -> IntSet

-- | <i>O(n)</i>. Build a set from an ascending list of distinct elements.
gr<i>The precondition (input list is strictly ascending) is not
grchecked.</i>
fromDistinctAscList :: [Key] -> IntSet

-- | <i>O(n)</i>. Show the tree that implements the set. The tree is shown
grin a compressed, hanging format.
showTree :: IntSet -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the set. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
showTreeWith :: Bool -> Bool -> IntSet -> String


-- | <h1>WARNING</h1>
gr
grThis module is considered <b>internal</b>.
gr
grThe Package Versioning Policy <b>does not apply</b>.
gr
grThis contents of this module may change <b>in any way whatsoever</b>
grand <b>without any warning</b> between minor versions of this package.
gr
grAuthors importing this module are expected to track development
grclosely.
gr
gr<h1>Description</h1>
gr
grThis defines the data structures and core (hidden) manipulations on
grrepresentations.
module Data.IntMap.Internal

-- | A map of integers to values <tt>a</tt>.
data IntMap a
Bin :: {-# UNPACK #-} !Prefix -> {-# UNPACK #-} !Mask -> !(IntMap a) -> !(IntMap a) -> IntMap a
Tip :: {-# UNPACK #-} !Key -> a -> IntMap a
Nil :: IntMap a
type Key = Int

-- | <i>O(min(n,W))</i>. Find the value at a key. Calls <a>error</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: IntMap a -> Key -> a

-- | <i>O(min(n,W))</i>. Find the value at a key. Returns <a>Nothing</a>
grwhen the element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] !? 1 == Nothing
grfromList [(5,'a'), (3,'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: IntMap a -> Key -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: IntMap a -> IntMap b -> IntMap a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.IntMap.null (empty)           == True
grData.IntMap.null (singleton 1 'a') == False
gr</pre>
null :: IntMap a -> Bool

-- | <i>O(n)</i>. Number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: IntMap a -> Int

-- | <i>O(min(n,W))</i>. Is the key a member of the map?
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Is the key not a member of the map?
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Lookup the value at a key in the map. See also
gr<a>lookup</a>.
lookup :: Key -> IntMap a -> Maybe a

-- | <i>O(min(n,W))</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns <tt>def</tt>
grwhen the key is not an element of the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: a -> Key -> IntMap a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: IntMap a

-- | <i>O(1)</i>. A map of one element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: Key -> a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert a new key/value pair in the map. If the key
gris already present in the map, the associated value is replaced with
grthe supplied value, i.e. <a>insert</a> is equivalent to
gr<tt><a>insertWith</a> <a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWith</a> f key value mp</tt> will insert the pair (key,
grvalue) into <tt>mp</tt> if key does not exist in the map. If the key
grdoes exist, the function will insert <tt>f new_value old_value</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWithKey</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert <tt>f key new_value
grold_value</tt>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>insertLookupWithKey</a> f k
grx map</tt>) is a pair where the first element is equal to
gr(<tt><a>lookup</a> k map</tt>) and the second element equal to
gr(<tt><a>insertWithKey</a> f k x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. Delete a key and its value from the map. When the
grkey is not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: (a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: (Key -> a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: (a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Lookup and update. The function returns original
grvalue, if it is updated. This is different behavior than
gr<a>updateLookupWithKey</a>. Returns the original key value if the map
grentry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>alter</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alter</a> can be used to insert, delete, or update a value in an
gr<a>IntMap</a>. In short : <tt><a>lookup</a> k (<a>alter</a> f k m) = f
gr(<a>lookup</a> k m)</tt>.
alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in an <a>IntMap</a>. In short : <tt><a>lookup</a> k <a>$</a>
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; IntMap String -&gt; IO (IntMap String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: Functor f => (Maybe a -> f (Maybe a)) -> Key -> IntMap a -> f (IntMap a)

-- | <i>O(n+m)</i>. The (left-biased) union of two maps. It prefers the
grfirst map when duplicate keys are encountered, i.e. (<tt><a>union</a>
gr== <a>unionWith</a> <a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | The union of a list of maps.
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: [IntMap a] -> IntMap a

-- | The union of a list of maps, with a combining operation.
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: (a -> a -> a) -> [IntMap a] -> IntMap a

-- | <i>O(n+m)</i>. Difference between two maps (based on keys).
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The (left-biased) intersection of two maps (based on
grkeys).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMissing x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; Maybe
grz</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMatched x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt;
grMaybe z</tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f x y z</tt> and <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f x y z -> Key -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f x y</tt> and <tt>Key -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f x y -> Key -> x -> f (Maybe y)

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: SimpleWhenMissing a c -> SimpleWhenMissing b c -> SimpleWhenMatched a b c -> IntMap a -> IntMap b -> IntMap c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched
gr  :: (Key -&gt; x -&gt; y -&gt; Maybe z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (Key -> x -> y -> Maybe z) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched
gr  :: (Key -&gt; x -&gt; y -&gt; z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMatched :: Applicative f => (Key -> x -> y -> z) -> WhenMatched f x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (Key -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (Key -> x -> Maybe y) -> WhenMissing f x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (Key -> x -> y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (Key -> x -> Bool) -> WhenMissing f x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt>WhenMissing f k x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; f (Maybe
grz)</tt>.
data WhenMissing f x y
WhenMissing :: IntMap x -> f (IntMap y) -> Key -> x -> f (Maybe y) -> WhenMissing f x y
[missingSubtree] :: WhenMissing f x y -> IntMap x -> f (IntMap y)
[missingKey] :: WhenMissing f x y -> Key -> x -> f (Maybe y)

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt>WhenMatched f x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
newtype WhenMatched f x y z
WhenMatched :: Key -> x -> y -> f (Maybe z) -> WhenMatched f x y z
[matchedKey] :: WhenMatched f x y z -> Key -> x -> y -> f (Maybe z)

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f) => WhenMissing f a c -> WhenMissing f b c -> WhenMatched f a b c -> IntMap a -> IntMap b -> f (IntMap c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: (Key -> x -> y -> f (Maybe z)) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (Key -> x -> y -> f z) -> WhenMatched f x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (Key -> x -> f (Maybe y)) -> WhenMissing f x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (Key -> x -> f y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  \k x -&gt; (\b -&gt; guard b *&gt; Just x) &lt;$&gt; f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (Key -> x -> f Bool) -> WhenMissing f x x

-- | <i>O(n+m)</i>. A high-performance universal combining function. Using
gr<a>mergeWithKey</a>, all combining functions can be defined without
grany loss of efficiency (with exception of <a>union</a>,
gr<a>difference</a> and <a>intersection</a>, where sharing of some nodes
gris lost with <a>mergeWithKey</a>).
gr
grPlease make sure you know what is going on when using
gr<a>mergeWithKey</a>, otherwise you can be surprised by unexpected code
grgrowth or even corruption of the data structure.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine your custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>IntMap</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt> or
gr<tt><a>filterWithKey</a> f</tt> could be used for any <tt>f</tt>.
mergeWithKey :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c) -> IntMap a -> IntMap b -> IntMap c
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

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f s == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (,) k <a>$</a> f k v)
gr(<a>toList</a> m)</tt> That is, behaves exactly like a regular
gr<a>traverse</a> except that the traversing function also has access to
grthe key associated with a value.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)

-- | <i>O(n)</i>. The function <tt><a>mapAccum</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><a>mapAccumWithKey</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><tt>mapAccumR</tt></tt> threads an
graccumulating argument through the map in descending order of keys.
mapAccumRWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained
grby applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: (a -> a -> a) -> (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeysMonotonic</a> f s ==
gr<a>mapKeys</a> f s</tt>, but works only when <tt>f</tt> is strictly
grmonotonic. That is, for any values <tt>x</tt> and <tt>y</tt>, if
gr<tt>x</tt> &lt; <tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The
grprecondition is not checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has slightly better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
gr</pre>
mapKeysMonotonic :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (Key -> a -> m) -> IntMap a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: IntMap a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: IntMap a -> [Key]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Returns all key/value
grpairs in the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: IntMap a -> [(Key, a)]

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

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.IntSet.empty == empty
gr</pre>
fromSet :: (Key -> a) -> IntSet -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: IntMap a -> [(Key, a)]

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs with
gra combining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "ab"), (5, "cba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Build a map from a list of key/value pairs with
gra combining function. See also fromAscListWithKey'.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order.
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
gr</pre>
fromAscList :: [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
gr</pre>
fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]
gr</pre>
fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order and all distinct. <i>The precondition (input
grlist is strictly ascending) is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
gr</pre>
fromDistinctAscList :: forall a. [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Filter all values that satisfy some predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Filter all keys/values that satisfy some predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The restriction of a map to the keys in a set.
gr
gr<pre>
grm <a>restrictKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
gr</pre>
restrictKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n+m)</i>. Remove all the keys in a given set from a map.
gr
gr<pre>
grm <a>withoutKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
gr</pre>
withoutKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where all keys in <tt>map1</tt> are lower
grthan <tt>k</tt> and all keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Key -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(min(n,W))</i>. Performs a <a>split</a> but also returns whether
grthe pivot key was found in the original map.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6::Int] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d'),(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than two
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: IntMap a -> [IntMap a]

