h$mfT      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                                                                                                                   $:(c) Clark Gaebel 2012 (c) Johan Tibel 2012 BSD-stylelibraries@haskell.orgportableSafej containers0Return a word where only the highest bit is set.(c) David Feuer 2016 BSD-stylelibraries@haskell.orgportable Safe-InferredA  containersCreate an empty bit queue builder. This is represented as a single guard bit in the most significant position.  containersEnqueue a bit. This works by shifting the queue right one bit, then setting the most significant bit as requested.  containersConvert a bit queue builder to a bit queue. This shifts in a new guard bit on the left, and shifts right until the old guard bit falls off.  containers3Dequeue an element, or discover the queue is empty. containersConvert a bit queue to a list of bits by unconsing. This is used to test that the queue functions properly.  Noneb containersCoerce the second argument of a function. Conceptually, can be thought of as:  (f .^# g) x y = f x (g y) :However it is most useful when coercing the arguments to : ) foldl f b . fmap g = foldl (f .^# g) b 89 None containersChecks if two pointers are equal. Yes means yes; no means maybe. The values should be forced to at least WHNF before comparison to get moderately reliable results. containersChecks if two pointers are equal, without requiring them to have the same type. The values should be forced to at least WHNF before comparison to get moderately reliable results.44 Safe-InferredQ  Safe-Inferred~Safe= containers'The same as a regular Haskell pair, but  (x :*: _|_) = (_|_ :*: y) = _|_  containers)Convert a strict pair to a standard pair.1(c) Daan Leijen 2002 BSD-stylelibraries@haskell.orgportable Trustworthy 23R containersA set of values a. containersO(m*log(n/m+1)), m <= n. See 8.  containersO(1). Is this the empty set?! containersO(1)$. The number of elements in the set." containersO(log n). Is the element in the set?# containersO(log n) . Is the element not in the set?$ containersO(log n)2. Find largest element smaller than the given one. lookupLT 3 (fromList [3, 5]) == Nothing lookupLT 5 (fromList [3, 5]) == Just 3% containersO(log n)3. Find smallest element greater than the given one. lookupGT 4 (fromList [3, 5]) == Just 5 lookupGT 5 (fromList [3, 5]) == Nothing& containersO(log n)9. Find largest element smaller or equal to the given one. lookupLE 2 (fromList [3, 5]) == Nothing lookupLE 4 (fromList [3, 5]) == Just 3 lookupLE 5 (fromList [3, 5]) == Just 5' containersO(log n):. Find smallest element greater or equal to the given one. lookupGE 3 (fromList [3, 5]) == Just 3 lookupGE 4 (fromList [3, 5]) == Just 5 lookupGE 6 (fromList [3, 5]) == Nothing( containersO(1). The empty set.) containersO(1). Create a singleton set.* containersO(log n). Insert an element in a set. If the set already contains an element equal to the given value, it is replaced with the new value.+ containersO(log n). Delete an element from a set., containersO(log n) (, f x s) can delete or insert x in s4 depending on whether an equal element is found in s. In short: " x <$> , f x s = f (" x s) Note that unlike *, , will not. replace an element equal to the given value.Note: , is a variant of the at combinator from Control.Lens.At.- containersO(m*log(n/m + 1)), m <= n. (s1 `isProperSubsetOf` s2) indicates whether s1 is a proper subset of s2. s1 `isProperSubsetOf` s2 = s1 . s2 && s1 /= s2 . containersO(m*log(n/m + 1)), m <= n. (s1 `isSubsetOf` s2) indicates whether s1 is a subset of s2. s1 `isSubsetOf` s2 = all ("& s2) s1 s1 `isSubsetOf` s2 = null (s1 8 s2) s1 `isSubsetOf` s2 = s1 7" s2 == s2 s1 `isSubsetOf` s2 = s1 9 s2 == s1 /  containersO(m*log(n/m + 1)), m <= n. Check whether two sets are disjoint (i.e., their intersection is empty). disjoint (fromList [2,4,6]) (fromList [1,3]) == True disjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False disjoint (fromList [1,2]) (fromList [1,2,3,4]) == False disjoint (fromList []) (fromList []) == True xs / ys = null (xs 9 ys) 0  containersO(log n). The minimal element of a set.1 containersO(log n). The minimal element of a set.2  containersO(log n). The maximal element of a set.3 containersO(log n). The maximal element of a set.4 containersO(log n). Delete the minimal element. Returns an empty set if the set is empty.5 containersO(log n). Delete the maximal element. Returns an empty set if the set is empty.6 containers1The union of the sets in a Foldable structure : (6 == A 7 ().7 containersO(m*log(n/m + 1)), m <= n. The union of two sets, preferring the first set when equal elements are encountered.8 containersO(m*log(n/m + 1)), m <= n. Difference of two sets.Return elements of the first set not existing in the second set. =difference (fromList [5, 3]) (fromList [5, 7]) == singleton 39 containersO(m*log(n/m + 1)), m <= n. The intersection of two sets. Elements of the result come from the first set, so for example import qualified Data.Set as S data AB = A | B deriving Show instance Ord AB where compare _ _ = EQ instance Eq AB where _ == _ = True main = print (S.singleton A `S.intersection` S.singleton B, S.singleton B `S.intersection` S.singleton A)prints (fromList [A],fromList [B]).: containersO(n)1. Filter all elements that satisfy the predicate.; containersO(n). Partition the set into two sets, one with all elements that satisfy the predicate and one with all elements that don't satisfy the predicate. See also L.