-- | <i>O(n+m)</i>. Is this a submap? Defined as (<tt><a>isSubmapOf</a> =
gr<a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. The expression (<tt><a>isSubmapOfBy</a> f m1 m2</tt>)
grreturns <a>True</a> if all keys in <tt>m1</tt> are in <tt>m2</tt>, and
grwhen <tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
gr</pre>
isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grDefined as (<tt><a>isProperSubmapOf</a> = <a>isProperSubmapOfBy</a>
gr(==)</tt>).
isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grThe expression (<tt><a>isProperSubmapOfBy</a> f m1 m2</tt>) returns
gr<a>True</a> when <tt>m1</tt> and <tt>m2</tt> are not equal, all keys
grin <tt>m1</tt> are in <tt>m2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(min(n,W))</i>. The minimal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMin :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMax :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The minimal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>minViewWithKey</a> if the map may be empty.
findMin :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>maxViewWithKey</a> if the map may be empty.
findMax :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. Delete the minimal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMin :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete the maximal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMax :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete and find the minimal element. This function
grthrows an error if the map is empty. Use <a>minViewWithKey</a> if the
grmap may be empty.
deleteFindMin :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Delete and find the maximal element. This function
grthrows an error if the map is empty. Use <a>maxViewWithKey</a> if the
grmap may be empty.
deleteFindMax :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Retrieves the minimal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
minView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
maxView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format.
showTree :: Show a => IntMap a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the map. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String
type Mask = Int
type Prefix = Int
type Nat = Word
natFromInt :: Key -> Nat
intFromNat :: Nat -> Key
link :: Prefix -> IntMap a -> Prefix -> IntMap a -> IntMap a
bin :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a
binCheckLeft :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a
binCheckRight :: Prefix -> Mask -> IntMap a -> IntMap a -> IntMap a

-- | Should this key follow the left subtree of a <a>Bin</a> with switching
grbit <tt>m</tt>? N.B., the answer is only valid when <tt>match i p
grm</tt> is true.
zero :: Key -> Mask -> Bool

-- | Does the key <tt>i</tt> differ from the prefix <tt>p</tt> before
grgetting to the switching bit <tt>m</tt>?
nomatch :: Key -> Prefix -> Mask -> Bool

-- | Does the key <tt>i</tt> match the prefix <tt>p</tt> (up to but not
grincluding bit <tt>m</tt>)?
match :: Key -> Prefix -> Mask -> Bool

-- | The prefix of key <tt>i</tt> up to (but not including) the switching
grbit <tt>m</tt>.
mask :: Key -> Mask -> Prefix

-- | The prefix of key <tt>i</tt> up to (but not including) the switching
grbit <tt>m</tt>.
maskW :: Nat -> Nat -> Prefix

-- | Does the left switching bit specify a shorter prefix?
shorter :: Mask -> Mask -> Bool

-- | The first switching bit where the two prefixes disagree.
branchMask :: Prefix -> Prefix -> Mask

-- | Return a word where only the highest bit is set.
highestBitMask :: Word -> Word

-- | Map covariantly over a <tt><a>WhenMissing</a> f x</tt>.
mapWhenMissing :: (Applicative f, Monad f) => (a -> b) -> WhenMissing f x a -> WhenMissing f x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b

-- | Map contravariantly over a <tt><a>WhenMissing</a> f _ x</tt>.
lmapWhenMissing :: (b -> a) -> WhenMissing f a x -> WhenMissing f b x

-- | Map contravariantly over a <tt><a>WhenMatched</a> f _ y z</tt>.
contramapFirstWhenMatched :: (b -> a) -> WhenMatched f a y z -> WhenMatched f b y z

-- | Map contravariantly over a <tt><a>WhenMatched</a> f x _ z</tt>.
contramapSecondWhenMatched :: (b -> a) -> WhenMatched f x a z -> WhenMatched f x b z

-- | Map covariantly over a <tt><a>WhenMissing</a> f x</tt>, using only a
gr'Functor f' constraint.
mapGentlyWhenMissing :: Functor f => (a -> b) -> WhenMissing f x a -> WhenMissing f x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f k x</tt>, using only a
gr'Functor f' constraint.
mapGentlyWhenMatched :: Functor f => (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b
instance GHC.Base.Functor f => GHC.Base.Functor (Data.IntMap.Internal.WhenMatched f x y)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => Control.Category.Category (Data.IntMap.Internal.WhenMatched f x)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => GHC.Base.Applicative (Data.IntMap.Internal.WhenMatched f x y)
instance (GHC.Base.Monad f, GHC.Base.Applicative f) => GHC.Base.Monad (Data.IntMap.Internal.WhenMatched f x y)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Functor (Data.IntMap.Internal.WhenMissing f x)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => Control.Category.Category (Data.IntMap.Internal.WhenMissing f)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Applicative (Data.IntMap.Internal.WhenMissing f x)
instance (GHC.Base.Applicative f, GHC.Base.Monad f) => GHC.Base.Monad (Data.IntMap.Internal.WhenMissing f x)
instance GHC.Base.Monoid (Data.IntMap.Internal.IntMap a)
instance GHC.Base.Semigroup (Data.IntMap.Internal.IntMap a)
instance Data.Foldable.Foldable Data.IntMap.Internal.IntMap
instance Data.Traversable.Traversable Data.IntMap.Internal.IntMap
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.IntMap.Internal.IntMap a)
instance Data.Data.Data a => Data.Data.Data (Data.IntMap.Internal.IntMap a)
instance GHC.Exts.IsList (Data.IntMap.Internal.IntMap a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.IntMap.Internal.IntMap a)
instance Data.Functor.Classes.Eq1 Data.IntMap.Internal.IntMap
instance GHC.Classes.Ord a => GHC.Classes.Ord (Data.IntMap.Internal.IntMap a)
instance Data.Functor.Classes.Ord1 Data.IntMap.Internal.IntMap
instance GHC.Base.Functor Data.IntMap.Internal.IntMap
instance GHC.Show.Show a => GHC.Show.Show (Data.IntMap.Internal.IntMap a)
instance Data.Functor.Classes.Show1 Data.IntMap.Internal.IntMap
instance GHC.Read.Read e => GHC.Read.Read (Data.IntMap.Internal.IntMap e)
instance Data.Functor.Classes.Read1 Data.IntMap.Internal.IntMap


-- | This module defines an API for writing functions that merge two maps.
grThe key functions are <a>merge</a> and <a>mergeA</a>. Each of these
grcan be used with several different "merge tactics".
gr
grThe <a>merge</a> and <a>mergeA</a> functions are shared by the lazy
grand strict modules. Only the choice of merge tactics determines
grstrictness. If you use <a>mapMissing</a> from this module then the
grresults will be forced before they are inserted. If you use
gr<a>mapMissing</a> from <a>Data.Map.Merge.Lazy</a> then they will not.
gr
gr<h2>Efficiency note</h2>
gr
grThe <tt>Category</tt>, <a>Applicative</a>, and <a>Monad</a> instances
grfor <a>WhenMissing</a> tactics are included because they are valid.
grHowever, they are inefficient in many cases and should usually be
gravoided. The instances for <a>WhenMatched</a> tactics should not pose
grany major efficiency problems.
module Data.IntMap.Merge.Strict

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMissing x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; Maybe
grz</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMatched x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt;
grMaybe z</tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: SimpleWhenMissing a c -> SimpleWhenMissing b c -> SimpleWhenMatched a b c -> IntMap a -> IntMap b -> IntMap c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched
gr  :: (Key -&gt; x -&gt; y -&gt; Maybe z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (Key -> x -> y -> Maybe z) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched
gr  :: (Key -&gt; x -&gt; y -&gt; z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMatched :: Applicative f => (Key -> x -> y -> z) -> WhenMatched f x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (Key -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (Key -> x -> Maybe y) -> WhenMissing f x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (Key -> x -> y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (Key -> x -> Bool) -> WhenMissing f x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt>WhenMissing f k x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; f (Maybe
grz)</tt>.
data WhenMissing f x y

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt>WhenMatched f x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
data WhenMatched f x y z

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f) => WhenMissing f a c -> WhenMissing f b c -> WhenMatched f a b c -> IntMap a -> IntMap b -> f (IntMap c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: (Key -> x -> y -> f (Maybe z)) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (Key -> x -> y -> f z) -> WhenMatched f x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (Key -> x -> f (Maybe y)) -> WhenMissing f x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (Key -> x -> f y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  \k x -&gt; (\b -&gt; guard b *&gt; Just x) &lt;$&gt; f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (Key -> x -> f Bool) -> WhenMissing f x x

-- | Map covariantly over a <tt><a>WhenMissing</a> f x</tt>.
mapWhenMissing :: (Applicative f, Monad f) => (a -> b) -> WhenMissing f x a -> WhenMissing f x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f x y z</tt> and <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f x y z -> Key -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f x y</tt> and <tt>Key -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f x y -> Key -> x -> f (Maybe y)


-- | This module defines an API for writing functions that merge two maps.
grThe key functions are <a>merge</a> and <a>mergeA</a>. Each of these
grcan be used with several different "merge tactics".
gr
grThe <a>merge</a> and <a>mergeA</a> functions are shared by the lazy
grand strict modules. Only the choice of merge tactics determines
grstrictness. If you use <a>mapMissing</a> from
gr<a>Data.Map.Merge.Strict</a> then the results will be forced before
grthey are inserted. If you use <a>mapMissing</a> from this module then
grthey will not.
gr
gr<h2>Efficiency note</h2>
gr
grThe <tt>Category</tt>, <a>Applicative</a>, and <a>Monad</a> instances
grfor <a>WhenMissing</a> tactics are included because they are valid.
grHowever, they are inefficient in many cases and should usually be
gravoided. The instances for <a>WhenMatched</a> tactics should not pose
grany major efficiency problems.
module Data.IntMap.Merge.Lazy

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMissing x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; Maybe
grz</tt>.
type SimpleWhenMissing = WhenMissing Identity

-- | A tactic for dealing with keys present in both maps in <a>merge</a>.
gr
grA tactic of type <tt>SimpleWhenMatched x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt;
grMaybe z</tt>.
type SimpleWhenMatched = WhenMatched Identity

-- | Merge two maps.
gr
gr<tt>merge</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>mapMaybeMissing</a> and <a>zipWithMaybeMatched</a>.
gr
grConsider
gr
gr<pre>
grmerge (mapMaybeMissing g1)
gr             (mapMaybeMissing g2)
gr             (zipWithMaybeMatched f)
gr             m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>merge</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
grmaybes = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grThis produces a <a>Maybe</a> for each key:
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>mapMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr</ul>
gr
grWhen <a>merge</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should typically use
gr<a>merge</a> to define your custom combining functions.
gr
grExamples:
gr
gr<pre>
grunionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grintersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)
gr</pre>
gr
gr<pre>
grdifferenceWith f = merge diffPreserve diffDrop f
gr</pre>
gr
gr<pre>
grsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -&gt; Nothing)
gr</pre>
gr
gr<pre>
grmapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)
gr</pre>
merge :: SimpleWhenMissing a c -> SimpleWhenMissing b c -> SimpleWhenMatched a b c -> IntMap a -> IntMap b -> IntMap c

-- | When a key is found in both maps, apply a function to the key and
grvalues and maybe use the result in the merged map.
gr
gr<pre>
grzipWithMaybeMatched
gr  :: (Key -&gt; x -&gt; y -&gt; Maybe z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMaybeMatched :: Applicative f => (Key -> x -> y -> Maybe z) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues and use the result in the merged map.
gr
gr<pre>
grzipWithMatched
gr  :: (Key -&gt; x -&gt; y -&gt; z)
gr  -&gt; SimpleWhenMatched x y z
gr</pre>
zipWithMatched :: Applicative f => (Key -> x -> y -> z) -> WhenMatched f x y z

-- | Map over the entries whose keys are missing from the other map,
groptionally removing some. This is the most powerful
gr<a>SimpleWhenMissing</a> tactic, but others are usually more
grefficient.
gr
gr<pre>
grmapMaybeMissing :: (Key -&gt; x -&gt; Maybe y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMaybeMissing f = traverseMaybeMissing (\k x -&gt; pure (f k x))
gr</pre>
gr
grbut <tt>mapMaybeMissing</tt> uses fewer unnecessary <a>Applicative</a>
groperations.
mapMaybeMissing :: Applicative f => (Key -> x -> Maybe y) -> WhenMissing f x y

-- | Drop all the entries whose keys are missing from the other map.
gr
gr<pre>
grdropMissing :: SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grdropMissing = mapMaybeMissing (\_ _ -&gt; Nothing)
gr</pre>
gr
grbut <tt>dropMissing</tt> is much faster.
dropMissing :: Applicative f => WhenMissing f x y

-- | Preserve, unchanged, the entries whose keys are missing from the other
grmap.
gr
gr<pre>
grpreserveMissing :: SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grpreserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -&gt; Just x)
gr</pre>
gr
grbut <tt>preserveMissing</tt> is much faster.
preserveMissing :: Applicative f => WhenMissing f x x

-- | Map over the entries whose keys are missing from the other map.
gr
gr<pre>
grmapMissing :: (k -&gt; x -&gt; y) -&gt; SimpleWhenMissing x y
gr</pre>
gr
gr<pre>
grmapMissing f = mapMaybeMissing (\k x -&gt; Just $ f k x)
gr</pre>
gr
grbut <tt>mapMissing</tt> is somewhat faster.
mapMissing :: Applicative f => (Key -> x -> y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map.
gr
gr<pre>
grfilterMissing :: (k -&gt; x -&gt; Bool) -&gt; SimpleWhenMissing x x
gr</pre>
gr
gr<pre>
grfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -&gt; guard (f k x) *&gt; Just x
gr</pre>
gr
grbut this should be a little faster.
filterMissing :: Applicative f => (Key -> x -> Bool) -> WhenMissing f x x

-- | A tactic for dealing with keys present in one map but not the other in
gr<a>merge</a> or <a>mergeA</a>.
gr
grA tactic of type <tt>WhenMissing f k x z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; f (Maybe
grz)</tt>.
data WhenMissing f x y

-- | A tactic for dealing with keys present in both maps in <a>merge</a> or
gr<a>mergeA</a>.
gr
grA tactic of type <tt>WhenMatched f x y z</tt> is an abstract
grrepresentation of a function of type <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
data WhenMatched f x y z

-- | An applicative version of <a>merge</a>.
gr
gr<tt>mergeA</tt> takes two <a>WhenMissing</a> tactics, a
gr<a>WhenMatched</a> tactic and two maps. It uses the tactics to merge
grthe maps. Its behavior is best understood via its fundamental tactics,
gr<a>traverseMaybeMissing</a> and <a>zipWithMaybeAMatched</a>.
gr
grConsider
gr
gr<pre>
grmergeA (traverseMaybeMissing g1)
gr              (traverseMaybeMissing g2)
gr              (zipWithMaybeAMatched f)
gr              m1 m2
gr</pre>
gr
grTake, for example,
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>), (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 = [(1, "one"), (2, "two"), (4, "three")]
gr</pre>
gr
gr<tt>mergeA</tt> will first '<tt>align'</tt> these maps by key:
gr
gr<pre>
grm1 = [(0, <tt>a</tt>), (1, <tt>b</tt>),               (3,<tt>c</tt>), (4, <tt>d</tt>)]
grm2 =           [(1, "one"), (2, "two"),          (4, "three")]
gr</pre>
gr
grIt will then pass the individual entries and pairs of entries to
gr<tt>g1</tt>, <tt>g2</tt>, or <tt>f</tt> as appropriate:
gr
gr<pre>
gractions = [g1 0 <tt>a</tt>, f 1 <tt>b</tt> "one", g2 2 "two", g1 3 <tt>c</tt>, f 4 <tt>d</tt> "three"]
gr</pre>
gr
grNext, it will perform the actions in the <tt>actions</tt> list in
grorder from left to right.
gr
gr<pre>
grkeys =     0        1          2           3        4
grresults = [Nothing, Just True, Just False, Nothing, Just True]
gr</pre>
gr
grFinally, the <tt>Just</tt> results are collected into a map:
gr
gr<pre>
grreturn value = [(1, True), (2, False), (4, True)]
gr</pre>
gr
grThe other tactics below are optimizations or simplifications of
gr<a>traverseMaybeMissing</a> for special cases. Most importantly,
gr
gr<ul>
gr<li><a>dropMissing</a> drops all the keys.</li>
gr<li><a>preserveMissing</a> leaves all the entries alone.</li>
gr<li><a>mapMaybeMissing</a> does not use the <a>Applicative</a>
grcontext.</li>
gr</ul>
gr
grWhen <a>mergeA</a> is given three arguments, it is inlined at the call
grsite. To prevent excessive inlining, you should generally only use
gr<a>mergeA</a> to define custom combining functions.
mergeA :: (Applicative f) => WhenMissing f a c -> WhenMissing f b c -> WhenMatched f a b c -> IntMap a -> IntMap b -> f (IntMap c)

-- | When a key is found in both maps, apply a function to the key and
grvalues, perform the resulting action, and maybe use the result in the
grmerged map.
gr
grThis is the fundamental <a>WhenMatched</a> tactic.
zipWithMaybeAMatched :: (Key -> x -> y -> f (Maybe z)) -> WhenMatched f x y z

-- | When a key is found in both maps, apply a function to the key and
grvalues to produce an action and use its result in the merged map.
zipWithAMatched :: Applicative f => (Key -> x -> y -> f z) -> WhenMatched f x y z

-- | Traverse over the entries whose keys are missing from the other map,
groptionally producing values to put in the result. This is the most
grpowerful <a>WhenMissing</a> tactic, but others are usually more
grefficient.
traverseMaybeMissing :: Applicative f => (Key -> x -> f (Maybe y)) -> WhenMissing f x y

-- | Traverse over the entries whose keys are missing from the other map.
traverseMissing :: Applicative f => (Key -> x -> f y) -> WhenMissing f x y

-- | Filter the entries whose keys are missing from the other map using
grsome <a>Applicative</a> action.
gr
gr<pre>
grfilterAMissing f = Merge.Lazy.traverseMaybeMissing $
gr  \k x -&gt; (\b -&gt; guard b *&gt; Just x) &lt;$&gt; f k x
gr</pre>
gr
grbut this should be a little faster.
filterAMissing :: Applicative f => (Key -> x -> f Bool) -> WhenMissing f x x

-- | Map covariantly over a <tt><a>WhenMissing</a> f x</tt>.
mapWhenMissing :: (Applicative f, Monad f) => (a -> b) -> WhenMissing f x a -> WhenMissing f x b