< containers O(n*log n). < f s! is the set obtained by applying f to each element of s.It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y= containersO(n). The= f s == < f s, but works only when f is strictly increasing.  The precondition is not checked. Semi-formally, we have: and [x < y ==> f x < f y | x <- ls, y <- ls] ==> mapMonotonic f s == map f s where ls = toList s> containersO(n). Fold the elements in the set using the given right-associative binary operator. This function is an equivalent of ?( and is present for compatibility only.Please note that fold will be deprecated in the future and removed.? containersO(n). Fold the elements in the set using the given right-associative binary operator, such that ? f z == !" f z . E. For example,  toAscList set = foldr (:) [] set@ containersO(n). A strict version of ?. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.A containersO(n). Fold the elements in the set using the given left-associative binary operator, such that A f z == !# f z . E. For example, (toDescList set = foldl (flip (:)) [] setB containersO(n). A strict version of A. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.C containersO(n). An alias of E. The elements of a set in ascending order. Subject to list fusion.D containersO(n). Convert the set to a list of elements. Subject to list fusion.E containersO(n). Convert the set to an ascending list of elements. Subject to list fusion.F containersO(n). Convert the set to a descending list of elements. Subject to list fusion.G containers O(n*log n)'. Create a set from a list of elements.If the elements are ordered, a linear-time implementation is used, with the performance equal to J.H containersO(n)6. Build a set from an ascending list in linear time. :The precondition (input list is ascending) is not checked.I containersO(n)6. Build a set from a descending list in linear time. ;The precondition (input list is descending) is not checked.J containersO(n). Build a set from an ascending list of distinct elements in linear time. The precondition (input list is strictly ascending) is not checked.K containersO(n). Build a set from a descending list of distinct elements in linear time. The precondition (input list is strictly descending) is not checked.L containersO(log n). The expression (L x set ) is a pair  (set1,set2) where set1 comprises the elements of set less than x and set2 comprises the elements of set greater than x.M containersO(log n) . Performs a L but also returns whether the pivot element was found in the original set.N containersO(log n) . Return the index of an element, which is its zero-based index in the sorted sequence of elements. The index is a number from 0 up to, but not including, the ! of the set. Calls  when the element is not a " of the set. findIndex 2 (fromList [5,3]) Error: element is not in the set findIndex 3 (fromList [5,3]) == 0 findIndex 5 (fromList [5,3]) == 1 findIndex 6 (fromList [5,3]) Error: element is not in the setO containersO(log n) . Lookup the index of an element, which is its zero-based index in the sorted sequence of elements. The index is a number from 0 up to, but not including, the ! of the set. isJust (lookupIndex 2 (fromList [5,3])) == False fromJust (lookupIndex 3 (fromList [5,3])) == 0 fromJust (lookupIndex 5 (fromList [5,3])) == 1 isJust (lookupIndex 6 (fromList [5,3])) == FalseP containersO(log n). Retrieve an element by its index, i.e. by its zero-based index in the sorted sequence of elements. If the index7 is out of range (less than zero, greater or equal to ! of the set),  is called. elemAt 0 (fromList [5,3]) == 3 elemAt 1 (fromList [5,3]) == 5 elemAt 2 (fromList [5,3]) Error: index out of rangeQ containersO(log n). Delete the element at index, i.e. by its zero-based index in the sorted sequence of elements. If the index7 is out of range (less than zero, greater or equal to ! of the set),  is called. deleteAt 0 (fromList [5,3]) == singleton 5 deleteAt 1 (fromList [5,3]) == singleton 3 deleteAt 2 (fromList [5,3]) Error: index out of range deleteAt (-1) (fromList [5,3]) Error: index out of rangeR containersTake a given number of elements in order, beginning with the smallest ones.  take n = J . !$ n . E S containersDrop a given number of elements in order, beginning with the smallest ones.  drop n = J . !% n . E T containersO(log n)$. Split a set at a particular index. splitAt !n !xs = (R n xs, S n xs) U containersO(log n). Take while a predicate on the elements holds. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k. See note at W. takeWhileAntitone p = J . &' p . D takeWhileAntitone p = : p V containersO(log n). Drop while a predicate on the elements holds. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k. See note at W. dropWhileAntitone p = J . &( p . D dropWhileAntitone p = : (not . p) W containersO(log n). Divide a set at the point where a predicate on the elements stops holding. The user is responsible for ensuring that for all elements j and k in the set, j < k ==> p j >= p k. spanAntitone p xs = (U p xs, V* p xs) spanAntitone p xs = partition p xs  Note: if p is not actually antitone, then  spanAntitone will split the set at some  unspecified point where the predicate switches from holding to not holding (where the predicate is seen to hold before the first element and to fail after the last element).Z containersO(log n)&. Delete and find the minimal element. 0deleteFindMin set = (findMin set, deleteMin set)[ containersO(log n)&. Delete and find the maximal element. 0deleteFindMax set = (findMax set, deleteMax set)\ containersO(log n). Retrieves the minimal key of the set, and the set stripped of that element, or  if passed an empty set.] containersO(log n). Retrieves the maximal key of the set, and the set stripped of that element, or  if passed an empty set._ containersO(1). Decompose a set into pieces based on the structure of the underlying tree. This function is useful for consuming a set in parallel.No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first subset less than all elements in the second, and so on). Examples: splitRoot (fromList [1..6]) == [fromList [1,2,3],fromList [4],fromList [5,6]] splitRoot empty == []Note that the current implementation does not return more than three subsets, but you should not depend on this behaviour because it can change in the future without notice.`  containers=Calculate the power set of a set: the set of all its subsets. t " powerSet s == t . s Example: powerSet (fromList [1,2,3]) = fromList $ map fromList [[],[1],[1,2],[1,2,3],[1,3],[2],[2,3],[3]] a  containersO(m*n)< (conjectured). Calculate the Cartesian product of two sets. cartesianProduct xs ys = fromList $ liftA2 (,) (toList xs) (toList ys) Example: cartesianProduct (fromList [1,2]) (fromList ['a','b']) = fromList [(1,'a'), (1,'b'), (2,'a'), (2,'b')] b  containers)Calculate the disjoint union of two sets. # disjointUnion xs ys = map Left xs 7 map Right ysExample: disjointUnion (fromList [1,2]) (fromList ["hi", "bye"]) = fromList [Left 1, Left 2, Right "hi", Right "bye"] c containersO(n). Show the tree that implements the set. The tree is shown in a compressed, hanging format.d containersO(n). The expression (showTreeWith hang wide map.) shows the tree that implements the set. If hang is True, a hanging6 tree is shown otherwise a rotated tree is shown. If wide is !, an extra wide version is shown. Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5] 4 +--2 | +--1 | +--3 +--5 Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5] 4 | +--2 | | | +--1 | | | +--3 | +--5 Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5] +--5 | 4 | | +--3 | | +--2 | +--1e containersO(n).. Test if the internal set structure is valid.i  containersj  containersk  containerso containersq containers!Folds in order of increasing key.r containers !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdef !"#$%&'.-/()*+,`7689ab:UVW;LM_ONPQRST<=?A@B>021345Z[]\CDGEFHJIKcde^fXY9 (c) Daan Leijen 2002 BSD-stylelibraries@haskell.orgportableSafeS !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWZ[\]_`abcde()GHIJK`*+,"#$%&' !.-/7689ab:UVW;LM_ONPQRST<=?A@B>021345Z[]\CDEFcde(c) Ross Paterson 2005 (c) Louis Wasserman 2009 (c) Bertram Felgenhauer, David Feuer, Ross Paterson, and Milan Straka 2014 BSD-stylelibraries@haskell.orgportable Trustworthy&238>v containers$View of the right end of a sequence.w containersempty sequencex containersthe sequence minus the rightmost element, and the rightmost elementy containers#View of the left end of a sequence.z containersempty sequence{ containers-leftmost element and the rest of the sequence containers!General-purpose finite sequences. containersA bidirectional pattern synonym viewing the rear of a non-empty sequence. containersA bidirectional pattern synonym viewing the front of a non-empty sequence. containers;A bidirectional pattern synonym matching an empty sequence. containers O(n) <. Intersperse an element between the elements of a sequence. intersperse a empty = empty intersperse a (singleton x) = singleton x intersperse a (fromList [x,y]) = fromList [x,a,y] intersperse a (fromList [x,y,z]) = fromList [x,a,y,a,z]  containers O(1) . The empty sequence. containers O(1) . A singleton sequence. containers O(\log n) .  replicate n x is a sequence consisting of n copies of x. containers is an  version of  , and makes  O(\log n)  calls to  and . *replicateA n x = sequenceA (replicate n x) containers is a sequence counterpart of )*. )replicateM n x = sequence (replicate n x)For  base >= 4.8.0 and containers >= 0.5.11,  is a synonym for . containersO(log k).  k xs forms a sequence of length k by repeatedly concatenating xs with itself. xs may only be empty if k is 0.2cycleTaking k = fromList . take k . cycle . toList containers O(1) . Add an element to the left end of a sequence. Mnemonic: a triangle with the single element at the pointy end. containers O(1) . Add an element to the right end of a sequence. Mnemonic: a triangle with the single element at the pointy end. containers O(\log(\min(n_1,n_2))) . Concatenate two sequences. containersBuilds a sequence from a seed value. Takes time linear in the number of generated elements. WARNING: If the number of generated elements is infinite, this method will not terminate. containers f x is equivalent to  ( ( swap . f) x). containers O(n) . Constructs a sequence by repeated application of a function to a seed value. iterateN n f x = fromList (Prelude.take n (Prelude.iterate f x)) containers O(1) . Is this the empty sequence? containers O(1) ). The number of elements in the sequence. containers O(1) %. Analyse the left end of a sequence. containers O(1) &. Analyse the right end of a sequence. containers is similar to :, but returns a sequence of reduced values from the left: scanl f z (fromList [x1, x2, ...]) = fromList [z, z `f` x1, (z `f` x1) `f` x2, ...] containers is a variant of % that has no starting value argument: scanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...] containers is the right-to-left dual of . containers is a variant of % that has no starting value argument. containers O(\log(\min(i,n-i))) . The element at the specified position, counting from 0. The argument should thus be a non-negative integer less than the size of the sequence. If the position is out of range,  fails with an error.xs `index` i = toList xs !! i Caution:  necessarily delays retrieving the requested element until the result is forced. It can therefore lead to a space leak if the result is stored, unforced, in another structure. To retrieve an element immediately without forcing it, use  or . containers O(\log(\min(i,n-i))) . The element at the specified position, counting from 0. If the specified position is negative or at least the length of the sequence,  returns .;0 <= i < length xs ==> lookup i xs == Just (toList xs !! i)1i < 0 || i >= length xs ==> lookup i xs = NothingUnlike , this can be used to retrieve an element without forcing it. For example, to insert the fifth element of a sequence xs into a + m at key k, you could use /case lookup 5 xs of Nothing -> m Just x -> , k x m  containers O(\log(\min(i,n-i))) . A flipped, infix version of . containers O(\log(\min(i,n-i))) . Replace the element at the specified position. If the position is out of range, the original sequence is returned. containers O(\log(\min(i,n-i))) . Update the element at the specified position. If the position is out of range, the original sequence is returned.  can lead to poor performance and even memory leaks, because it does not force the new value before installing it in the sequence.  should usually be preferred. containers O(\log(\min(i,n-i))) . Update the element at the specified position. If the position is out of range, the original sequence is returned. The new value is forced before it is installed in the sequence. adjust' f i xs = case xs !? i of Nothing -> xs Just x -> let !x' = f x in update i x' xs  containers O(\log(\min(i,n-i))) .  i x xs inserts x into xs at the index i), shifting the rest of the sequence over. insertAt 2 x (fromList [a,b,c,d]) = fromList [a,b,x,c,d] insertAt 4 x (fromList [a,b,c,d]) = insertAt 10 x (fromList [a,b,c,d]) = fromList [a,b,c,d,x] 7insertAt i x xs = take i xs >< singleton x >< drop i xs containers O(\log(\min(i,n-i))) . Delete the element of a sequence at a given index. Return the original sequence if the index is out of range. deleteAt 2 [a,b,c,d] = [a,b,d] deleteAt 4 [a,b,c,d] = deleteAt (-1) [a,b,c,d] = [a,b,c,d]  containersA generalization of ,  takes a mapping function that also depends on the element's index, and applies it to every element in the sequence. containers is a version of 7 that also offers access to the index of each element. containers O(n) . Convert a given sequence length and a function representing that sequence into a sequence. containers O(n) 5. Create a sequence consisting of the elements of an . Note that the resulting sequence elements may be evaluated lazily (as on GHC), so you must force the entire structure to be sure that the original array can be garbage-collected. containers O(\log(\min(i,n-i)))  . The first i elements of a sequence. If i is negative,  i s yields the empty sequence. If the sequence contains fewer than i+ elements, the whole sequence is returned. containers O(\log(\min(i,n-i))) ). Elements of a sequence after the first i. If i is negative,  i s yields the whole sequence. If the sequence contains fewer than i+ elements, the empty sequence is returned. containers O(\log(\min(i,n-i))) ). Split a sequence at a given position.  i s = ( i s,  i s). containers,O \Bigl(\bigl(\frac{n}{c}\bigr) \log c\Bigr).  chunksOf c xs splits xs into chunks of size c>0. If c does not divide the length of xs< evenly, then the last element of the result will be short.Side note: the given performance bound is missing some messy terms that only really affect edge cases. Performance degrades smoothly from  O(1)  (for  c = n ) to  O(n)  (for  c = 1  ). The true bound is more like  O \Bigl( \bigl(\frac{n}{c} - 1\bigr) (\log (c + 1)) + 1 \Bigr)  containers O(n) . Returns a sequence of all suffixes of this sequence, longest first. For example, tails (fromList "abc") = fromList [fromList "abc", fromList "bc", fromList "c", fromList ""]Evaluating the  i th suffix takes  O(\log(\min(i, n-i))) 5, but evaluating every suffix in the sequence takes  O(n)  due to sharing. containers O(n) . Returns a sequence of all prefixes of this sequence, shortest first. For example, inits (fromList "abc") = fromList [fromList "", fromList "a", fromList "ab", fromList "abc"]Evaluating the  i th prefix takes  O(\log(\min(i, n-i))) 5, but evaluating every prefix in the sequence takes  O(n)  due to sharing. containers is a version of 9 that also provides access to the index of each element. containers is a version of 9 that also provides access to the index of each element. containers O(i)  where  i  is the prefix length. , applied to a predicate p and a sequence xs2, returns the longest prefix (possibly empty) of xs of elements that satisfy p. containers O(i)  where  i  is the suffix length. , applied to a predicate p and a sequence xs2, returns the longest suffix (possibly empty) of xs of elements that satisfy p. p xs is equivalent to  ( p ( xs)). containers O(i)  where  i  is the prefix length.  p xs% returns the suffix remaining after  p xs. containers O(i)  where  i  is the suffix length.  