-- | Map covariantly over a <tt><a>WhenMatched</a> f x y</tt>.
mapWhenMatched :: Functor f => (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b

-- | Map contravariantly over a <tt><a>WhenMissing</a> f _ x</tt>.
lmapWhenMissing :: (b -> a) -> WhenMissing f a x -> WhenMissing f b x

-- | Map contravariantly over a <tt><a>WhenMatched</a> f _ y z</tt>.
contramapFirstWhenMatched :: (b -> a) -> WhenMatched f a y z -> WhenMatched f b y z

-- | Map contravariantly over a <tt><a>WhenMatched</a> f x _ z</tt>.
contramapSecondWhenMatched :: (b -> a) -> WhenMatched f x a z -> WhenMatched f x b z

-- | Along with zipWithMaybeAMatched, witnesses the isomorphism between
gr<tt>WhenMatched f x y z</tt> and <tt>Key -&gt; x -&gt; y -&gt; f
gr(Maybe z)</tt>.
runWhenMatched :: WhenMatched f x y z -> Key -> x -> y -> f (Maybe z)

-- | Along with traverseMaybeMissing, witnesses the isomorphism between
gr<tt>WhenMissing f x y</tt> and <tt>Key -&gt; x -&gt; f (Maybe y)</tt>.
runWhenMissing :: WhenMissing f x y -> Key -> x -> f (Maybe y)


-- | An efficient implementation of maps from integer keys to values
gr(dictionaries).
gr
grAPI of this module is strict in both the keys and the values. If you
grneed value-lazy maps, use <a>Data.IntMap.Lazy</a> instead. The
gr<a>IntMap</a> type itself is shared between the lazy and strict
grmodules, meaning that the same <a>IntMap</a> value can be passed to
grfunctions in both modules (although that is rarely needed).
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.IntMap.Strict (IntMap)
grimport qualified Data.IntMap.Strict as IntMap
gr</pre>
gr
grThe implementation is based on <i>big-endian patricia trees</i>. This
grdata structure performs especially well on binary operations like
gr<a>union</a> and <a>intersection</a>. However, my benchmarks show that
grit is also (much) faster on insertions and deletions when compared to
gra generic size-balanced map implementation (see <a>Data.Map</a>).
gr
gr<ul>
gr<li>Chris Okasaki and Andy Gill, "<i>Fast Mergeable Integer Maps</i>",
grWorkshop on ML, September 1998, pages 77-86,
gr<a>http://citeseer.ist.psu.edu/okasaki98fast.html</a></li>
gr<li>D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve
grInformation Coded In Alphanumeric/", Journal of the ACM, 15(4),
grOctober 1968, pages 514-534.</li>
gr</ul>
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation <a>http://en.wikipedia.org/wiki/Big_O_notation</a>. Many
groperations have a worst-case complexity of <i>O(min(n,W))</i>. This
grmeans that the operation can become linear in the number of elements
grwith a maximum of <i>W</i> -- the number of bits in an <a>Int</a> (32
gror 64).
gr
grBe aware that the <a>Functor</a>, <a>Traversable</a> and <tt>Data</tt>
grinstances are the same as for the <a>Data.IntMap.Lazy</a> module, so
grif they are used on strict maps, the resulting maps will be lazy.
module Data.IntMap.Strict

-- | A map of integers to values <tt>a</tt>.
data IntMap a
type Key = Int

-- | <i>O(min(n,W))</i>. Find the value at a key. Calls <a>error</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: IntMap a -> Key -> a

-- | <i>O(min(n,W))</i>. Find the value at a key. Returns <a>Nothing</a>
grwhen the element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] !? 1 == Nothing
grfromList [(5,'a'), (3,'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: IntMap a -> Key -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: IntMap a -> IntMap b -> IntMap a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.IntMap.null (empty)           == True
grData.IntMap.null (singleton 1 'a') == False
gr</pre>
null :: IntMap a -> Bool

-- | <i>O(n)</i>. Number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: IntMap a -> Int

-- | <i>O(min(n,W))</i>. Is the key a member of the map?
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Is the key not a member of the map?
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Lookup the value at a key in the map. See also
gr<a>lookup</a>.
lookup :: Key -> IntMap a -> Maybe a

-- | <i>O(min(n,W))</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns <tt>def</tt>
grwhen the key is not an element of the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: a -> Key -> IntMap a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: IntMap a

-- | <i>O(1)</i>. A map of one element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: Key -> a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert a new key/value pair in the map. If the key
gris already present in the map, the associated value is replaced with
grthe supplied value, i.e. <a>insert</a> is equivalent to
gr<tt><a>insertWith</a> <a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWith</a> f key value mp</tt> will insert the pair (key,
grvalue) into <tt>mp</tt> if key does not exist in the map. If the key
grdoes exist, the function will insert <tt>f new_value old_value</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWithKey</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert <tt>f key new_value
grold_value</tt>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
gr
grIf the key exists in the map, this function is lazy in <tt>x</tt> but
grstrict in the result of <tt>f</tt>.
insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>insertLookupWithKey</a> f k
grx map</tt>) is a pair where the first element is equal to
gr(<tt><a>lookup</a> k map</tt>) and the second element equal to
gr(<tt><a>insertWithKey</a> f k x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. Delete a key and its value from the map. When the
grkey is not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: (a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: (Key -> a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: (a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Lookup and update. The function returns original
grvalue, if it is updated. This is different behavior than
gr<a>updateLookupWithKey</a>. Returns the original key value if the map
grentry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>alter</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alter</a> can be used to insert, delete, or update a value in an
gr<a>IntMap</a>. In short : <tt><a>lookup</a> k (<a>alter</a> f k m) = f
gr(<a>lookup</a> k m)</tt>.
alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in an <a>IntMap</a>. In short : <tt><a>lookup</a> k <a>$</a>
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; IntMap String -&gt; IO (IntMap String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map.
alterF :: Functor f => (Maybe a -> f (Maybe a)) -> Key -> IntMap a -> f (IntMap a)

-- | <i>O(n+m)</i>. The (left-biased) union of two maps. It prefers the
grfirst map when duplicate keys are encountered, i.e. (<tt><a>union</a>
gr== <a>unionWith</a> <a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | The union of a list of maps.
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: [IntMap a] -> IntMap a

-- | The union of a list of maps, with a combining operation.
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: (a -> a -> a) -> [IntMap a] -> IntMap a

-- | <i>O(n+m)</i>. Difference between two maps (based on keys).
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The (left-biased) intersection of two maps (based on
grkeys).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n+m)</i>. A high-performance universal combining function. Using
gr<a>mergeWithKey</a>, all combining functions can be defined without
grany loss of efficiency (with exception of <a>union</a>,
gr<a>difference</a> and <a>intersection</a>, where sharing of some nodes
gris lost with <a>mergeWithKey</a>).
gr
grPlease make sure you know what is going on when using
gr<a>mergeWithKey</a>, otherwise you can be surprised by unexpected code
grgrowth or even corruption of the data structure.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine your custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>IntMap</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt> or
gr<tt><a>filterWithKey</a> f</tt> could be used for any <tt>f</tt>.
mergeWithKey :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f s == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (,) k <a>$</a> f k v)
gr(<a>toList</a> m)</tt> That is, behaves exactly like a regular
gr<a>traverse</a> except that the traversing function also has access to
grthe key associated with a value.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)

-- | <i>O(n)</i>. The function <tt><a>mapAccum</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><a>mapAccumWithKey</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><tt>mapAccumR</tt></tt> threads an
graccumulating argument through the map in descending order of keys.
mapAccumRWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained
grby applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*log n)</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: (a -> a -> a) -> (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeysMonotonic</a> f s ==
gr<a>mapKeys</a> f s</tt>, but works only when <tt>f</tt> is strictly
grmonotonic. That is, for any values <tt>x</tt> and <tt>y</tt>, if
gr<tt>x</tt> &lt; <tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The
grprecondition is not checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has slightly better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
gr</pre>
mapKeysMonotonic :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (Key -> a -> m) -> IntMap a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: IntMap a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: IntMap a -> [Key]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Returns all key/value
grpairs in the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: IntMap a -> [(Key, a)]

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

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.IntSet.empty == empty
gr</pre>
fromSet :: (Key -> a) -> IntSet -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: IntMap a -> [(Key, a)]

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs with
gra combining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Build a map from a list of key/value pairs with
gra combining function. See also fromAscListWithKey'.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order.
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
gr</pre>
fromAscList :: [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
gr</pre>
fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
gr</pre>
fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order and all distinct. <i>The precondition (input
grlist is strictly ascending) is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
gr</pre>
fromDistinctAscList :: [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Filter all values that satisfy some predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Filter all keys/values that satisfy some predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The restriction of a map to the keys in a set.
gr
gr<pre>
grm <a>restrictKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
gr</pre>
restrictKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n+m)</i>. Remove all the keys in a given set from a map.
gr
gr<pre>
grm <a>withoutKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
gr</pre>
withoutKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where all keys in <tt>map1</tt> are lower
grthan <tt>k</tt> and all keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Key -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(min(n,W))</i>. Performs a <a>split</a> but also returns whether
grthe pivot key was found in the original map.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6::Int] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d'),(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than two
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: IntMap a -> [IntMap a]

-- | <i>O(n+m)</i>. Is this a submap? Defined as (<tt><a>isSubmapOf</a> =
gr<a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. The expression (<tt><a>isSubmapOfBy</a> f m1 m2</tt>)
grreturns <a>True</a> if all keys in <tt>m1</tt> are in <tt>m2</tt>, and
grwhen <tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
gr</pre>
isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grDefined as (<tt><a>isProperSubmapOf</a> = <a>isProperSubmapOfBy</a>
gr(==)</tt>).
isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grThe expression (<tt><a>isProperSubmapOfBy</a> f m1 m2</tt>) returns
gr<a>True</a> when <tt>m1</tt> and <tt>m2</tt> are not equal, all keys
grin <tt>m1</tt> are in <tt>m2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(min(n,W))</i>. The minimal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMin :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMax :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The minimal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>minViewWithKey</a> if the map may be empty.
findMin :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>maxViewWithKey</a> if the map may be empty.
findMax :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. Delete the minimal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMin :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete the maximal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMax :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete and find the minimal element. This function
grthrows an error if the map is empty. Use <a>minViewWithKey</a> if the
grmap may be empty.
deleteFindMin :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Delete and find the maximal element. This function
grthrows an error if the map is empty. Use <a>maxViewWithKey</a> if the
grmap may be empty.
deleteFindMax :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Retrieves the minimal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
minView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
maxView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format.

-- | <i>Deprecated: These debugging functions will be removed from this
grmodule. They are available from Data.IntMap.Internal.Debug.</i>
showTree :: Show a => IntMap a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the map. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.

-- | <i>Deprecated: These debugging functions will be removed from this
grmodule. They are available from Data.IntMap.Internal.Debug.</i>
showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String


-- | An efficient implementation of maps from integer keys to values
gr(dictionaries).
gr
grAPI of this module is strict in the keys, but lazy in the values. If
gryou need value-strict maps, use <a>Data.IntMap.Strict</a> instead. The
gr<a>IntMap</a> type itself is shared between the lazy and strict
grmodules, meaning that the same <a>IntMap</a> value can be passed to
grfunctions in both modules (although that is rarely needed).
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.IntMap.Lazy (IntMap)
grimport qualified Data.IntMap.Lazy as IntMap
gr</pre>
gr
grThe implementation is based on <i>big-endian patricia trees</i>. This
grdata structure performs especially well on binary operations like
gr<a>union</a> and <a>intersection</a>. However, my benchmarks show that
grit is also (much) faster on insertions and deletions when compared to
gra generic size-balanced map implementation (see <a>Data.Map</a>).
gr
gr<ul>
gr<li>Chris Okasaki and Andy Gill, "<i>Fast Mergeable Integer Maps</i>",
grWorkshop on ML, September 1998, pages 77-86,
gr<a>http://citeseer.ist.psu.edu/okasaki98fast.html</a></li>
gr<li>D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve
grInformation Coded In Alphanumeric/", Journal of the ACM, 15(4),
grOctober 1968, pages 514-534.</li>
gr</ul>
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation <a>http://en.wikipedia.org/wiki/Big_O_notation</a>. Many
groperations have a worst-case complexity of <i>O(min(n,W))</i>. This
grmeans that the operation can become linear in the number of elements
grwith a maximum of <i>W</i> -- the number of bits in an <a>Int</a> (32
gror 64).
module Data.IntMap.Lazy

-- | A map of integers to values <tt>a</tt>.
data IntMap a
type Key = Int

-- | <i>O(min(n,W))</i>. Find the value at a key. Calls <a>error</a> when
grthe element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] ! 1    Error: element not in the map
grfromList [(5,'a'), (3,'b')] ! 5 == 'a'
gr</pre>
(!) :: IntMap a -> Key -> a

-- | <i>O(min(n,W))</i>. Find the value at a key. Returns <a>Nothing</a>
grwhen the element can not be found.
gr
gr<pre>
grfromList [(5,'a'), (3,'b')] !? 1 == Nothing
grfromList [(5,'a'), (3,'b')] !? 5 == Just 'a'
gr</pre>
(!?) :: IntMap a -> Key -> Maybe a
infixl 9 !?

-- | Same as <a>difference</a>.
(\\) :: IntMap a -> IntMap b -> IntMap a
infixl 9 \\

-- | <i>O(1)</i>. Is the map empty?
gr
gr<pre>
grData.IntMap.null (empty)           == True
grData.IntMap.null (singleton 1 'a') == False
gr</pre>
null :: IntMap a -> Bool

-- | <i>O(n)</i>. Number of elements in the map.
gr
gr<pre>
grsize empty                                   == 0
grsize (singleton 1 'a')                       == 1
grsize (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3
gr</pre>
size :: IntMap a -> Int

-- | <i>O(min(n,W))</i>. Is the key a member of the map?
gr
gr<pre>
grmember 5 (fromList [(5,'a'), (3,'b')]) == True
grmember 1 (fromList [(5,'a'), (3,'b')]) == False
gr</pre>
member :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Is the key not a member of the map?
gr
gr<pre>
grnotMember 5 (fromList [(5,'a'), (3,'b')]) == False
grnotMember 1 (fromList [(5,'a'), (3,'b')]) == True
gr</pre>
notMember :: Key -> IntMap a -> Bool

-- | <i>O(min(n,W))</i>. Lookup the value at a key in the map. See also
gr<a>lookup</a>.
lookup :: Key -> IntMap a -> Maybe a

-- | <i>O(min(n,W))</i>. The expression <tt>(<a>findWithDefault</a> def k
grmap)</tt> returns the value at key <tt>k</tt> or returns <tt>def</tt>
grwhen the key is not an element of the map.
gr
gr<pre>
grfindWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'
grfindWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'
gr</pre>
findWithDefault :: a -> Key -> IntMap a -> a

-- | <i>O(log n)</i>. Find largest key smaller than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
gr</pre>
lookupLT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater than the given one and
grreturn the corresponding (key, value) pair.
gr
gr<pre>
grlookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGT :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find largest key smaller or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing
grlookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
gr</pre>
lookupLE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(log n)</i>. Find smallest key greater or equal to the given one
grand return the corresponding (key, value) pair.
gr
gr<pre>
grlookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')
grlookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')
grlookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing
gr</pre>
lookupGE :: Key -> IntMap a -> Maybe (Key, a)

-- | <i>O(1)</i>. The empty map.
gr
gr<pre>
grempty      == fromList []
grsize empty == 0
gr</pre>
empty :: IntMap a

-- | <i>O(1)</i>. A map of one element.
gr
gr<pre>
grsingleton 1 'a'        == fromList [(1, 'a')]
grsize (singleton 1 'a') == 1
gr</pre>
singleton :: Key -> a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert a new key/value pair in the map. If the key
gris already present in the map, the associated value is replaced with
grthe supplied value, i.e. <a>insert</a> is equivalent to
gr<tt><a>insertWith</a> <a>const</a></tt>.
gr
gr<pre>
grinsert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]
grinsert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]
grinsert 5 'x' empty                         == singleton 5 'x'
gr</pre>
insert :: Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWith</a> f key value mp</tt> will insert the pair (key,
grvalue) into <tt>mp</tt> if key does not exist in the map. If the key
grdoes exist, the function will insert <tt>f new_value old_value</tt>.
gr
gr<pre>
grinsertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]
grinsertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWith (++) 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWith :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Insert with a combining function.
gr<tt><a>insertWithKey</a> f key value mp</tt> will insert the pair
gr(key, value) into <tt>mp</tt> if key does not exist in the map. If the
grkey does exist, the function will insert <tt>f key new_value
grold_value</tt>.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]
grinsertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]
grinsertWithKey f 5 "xxx" empty                         == singleton 5 "xxx"
gr</pre>
insertWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>insertLookupWithKey</a> f k
grx map</tt>) is a pair where the first element is equal to
gr(<tt><a>lookup</a> k map</tt>) and the second element equal to
gr(<tt><a>insertWithKey</a> f k x map</tt>).
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grinsertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])
grinsertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "xxx")])
grinsertLookupWithKey f 5 "xxx" empty                         == (Nothing,  singleton 5 "xxx")
gr</pre>
gr
grThis is how to define <tt>insertLookup</tt> using
gr<tt>insertLookupWithKey</tt>:
gr
gr<pre>
grlet insertLookup kx x t = insertLookupWithKey (\_ a _ -&gt; a) kx x t
grinsertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])
grinsertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a"), (7, "x")])
gr</pre>
insertLookupWithKey :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. Delete a key and its value from the map. When the
grkey is not a member of the map, the original map is returned.
gr
gr<pre>
grdelete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grdelete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grdelete 5 empty                         == empty
gr</pre>
delete :: Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
gradjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
gradjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjust ("new " ++) 7 empty                         == empty
gr</pre>
adjust :: (a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Adjust a value at a specific key. When the key is
grnot a member of the map, the original map is returned.
gr
gr<pre>
grlet f key x = (show key) ++ ":new " ++ x
gradjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
gradjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
gradjustWithKey f 7 empty                         == empty
gr</pre>
adjustWithKey :: (Key -> a -> a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grupdate f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]
grupdate f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdate f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
update :: (a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. The expression (<tt><a>update</a> f k map</tt>)
grupdates the value <tt>x</tt> at <tt>k</tt> (if it is in the map). If
gr(<tt>f k x</tt>) is <a>Nothing</a>, the element is deleted. If it is
gr(<tt><a>Just</a> y</tt>), the key <tt>k</tt> is bound to the new value
gr<tt>y</tt>.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]
grupdateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]
grupdateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Lookup and update. The function returns original
grvalue, if it is updated. This is different behavior than
gr<a>updateLookupWithKey</a>. Returns the original key value if the map
grentry is deleted.
gr
gr<pre>
grlet f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing
grupdateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")])
grupdateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing,  fromList [(3, "b"), (5, "a")])
grupdateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")
gr</pre>
updateLookupWithKey :: (Key -> a -> Maybe a) -> Key -> IntMap a -> (Maybe a, IntMap a)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>alter</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alter</a> can be used to insert, delete, or update a value in an
gr<a>IntMap</a>. In short : <tt><a>lookup</a> k (<a>alter</a> f k m) = f
gr(<a>lookup</a> k m)</tt>.
alter :: (Maybe a -> Maybe a) -> Key -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. The expression (<tt><a>alterF</a> f k map</tt>)
gralters the value <tt>x</tt> at <tt>k</tt>, or absence thereof.
gr<a>alterF</a> can be used to inspect, insert, delete, or update a
grvalue in an <a>IntMap</a>. In short : <tt><a>lookup</a> k <a>$</a>
gr<a>alterF</a> f k m = f (<a>lookup</a> k m)</tt>.
gr
grExample:
gr
gr<pre>
grinteractiveAlter :: Int -&gt; IntMap String -&gt; IO (IntMap String)
grinteractiveAlter k m = alterF f k m where
gr  f Nothing -&gt; do
gr     putStrLn $ show k ++
gr         " was not found in the map. Would you like to add it?"
gr     getUserResponse1 :: IO (Maybe String)
gr  f (Just old) -&gt; do
gr     putStrLn "The key is currently bound to " ++ show old ++
gr         ". Would you like to change or delete it?"
gr     getUserresponse2 :: IO (Maybe String)
gr</pre>
gr
gr<a>alterF</a> is the most general operation for working with an
grindividual key that may or may not be in a given map.
gr
grNote: <a>alterF</a> is a flipped version of the <tt>at</tt> combinator
grfrom <a>At</a>.
alterF :: Functor f => (Maybe a -> f (Maybe a)) -> Key -> IntMap a -> f (IntMap a)