p xs% returns the prefix remaining after  p xs. p xs is equivalent to  ( p ( xs)). containers O(i)  where  i  is the prefix length. , applied to a predicate p and a sequence xs, returns a pair whose first element is the longest prefix (possibly empty) of xs of elements that satisfy p9 and the second element is the remainder of the sequence. containers O(i)  where  i  is the suffix length. , applied to a predicate p and a sequence xs, returns a pair whose first element is the longest suffix (possibly empty) of xs of elements that satisfy p9 and the second element is the remainder of the sequence. containers O(i)  where  i  is the breakpoint index. , applied to a predicate p and a sequence xs, returns a pair whose first element is the longest prefix (possibly empty) of xs of elements that do not satisfy p: and the second element is the remainder of the sequence. p is equivalent to  (not . p). containers p is equivalent to  (not . p). containers O(n) . The  function takes a predicate p and a sequence xs and returns sequences of those elements which do and do not satisfy the predicate. containers O(n) . The  function takes a predicate p and a sequence xs and returns a sequence of those elements which satisfy the predicate. containers finds the leftmost index of the specified element, if it is present, and otherwise . containers finds the rightmost index of the specified element, if it is present, and otherwise . containers finds the indices of the specified element, from left to right (i.e. in ascending order). containers finds the indices of the specified element, from right to left (i.e. in descending order). containers p xs9 finds the index of the leftmost element that satisfies p, if any exist. containers p xs: finds the index of the rightmost element that satisfies p, if any exist. containers p, finds all indices of elements that satisfy p, in ascending order. containers p, finds all indices of elements that satisfy p, in descending order. containers O(n) . Create a sequence from a finite list of elements. There is a function 5 in the opposite direction for all instances of the  class, including . containers O(n) . The reverse of a sequence.  containersUnzip a sequence of pairs. unzip ps = ps  (  ps) (  ps) Example: unzip $ fromList [(1,"a"), (2,"b"), (3,"c")] = (fromList [1,2,3], fromList ["a", "b", "c"]) !See the note about efficiency at .  containers O(n) 7. Unzip a sequence using a function to divide elements.  unzipWith f xs ==  ( f xs)Efficiency note: unzipWith9 produces its two results in lockstep. If you calculate  unzipWith f xs  and fully force either3 of the results, then the entire structure of the other one will be built as well. This behavior allows the garbage collector to collect each calculated pair component as soon as it dies, without having to wait for its mate to die. If you do not need this behavior, you may be better off simply calculating the sequence of pairs and using % to extract each component sequence. containers O(\min(n_1,n_2)) .  takes two sequences and returns a sequence of corresponding pairs. If one input is short, excess elements are discarded from the right end of the longer sequence. containers O(\min(n_1,n_2)) .  generalizes  by zipping with the function given as the first argument, instead of a tupling function. For example,  zipWith (+) is applied to two sequences to take the sequence of corresponding sums. containers O(\min(n_1,n_2,n_3)) .  takes three sequences and returns a sequence of triples, analogous to . containers O(\min(n_1,n_2,n_3)) .  takes a function which combines three elements, as well as three sequences and returns a sequence of their point-wise combinations, analogous to . containers O(\min(n_1,n_2,n_3,n_4)) .  takes four sequences and returns a sequence of quadruples, analogous to . containers O(\min(n_1,n_2,n_3,n_4)) .  takes a function which combines four elements, as well as four sequences and returns a sequence of their point-wise combinations, analogous to . containers   =    =  containers containers  containers  containers  containers  containers containers containers  containers containers containers containers containers containers containers containers containers containers containers containers containersvxwy{z|}~|}~y{zvxw x5{55560555555  Safe-Inferred containersA pairing heap tagged with both a key and the original position of its elements, for use in . containersA pairing heap tagged with some key for sorting elements, for use in . containersA pairing heap tagged with the original position of elements, to allow for stable sorting. containersA simple pairing heap. containers O(n \log n) .  sorts the specified  by the natural ordering of its elements. The sort is stable. If stability is not required,  can be slightly faster. containers O(n \log n) .  sorts the specified  according to the specified comparator. The sort is stable. If stability is not required,  can be slightly faster.  containers O(n \log n) .  sorts the specified  by comparing the results of a key function applied to each element.  f is equivalent to  ( -. f)8, but has the performance advantage of only evaluating f once for each element in the input list. This is called the decorate-sort-undecorate paradigm, or Schwartzian transform.An example of using  might be to sort a ' of strings according to their length: sortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"] If, instead,  had been used, 1 would be evaluated on every comparison, giving  O(n \log n)  evaluations, rather than  O(n) .If f2 is very cheap (for example a record selector, or ),  ( -. f) will be faster than  f. containers O(n \log n) .  