-- | <i>O(n+m)</i>. The (left-biased) union of two maps. It prefers the
grfirst map when duplicate keys are encountered, i.e. (<tt><a>union</a>
gr== <a>unionWith</a> <a>const</a></tt>).
gr
gr<pre>
grunion (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]
gr</pre>
union :: IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grunionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]
gr</pre>
unionWith :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The union with a combining function.
gr
gr<pre>
grlet f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value
grunionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]
gr</pre>
unionWithKey :: (Key -> a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

-- | The union of a list of maps.
gr
gr<pre>
grunions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "b"), (5, "a"), (7, "C")]
grunions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]
gr    == fromList [(3, "B3"), (5, "A3"), (7, "C")]
gr</pre>
unions :: [IntMap a] -> IntMap a

-- | The union of a list of maps, with a combining operation.
gr
gr<pre>
grunionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]
gr    == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]
gr</pre>
unionsWith :: (a -> a -> a) -> [IntMap a] -> IntMap a

-- | <i>O(n+m)</i>. Difference between two maps (based on keys).
gr
gr<pre>
grdifference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"
gr</pre>
difference :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function.
gr
gr<pre>
grlet f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing
grdifferenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])
gr    == singleton 3 "b:B"
gr</pre>
differenceWith :: (a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. Difference with a combining function. When two equal
grkeys are encountered, the combining function is applied to the key and
grboth values. If it returns <a>Nothing</a>, the element is discarded
gr(proper set difference). If it returns (<tt><a>Just</a> y</tt>), the
grelement is updated with a new value <tt>y</tt>.
gr
gr<pre>
grlet f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing
grdifferenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])
gr    == singleton 3 "3:b|B"
gr</pre>
differenceWithKey :: (Key -> a -> b -> Maybe a) -> IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The (left-biased) intersection of two maps (based on
grkeys).
gr
gr<pre>
grintersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"
gr</pre>
intersection :: IntMap a -> IntMap b -> IntMap a

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grintersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"
gr</pre>
intersectionWith :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n+m)</i>. The intersection with a combining function.
gr
gr<pre>
grlet f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar
grintersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"
gr</pre>
intersectionWithKey :: (Key -> a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n+m)</i>. A high-performance universal combining function. Using
gr<a>mergeWithKey</a>, all combining functions can be defined without
grany loss of efficiency (with exception of <a>union</a>,
gr<a>difference</a> and <a>intersection</a>, where sharing of some nodes
gris lost with <a>mergeWithKey</a>).
gr
grPlease make sure you know what is going on when using
gr<a>mergeWithKey</a>, otherwise you can be surprised by unexpected code
grgrowth or even corruption of the data structure.
gr
grWhen <a>mergeWithKey</a> is given three arguments, it is inlined to
grthe call site. You should therefore use <a>mergeWithKey</a> only to
grdefine your custom combining functions. For example, you could define
gr<a>unionWithKey</a>, <a>differenceWithKey</a> and
gr<a>intersectionWithKey</a> as
gr
gr<pre>
grmyUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) id id m1 m2
grmyDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2
grmyIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -&gt; Just (f k x1 x2)) (const empty) (const empty) m1 m2
gr</pre>
gr
grWhen calling <tt><a>mergeWithKey</a> combine only1 only2</tt>, a
grfunction combining two <a>IntMap</a>s is created, such that
gr
gr<ul>
gr<li>if a key is present in both maps, it is passed with both
grcorresponding values to the <tt>combine</tt> function. Depending on
grthe result, the key is either present in the result with specified
grvalue, or is left out;</li>
gr<li>a nonempty subtree present only in the first map is passed to
gr<tt>only1</tt> and the output is added to the result;</li>
gr<li>a nonempty subtree present only in the second map is passed to
gr<tt>only2</tt> and the output is added to the result.</li>
gr</ul>
gr
grThe <tt>only1</tt> and <tt>only2</tt> methods <i>must return a map
grwith a subset (possibly empty) of the keys of the given map</i>. The
grvalues can be modified arbitrarily. Most common variants of
gr<tt>only1</tt> and <tt>only2</tt> are <a>id</a> and <tt><a>const</a>
gr<a>empty</a></tt>, but for example <tt><a>map</a> f</tt> or
gr<tt><a>filterWithKey</a> f</tt> could be used for any <tt>f</tt>.
mergeWithKey :: (Key -> a -> b -> Maybe c) -> (IntMap a -> IntMap c) -> (IntMap b -> IntMap c) -> IntMap a -> IntMap b -> IntMap c

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grmap (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]
gr</pre>
map :: (a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map a function over all values in the map.
gr
gr<pre>
grlet f key x = (show key) ++ ":" ++ x
grmapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]
gr</pre>
mapWithKey :: (Key -> a -> b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. <tt><a>traverseWithKey</a> f s == <a>fromList</a>
gr<a>$</a> <a>traverse</a> ((k, v) -&gt; (,) k <a>$</a> f k v)
gr(<a>toList</a> m)</tt> That is, behaves exactly like a regular
gr<a>traverse</a> except that the traversing function also has access to
grthe key associated with a value.
gr
gr<pre>
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])
grtraverseWithKey (\k v -&gt; if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')])           == Nothing
gr</pre>
traverseWithKey :: Applicative t => (Key -> a -> t b) -> IntMap a -> t (IntMap b)

-- | <i>O(n)</i>. The function <tt><a>mapAccum</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a b = (a ++ b, b ++ "X")
grmapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccum :: (a -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><a>mapAccumWithKey</a></tt> threads an
graccumulating argument through the map in ascending order of keys.
gr
gr<pre>
grlet f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
grmapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
gr</pre>
mapAccumWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n)</i>. The function <tt><tt>mapAccumR</tt></tt> threads an
graccumulating argument through the map in descending order of keys.
mapAccumRWithKey :: (a -> Key -> b -> (a, c)) -> a -> IntMap b -> (a, IntMap c)

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeys</a> f s</tt> is the map obtained
grby applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the value at the
grgreatest of the original keys is retained.
gr
gr<pre>
grmapKeys (+ 1) (fromList [(5,"a"), (3,"b")])                        == fromList [(4, "b"), (6, "a")]
grmapKeys (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
grmapKeys (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
gr</pre>
mapKeys :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeysWith</a> c f s</tt> is the map
grobtained by applying <tt>f</tt> to each key of <tt>s</tt>.
gr
grThe size of the result may be smaller if <tt>f</tt> maps two or more
grdistinct keys to the same new key. In this case the associated values
grwill be combined using <tt>c</tt>.
gr
gr<pre>
grmapKeysWith (++) (\ _ -&gt; 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
grmapKeysWith (++) (\ _ -&gt; 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
gr</pre>
mapKeysWith :: (a -> a -> a) -> (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n*min(n,W))</i>. <tt><a>mapKeysMonotonic</a> f s ==
gr<a>mapKeys</a> f s</tt>, but works only when <tt>f</tt> is strictly
grmonotonic. That is, for any values <tt>x</tt> and <tt>y</tt>, if
gr<tt>x</tt> &lt; <tt>y</tt> then <tt>f x</tt> &lt; <tt>f y</tt>. <i>The
grprecondition is not checked.</i> Semi-formally, we have:
gr
gr<pre>
grand [x &lt; y ==&gt; f x &lt; f y | x &lt;- ls, y &lt;- ls]
gr                    ==&gt; mapKeysMonotonic f s == mapKeys f s
gr    where ls = keys s
gr</pre>
gr
grThis means that <tt>f</tt> maps distinct original keys to distinct
grresulting keys. This function has slightly better performance than
gr<a>mapKeys</a>.
gr
gr<pre>
grmapKeysMonotonic (\ k -&gt; k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
gr</pre>
mapKeysMonotonic :: (Key -> Key) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator, such that <tt><a>foldr</a> f z ==
gr<a>foldr</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems map = foldr (:) [] map
gr</pre>
gr
gr<pre>
grlet f a len = len + (length a)
grfoldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldr :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the values in the map using the given
grleft-associative binary operator, such that <tt><a>foldl</a> f z ==
gr<a>foldl</a> f z . <a>elems</a></tt>.
gr
grFor example,
gr
gr<pre>
grelems = reverse . foldl (flip (:)) []
gr</pre>
gr
gr<pre>
grlet f len a = len + (length a)
grfoldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
gr</pre>
foldl :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator, such that <tt><a>foldrWithKey</a> f
grz == <a>foldr</a> (<a>uncurry</a> f) z . <a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys map = foldrWithKey (\k x ks -&gt; k:ks) [] map
gr</pre>
gr
gr<pre>
grlet f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"
gr</pre>
foldrWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grleft-associative binary operator, such that <tt><a>foldlWithKey</a> f
grz == <a>foldl</a> (\z' (kx, x) -&gt; f z' kx x) z .
gr<a>toAscList</a></tt>.
gr
grFor example,
gr
gr<pre>
grkeys = reverse . foldlWithKey (\ks k x -&gt; k:ks) []
gr</pre>
gr
gr<pre>
grlet f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"
grfoldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"
gr</pre>
foldlWithKey :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grmonoid, such that
gr
gr<pre>
gr<a>foldMapWithKey</a> f = <a>fold</a> . <a>mapWithKey</a> f
gr</pre>
gr
grThis can be an asymptotically faster than <a>foldrWithKey</a> or
gr<a>foldlWithKey</a> for some monoids.
foldMapWithKey :: Monoid m => (Key -> a -> m) -> IntMap a -> m

-- | <i>O(n)</i>. A strict version of <a>foldr</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldr' :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldl</a>. Each application of the
groperator is evaluated before using the result in the next application.
grThis function is strict in the starting value.
foldl' :: (a -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. A strict version of <a>foldrWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldrWithKey' :: (Key -> a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. A strict version of <a>foldlWithKey</a>. Each application
grof the operator is evaluated before using the result in the next
grapplication. This function is strict in the starting value.
foldlWithKey' :: (a -> Key -> b -> a) -> a -> IntMap b -> a

-- | <i>O(n)</i>. Return all elements of the map in the ascending order of
grtheir keys. Subject to list fusion.
gr
gr<pre>
grelems (fromList [(5,"a"), (3,"b")]) == ["b","a"]
grelems empty == []
gr</pre>
elems :: IntMap a -> [a]

-- | <i>O(n)</i>. Return all keys of the map in ascending order. Subject to
grlist fusion.
gr
gr<pre>
grkeys (fromList [(5,"a"), (3,"b")]) == [3,5]
grkeys empty == []
gr</pre>
keys :: IntMap a -> [Key]

-- | <i>O(n)</i>. An alias for <a>toAscList</a>. Returns all key/value
grpairs in the map in ascending key order. Subject to list fusion.
gr
gr<pre>
grassocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grassocs empty == []
gr</pre>
assocs :: IntMap a -> [(Key, a)]

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

-- | <i>O(n)</i>. Build a map from a set of keys and a function which for
greach key computes its value.
gr
gr<pre>
grfromSet (\k -&gt; replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]
grfromSet undefined Data.IntSet.empty == empty
gr</pre>
fromSet :: (Key -> a) -> IntSet -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs. Subject to
grlist fusion.
gr
gr<pre>
grtoList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
grtoList empty == []
gr</pre>
toList :: IntMap a -> [(Key, a)]

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs.
gr
gr<pre>
grfromList [] == empty
grfromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]
grfromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]
gr</pre>
fromList :: [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Create a map from a list of key/value pairs with
gra combining function. See also <a>fromAscListWith</a>.
gr
gr<pre>
grfromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "ab"), (5, "cba")]
grfromListWith (++) [] == empty
gr</pre>
fromListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n*min(n,W))</i>. Build a map from a list of key/value pairs with
gra combining function. See also fromAscListWithKey'.
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grfromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")]
grfromListWithKey f [] == empty
gr</pre>
fromListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in ascending order. Subject to list fusion.
gr
gr<pre>
grtoAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]
gr</pre>
toAscList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Convert the map to a list of key/value pairs where the
grkeys are in descending order. Subject to list fusion.
gr
gr<pre>
grtoDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]
gr</pre>
toDescList :: IntMap a -> [(Key, a)]

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order.
gr
gr<pre>
grfromAscList [(3,"b"), (5,"a")]          == fromList [(3, "b"), (5, "a")]
grfromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]
gr</pre>
fromAscList :: [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grfromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]
gr</pre>
fromAscListWith :: (a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order, with a combining function on equal keys.
gr<i>The precondition (input list is ascending) is not checked.</i>
gr
gr<pre>
grlet f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value
grfromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")]
gr</pre>
fromAscListWithKey :: (Key -> a -> a -> a) -> [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Build a map from a list of key/value pairs where the keys
grare in ascending order and all distinct. <i>The precondition (input
grlist is strictly ascending) is not checked.</i>
gr
gr<pre>
grfromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]
gr</pre>
fromDistinctAscList :: forall a. [(Key, a)] -> IntMap a

-- | <i>O(n)</i>. Filter all values that satisfy some predicate.
gr
gr<pre>
grfilter (&gt; "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
grfilter (&gt; "x") (fromList [(5,"a"), (3,"b")]) == empty
grfilter (&lt; "a") (fromList [(5,"a"), (3,"b")]) == empty
gr</pre>
filter :: (a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Filter all keys/values that satisfy some predicate.
gr
gr<pre>
grfilterWithKey (\k _ -&gt; k &gt; 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
filterWithKey :: (Key -> a -> Bool) -> IntMap a -> IntMap a

-- | <i>O(n+m)</i>. The restriction of a map to the keys in a set.
gr
gr<pre>
grm <a>restrictKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>member'</a> s) m
gr</pre>
restrictKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n+m)</i>. Remove all the keys in a given set from a map.
gr
gr<pre>
grm <a>withoutKeys</a> s = <a>filterWithKey</a> (k _ -&gt; k `<a>notMember'</a> s) m
gr</pre>
withoutKeys :: IntMap a -> IntSet -> IntMap a

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartition (&gt; "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grpartition (&lt; "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartition (&gt; "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partition :: (a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Partition the map according to some predicate. The first
grmap contains all elements that satisfy the predicate, the second all
grelements that fail the predicate. See also <a>split</a>.
gr
gr<pre>
grpartitionWithKey (\ k _ -&gt; k &gt; 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")
grpartitionWithKey (\ k _ -&gt; k &lt; 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)
grpartitionWithKey (\ k _ -&gt; k &gt; 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])
gr</pre>
partitionWithKey :: (Key -> a -> Bool) -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(n)</i>. Map values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f x = if x == "a" then Just "new a" else Nothing
grmapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"
gr</pre>
mapMaybe :: (a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map keys/values and collect the <a>Just</a> results.
gr
gr<pre>
grlet f k _ = if k &lt; 5 then Just ("key : " ++ (show k)) else Nothing
grmapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"
gr</pre>
mapMaybeWithKey :: (Key -> a -> Maybe b) -> IntMap a -> IntMap b

-- | <i>O(n)</i>. Map values and separate the <a>Left</a> and <a>Right</a>
grresults.
gr
gr<pre>
grlet f a = if a &lt; "c" then Left a else Right a
grmapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
gr
grmapEither (\ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr</pre>
mapEither :: (a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(n)</i>. Map keys/values and separate the <a>Left</a> and
gr<a>Right</a> results.
gr
gr<pre>
grlet f k a = if k &lt; 5 then Left (k * 2) else Right (a ++ a)
grmapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
gr
grmapEitherWithKey (\_ a -&gt; Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
gr    == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
gr</pre>
mapEitherWithKey :: (Key -> a -> Either b c) -> IntMap a -> (IntMap b, IntMap c)

-- | <i>O(min(n,W))</i>. The expression (<tt><a>split</a> k map</tt>) is a
grpair <tt>(map1,map2)</tt> where all keys in <tt>map1</tt> are lower
grthan <tt>k</tt> and all keys in <tt>map2</tt> larger than <tt>k</tt>.
grAny key equal to <tt>k</tt> is found in neither <tt>map1</tt> nor
gr<tt>map2</tt>.
gr
gr<pre>
grsplit 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])
grsplit 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")
grsplit 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")
grsplit 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)
grsplit 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)
gr</pre>
split :: Key -> IntMap a -> (IntMap a, IntMap a)