sorts the specified  by the natural ordering of its elements, but the sort is not stable. This algorithm is frequently faster and uses less memory than . containers O(n \log n) . A generalization of ,  takes an arbitrary comparator and sorts the specified sequence. The sort is not stable. This algorithm is frequently faster and uses less memory than .  containers O(n \log n) .  sorts the specified  by comparing the results of a key function applied to each element.  f is equivalent to  ( -. f)8, but has the performance advantage of only evaluating f once for each element in the input list. This is called the decorate-sort-undecorate paradigm, or Schwartzian transform.An example of using  might be to sort a ' of strings according to their length: unstableSortOn length (fromList ["alligator", "monkey", "zebra"]) == fromList ["zebra", "monkey", "alligator"] If, instead,  had been used, 1 would be evaluated on every comparison, giving  O(n \log n)  evaluations, rather than  O(n) .If f2 is very cheap (for example a record selector, or ),  ( -. f) will be faster than  f. containers merges two s. containers merges two s, based on the tag value. containers merges two >s, taking into account the original position of the elements. containers merges two s, based on the tag value, taking into account the original position of the elements. containersPop the smallest element from the queue, using the supplied comparator. containersPop the smallest element from the queue, using the supplied comparator, deferring to the item's original position when the comparator returns . containersPop the smallest element from the queue, using the supplied comparator on the tag. containersPop the smallest element from the queue, using the supplied comparator on the tag, deferring to the item's original position when the comparator returns . containersA $-like function, specialized to the /0= monoid, which takes advantage of the internal structure of  to avoid wrapping in  at certain points. containersA 1$-like function, specialized to the /0= monoid, which takes advantage of the internal structure of  to avoid wrapping in  at certain points.((8888(c) Ross Paterson 2005 (c) Louis Wasserman 2009 (c) Bertram Felgenhauer, David Feuer, Ross Paterson, and Milan Straka 2014 BSD-stylelibraries@haskell.orgportable Safe-Inferredbvxwy{zy{zvxw "(c) The University of Glasgow 2002/BSD-style (see the file libraries/base/LICENSE)libraries@haskell.orgportable Trustworthy238 containers;This type synonym exists primarily for historical reasons. containers=Non-empty, possibly infinite, multi-way trees; also known as  rose trees. containers label value containerszero or more child trees containers&2-dimensional ASCII drawing of a tree.Examples =putStr $ drawTree $ fmap show (Node 1 [Node 2 [], Node 3 []]) 1 | +- 2 | `- 3  containers(2-dimensional ASCII drawing of a forest.Examples putStr $ drawForest $ map (fmap show) [(Node 1 [Node 2 [], Node 3 []]), (Node 10 [Node 20 []])] 1 | +- 2 | `- 3 10 | `- 20  containers,Returns the elements of a tree in pre-order.  a / \ => [a,b,c] b c Examples 2flatten (Node 1 [Node 2 [], Node 3 []]) == [1,2,3] containers4Returns the list of nodes at each level of the tree. # a / \ => [[a], [b,c]] b c Examples 5levels (Node 1 [Node 2 [], Node 3 []]) == [[1],[2,3]] containers8Fold a tree into a "summary" value in depth-first order.!For each node in the tree, apply f to the  rootLabel and the result of applying f to each  subForest.0This is also known as the catamorphism on trees.ExamplesSum the values in a tree: foldTree (\x xs -> sum (x:xs)) (Node 1 [Node 2 [], Node 3 []]) == 6#Find the maximum value in the tree: foldTree (\x xs -> maximum (x:xs)) (Node 1 [Node 2 [], Node 3 []]) == 3'Count the number of leaves in the tree: foldTree (\_ xs -> if null xs then 1 else sum xs) (Node 1 [Node 2 [], Node 3 []]) == 2Find depth of the tree; i.e. the number of branches from the root of the tree to the furthest leaf: foldTree (\_ xs -> if null xs then 0 else 1 + maximum xs) (Node 1 [Node 2 [], Node 3 []]) == 1/You can even implement traverse using foldTree: traverse' f = foldTree (\x xs -> liftA2 Node (f x) (sequenceA xs)) containersBuild a (possibly infinite) tree from a seed value in breadth-first order.unfoldTree f b. constructs a tree by starting with the tree "Node { rootLabel=b, subForest=[] } and repeatedly applying f to each , value in the tree's leaves to generate its .For a monadic version see .ExamplesConstruct the tree of Integer%s where each node has two children:  left = 2*x and right = 2*x + 1, where x is the - of the node. Stop when the values exceed 7. let buildNode x = if 2*x + 1 > 7 then (x, []) else (x, [2*x, 2*x+1]) putStr $ drawTree $ fmap show $ unfoldTree buildNode 1  1 | +- 2 | | | +- 4 | | | `- 5 | `- 3 | +- 6 | `- 7  containersBuild a (possibly infinite) forest from a list of seed values in breadth-first order.unfoldForest f seeds invokes  on each seed value.For a monadic version see . containers+Monadic tree builder, in depth-first order. containers,Monadic forest builder, in depth-first order containers-Monadic tree builder, in breadth-first order.See  for more info.Implemented using an algorithm adapted from /Breadth-First Numbering: Lessons from a Small Exercise in Algorithm Design, by Chris Okasaki,  ICFP'00/. containers.Monadic forest builder, in breadth-first orderSee  for more info.Implemented using an algorithm adapted from /Breadth-First Numbering: Lessons from a Small Exercise in Algorithm Design, by Chris Okasaki,  ICFP'00/.  