-- | <i>O(min(n,W))</i>. Performs a <a>split</a> but also returns whether
grthe pivot key was found in the original map.
gr
gr<pre>
grsplitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])
grsplitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")
grsplitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")
grsplitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)
grsplitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)
gr</pre>
splitLookup :: Key -> IntMap a -> (IntMap a, Maybe a, IntMap a)

-- | <i>O(1)</i>. Decompose a map into pieces based on the structure of the
grunderlying tree. This function is useful for consuming a map in
grparallel.
gr
grNo guarantee is made as to the sizes of the pieces; an internal, but
grdeterministic process determines this. However, it is guaranteed that
grthe pieces returned will be in ascending order (all elements in the
grfirst submap less than all elements in the second, and so on).
gr
grExamples:
gr
gr<pre>
grsplitRoot (fromList (zip [1..6::Int] ['a'..])) ==
gr  [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d'),(5,'e'),(6,'f')]]
gr</pre>
gr
gr<pre>
grsplitRoot empty == []
gr</pre>
gr
grNote that the current implementation does not return more than two
grsubmaps, but you should not depend on this behaviour because it can
grchange in the future without notice.
splitRoot :: IntMap a -> [IntMap a]

-- | <i>O(n+m)</i>. Is this a submap? Defined as (<tt><a>isSubmapOf</a> =
gr<a>isSubmapOfBy</a> (==)</tt>).
isSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. The expression (<tt><a>isSubmapOfBy</a> f m1 m2</tt>)
grreturns <a>True</a> if all keys in <tt>m1</tt> are in <tt>m2</tt>, and
grwhen <tt>f</tt> returns <a>True</a> when applied to their respective
grvalues. For example, the following expressions are all <a>True</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (&lt;) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
gr</pre>
isSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grDefined as (<tt><a>isProperSubmapOf</a> = <a>isProperSubmapOfBy</a>
gr(==)</tt>).
isProperSubmapOf :: Eq a => IntMap a -> IntMap a -> Bool

-- | <i>O(n+m)</i>. Is this a proper submap? (ie. a submap but not equal).
grThe expression (<tt><a>isProperSubmapOfBy</a> f m1 m2</tt>) returns
gr<a>True</a> when <tt>m1</tt> and <tt>m2</tt> are not equal, all keys
grin <tt>m1</tt> are in <tt>m2</tt>, and when <tt>f</tt> returns
gr<a>True</a> when applied to their respective values. For example, the
grfollowing expressions are all <a>True</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (&lt;=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
gr</pre>
gr
grBut the following are all <a>False</a>:
gr
gr<pre>
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
grisProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
grisProperSubmapOfBy (&lt;)  (fromList [(1,1)])       (fromList [(1,1),(2,2)])
gr</pre>
isProperSubmapOfBy :: (a -> b -> Bool) -> IntMap a -> IntMap b -> Bool

-- | <i>O(min(n,W))</i>. The minimal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMin :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Returns <a>Nothing</a>
grif the map is empty.
lookupMax :: IntMap a -> Maybe (Key, a)

-- | <i>O(min(n,W))</i>. The minimal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>minViewWithKey</a> if the map may be empty.
findMin :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. The maximal key of the map. Calls <a>error</a> if
grthe map is empty. Use <a>maxViewWithKey</a> if the map may be empty.
findMax :: IntMap a -> (Key, a)

-- | <i>O(min(n,W))</i>. Delete the minimal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMin :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete the maximal key. Returns an empty map if
grthe map is empty.
gr
grNote that this is a change of behaviour for consistency with
gr<a>Map</a> – versions prior to 0.5 threw an error if the <a>IntMap</a>
grwas already empty.
deleteMax :: IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Delete and find the minimal element. This function
grthrows an error if the map is empty. Use <a>minViewWithKey</a> if the
grmap may be empty.
deleteFindMin :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Delete and find the maximal element. This function
grthrows an error if the map is empty. Use <a>maxViewWithKey</a> if the
grmap may be empty.
deleteFindMax :: IntMap a -> ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMin (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]
grupdateMin (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMin :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMax (\ a -&gt; Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]
grupdateMax (\ _ -&gt; Nothing)         (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMax :: (a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the minimal key.
gr
gr<pre>
grupdateMinWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]
grupdateMinWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"
gr</pre>
updateMinWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Update the value at the maximal key.
gr
gr<pre>
grupdateMaxWithKey (\ k a -&gt; Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]
grupdateMaxWithKey (\ _ _ -&gt; Nothing)                     (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"
gr</pre>
updateMaxWithKey :: (Key -> a -> Maybe a) -> IntMap a -> IntMap a

-- | <i>O(min(n,W))</i>. Retrieves the minimal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
minView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal key of the map, and the map
grstripped of that element, or <a>Nothing</a> if passed an empty map.
maxView :: IntMap a -> Maybe (a, IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the minimal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grminViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")
grminViewWithKey empty == Nothing
gr</pre>
minViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(min(n,W))</i>. Retrieves the maximal (key,value) pair of the map,
grand the map stripped of that element, or <a>Nothing</a> if passed an
grempty map.
gr
gr<pre>
grmaxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")
grmaxViewWithKey empty == Nothing
gr</pre>
maxViewWithKey :: IntMap a -> Maybe ((Key, a), IntMap a)

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format.

-- | <i>Deprecated: These debugging functions will be removed from this
grmodule. They are available from Data.IntMap.Internal.Debug.</i>
showTree :: Show a => IntMap a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the map. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.

-- | <i>Deprecated: These debugging functions will be removed from this
grmodule. They are available from Data.IntMap.Internal.Debug.</i>
showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String


-- | An efficient implementation of maps from integer keys to values
gr(dictionaries).
gr
grThis module re-exports the value lazy <a>Data.IntMap.Lazy</a> API,
grplus several deprecated value strict functions. Please note that these
grfunctions have different strictness properties than those in
gr<a>Data.IntMap.Strict</a>: they only evaluate the result of the
grcombining function. For example, the default value to
gr<a>insertWith'</a> is only evaluated if the combining function is
grcalled and uses it.
gr
grThese modules are intended to be imported qualified, to avoid name
grclashes with Prelude functions, e.g.
gr
gr<pre>
grimport Data.IntMap (IntMap)
grimport qualified Data.IntMap as IntMap
gr</pre>
gr
grThe implementation is based on <i>big-endian patricia trees</i>. This
grdata structure performs especially well on binary operations like
gr<a>union</a> and <a>intersection</a>. However, my benchmarks show that
grit is also (much) faster on insertions and deletions when compared to
gra generic size-balanced map implementation (see <a>Data.Map</a>).
gr
gr<ul>
gr<li>Chris Okasaki and Andy Gill, "<i>Fast Mergeable Integer Maps</i>",
grWorkshop on ML, September 1998, pages 77-86,
gr<a>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.5452</a></li>
gr<li>D.R. Morrison, "/PATRICIA -- Practical Algorithm To Retrieve
grInformation Coded In Alphanumeric/", Journal of the ACM, 15(4),
grOctober 1968, pages 514-534.</li>
gr</ul>
gr
grOperation comments contain the operation time complexity in the Big-O
grnotation <a>http://en.wikipedia.org/wiki/Big_O_notation</a>. Many
groperations have a worst-case complexity of <i>O(min(n,W))</i>. This
grmeans that the operation can become linear in the number of elements
grwith a maximum of <i>W</i> -- the number of bits in an <tt>Int</tt>
gr(32 or 64).
module Data.IntMap

-- | <i>O(log n)</i>. Same as <a>insertWith</a>, but the result of the
grcombining function is evaluated to WHNF before inserted to the map.

-- | <i>Deprecated: As of version 0.5, replaced by <a>insertWith</a>.</i>
insertWith' :: (a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(log n)</i>. Same as <a>insertWithKey</a>, but the result of the
grcombining function is evaluated to WHNF before inserted to the map.

-- | <i>Deprecated: As of version 0.5, replaced by
gr<a>insertWithKey</a>.</i>
insertWithKey' :: (Key -> a -> a -> a) -> Key -> a -> IntMap a -> IntMap a

-- | <i>O(n)</i>. Fold the values in the map using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldr</a> and is present for compatibility only.

-- | <i>Deprecated: As of version 0.5, replaced by <a>foldr</a>.</i>
fold :: (a -> b -> b) -> b -> IntMap a -> b

-- | <i>O(n)</i>. Fold the keys and values in the map using the given
grright-associative binary operator. This function is an equivalent of
gr<a>foldrWithKey</a> and is present for compatibility only.

-- | <i>Deprecated: As of version 0.5, replaced by <a>foldrWithKey</a>.</i>
foldWithKey :: (Key -> a -> b -> b) -> b -> IntMap a -> b

module Data.IntMap.Internal.Debug

-- | <i>O(n)</i>. Show the tree that implements the map. The tree is shown
grin a compressed, hanging format.
showTree :: Show a => IntMap a -> String

-- | <i>O(n)</i>. The expression (<tt><a>showTreeWith</a> hang wide
grmap</tt>) shows the tree that implements the map. If <tt>hang</tt> is
gr<a>True</a>, a <i>hanging</i> tree is shown otherwise a rotated tree
gris shown. If <tt>wide</tt> is <a>True</a>, an extra wide version is
grshown.
showTreeWith :: Show a => Bool -> Bool -> IntMap a -> String