containers  containers  containers  containers  containers containers containers containers "(c) The University of Glasgow 2002/BSD-style (see the file libraries/base/LICENSE)libraries@haskell.orgportableSafe238ӡ) containers,An edge from the first vertex to the second. containersThe bounds of an Array. containersAdjacency list representation of a graph, mapping each vertex to its list of successors. containers.Table indexed by a contiguous set of vertices.3Note: This is included for backwards compatibility. containers$Abstract representation of vertices. containersStrongly connected component. containers*A single vertex that is not in any cycle. containers.A maximal set of mutually reachable vertices. containers8The vertices of a list of strongly connected components. containers/The vertices of a strongly connected component. containersThe strongly connected components of a directed graph, reverse topologically sorted.Examples stronglyConnComp [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])] == [CyclicSCC ["d"],CyclicSCC ["b","c"],AcyclicSCC "a"] containersThe strongly connected components of a directed graph, reverse topologically sorted. The function is the same as , except that all the information about each node retained. This interface is used when you expect to apply  to (some of) the result of 8, so you don't want to lose the dependency information.Examples stronglyConnCompR [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])] == [CyclicSCC [("d",3,[3])],CyclicSCC [("b",1,[2,3]),("c",2,[1])],AcyclicSCC ("a",0,[1])] containers*Returns the list of vertices in the graph.Examples !vertices (buildG (0,-1) []) == [] 0vertices (buildG (0,2) [(0,1),(1,2)]) == [0,1,2] containers'Returns the list of edges in the graph.Examples edges (buildG (0,-1) []) == [] 3edges (buildG (0,2) [(0,1),(1,2)]) == [(0,1),(1,2)] containers#Build a graph from a list of edges.Warning: This function will cause a runtime exception if a vertex in the edge list is not within the given Bounds.Examples buildG (0,-1) [] == array (0,-1) [] buildG (0,2) [(0,1), (1,2)] == array (0,1) [(0,[1]),(1,[2])] buildG (0,2) [(0,1), (0,2), (1,2)] == array (0,2) [(0,[2,1]),(1,[2]),(2,[])] containers*The graph obtained by reversing all edges.Examples transposeG (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,[]),(1,[0]),(2,[1])] containers-A table of the count of edges from each node.Examples /outdegree (buildG (0,-1) []) == array (0,-1) [] outdegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,1),(1,1),(2,0)] containers-A table of the count of edges into each node.Examples .indegree (buildG (0,-1) []) == array (0,-1) [] indegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,0),(1,1),(2,1)] containers Identical to , except that the return value does not include the function which maps keys to vertices. This version of  is for backwards compatibility. containersBuild a graph from a list of nodes uniquely identified by keys, with a list of keys of nodes this node should have edges to.This function takes an adjacency list representing a graph with vertices of type key labeled by values of type node and produces a Graph)-based representation of that list. The Graph result represents the shape of the graph, and the functions describe a) how to retrieve the label and adjacent vertices of a given vertex, and b) how to retrieve a vertex given a key. (graph, nodeFromVertex, vertexFromKey) = graphFromEdges edgeListgraph :: Graph6 is the raw, array based adjacency list for the graph..nodeFromVertex :: Vertex -> (node, key, [key])7 returns the node associated with the given 0-based Int vertex; see warning below.$vertexFromKey :: key -> Maybe Vertex returns the Int2 vertex for the key if it exists in the graph, Nothing otherwise.To safely use this API you must either extract the list of vertices directly from the graph or first call vertexFromKey k. to check if a vertex corresponds to the key k5. Once it is known that a vertex exists you can use nodeFromVertex to access the labelled node and adjacent vertices. See below for examples.Note: The out-list may contain keys that don't correspond to nodes of the graph; they are ignored. Warning: The nodeFromVertex7 function will cause a runtime exception if the given Vertex does not exist.ExamplesAn empty graph. (graph, nodeFromVertex, vertexFromKey) = graphFromEdges [] graph = array (0,-1) []9A graph where the out-list references unspecified nodes ('c'), these are ignored. (graph, _, _) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c'])] array (0,1) [(0,[1]),(1,[])]0A graph with 3 vertices: ("a") -> ("b") -> ("c") (graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])] graph == array (0,2) [(0,[1]),(1,[2]),(2,[])] nodeFromVertex 0 == ("a",'a',"b") vertexFromKey 'a' == Just 0Get the label for a given key. let getNodePart (n, _, _) = n (graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])] getNodePart . nodeFromVertex <$> vertexFromKey 'a' == Just "A" containersA spanning forest of the graph, obtained from a depth-first search of the graph starting from each vertex in an unspecified order. containersA spanning forest of the part of the graph reachable from the listed vertices, obtained from a depth-first search of the graph starting at each of the listed vertices in order. containersA topological sort of the graph. The order is partially specified by the condition that a vertex i precedes j whenever j is reachable from i but not vice versa. containersReverse ordering of . containersThe connected components of a graph. Two vertices are connected if there is a path between them, traversing edges in either direction. containersThe strongly connected components of a graph, in reverse topological order.Examples scc (buildG (0,3) [(3,1),(1,2),(2,0),(0,1)]) == [Node {rootLabel = 0, subForest = [Node {rootLabel = 1, subForest = [Node {rootLabel = 2, subForest = []}]}]} ,Node {rootLabel = 3, subForest = []}] containers;Returns the list of vertices reachable from a given vertex.Examples $reachable (buildG (0,0) []) 0 == [0] 4reachable (buildG (0,2) [(0,1), (1,2)]) 0 == [0,1,2] containersReturns True/ if the second vertex reachable from the first.Examples "path (buildG (0,0) []) 0 0 == True .path (buildG (0,2) [(0,1), (1,2)]) 0 2 == True /path (buildG (0,2) [(0,1), (1,2)]) 2 0 == False containersThe biconnected components of a graph. An undirected graph is biconnected if the deletion of any vertex leaves it connected. containers  containers  containers  containers  containers  containers  containers  containers  containers  containers  containers  containers containersThe graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored. containersThe graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored. containersReverse topologically sorted  (c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable Trustworthy 23O  containers7A tactic for dealing with keys present in both maps in .A tactic of type  SimpleWhenMatched k x y z 6 is an abstract representation of a function of type  k -> x -> y -> Maybe z .  containers8A tactic for dealing with keys present in both maps in  or .A tactic of type  WhenMatched f k x y z 6 is an abstract representation of a function of type  k -> x -> y -> f (Maybe z) .  containersA tactic for dealing with keys present in one map but not the other in .A tactic of type  SimpleWhenMissing k x z 6 is an abstract representation of a function of type  k -> x -> Maybe z .  containersA tactic for dealing with keys present in one map but not the other in  or .A tactic of type  WhenMissing f k x z 6 is an abstract representation of a function of type  k -> x -> f (Maybe z) . containersA Map from keys k to values a.The  operation for  is 2, which prefers values from the left operand. If m1 maps a key k to a value a1, and m2( maps the same key to a different value a2, then their union m1 <> m2 maps k to a1. containersO(log n)". Find the value at a key. Calls # when the element can not be found. fromList [(5,'a'), (3,'b')] ! 1 Error: element not in the map fromList [(5,'a'), (3,'b')] ! 5 == 'a'  containersO(log n)$. Find the value at a key. Returns # when the element can not be found.-fromList [(5, 'a'), (3, 'b')] !? 1 == Nothing.fromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a' containersSame as . containersO(1). Is the map empty? Data.Map.null (empty) == True Data.Map.null (singleton 1 'a') == False containersO(1)$. The number of elements in the map. size empty == 0 size (singleton 1 'a') == 1 size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3 containersO(log n)'. Lookup the value at a key in the map.4The function will return the corresponding value as ( value), or  if the key isn't in the map.An example of using lookup: import Prelude hiding (lookup) import Data.Map employeeDept = fromList([("John","Sales"), ("Bob","IT")]) deptCountry = fromList([("IT","USA"), ("Sales","France")]) countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")]) employeeCurrency :: String -> Maybe String employeeCurrency name = do dept <- lookup name employeeDept country <- lookup dept deptCountry lookup country countryCurrency main = do putStrLn $ "John's currency: " ++ (show (employeeCurrency "John")) putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))The output of this program: 9 John's currency: Just "Euro" Pete's currency: Nothing containersO(log n)+. Is the key a member of the map? See also . member 5 (fromList [(5,'a'), (3,'b')]) == True member 1 (fromList [(5,'a'), (3,'b')]) == False containersO(log n)/. Is the key not a member of the map? See also . notMember 5 (fromList [(5,'a'), (3,'b')]) == False notMember 1 (fromList [(5,'a'), (3,'b')]) == True containersO(log n). The expression ( def k map) returns the value at key k or returns default value def! when the key is not in the map. findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x' findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a' containersO(log n). Find largest key smaller than the given one and return the corresponding (key, value) pair. lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') containersO(log n). Find smallest key greater than the given one and return the corresponding (key, value) pair. lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing containersO(log n). Find largest key smaller or equal to the given one and return the corresponding (key, value) pair. lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') containersO(log n). Find smallest key greater or equal to the given one and return the corresponding (key, value) pair. lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing containersO(1). The empty map. )empty == fromList [] size empty == 0 containersO(1). A map with a single element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1 containersO(log n). Insert a new key and value in the map. If the key is already present in the map, the associated value is replaced with the supplied value.  is equivalent to  . insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')] insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')] insert 5 'x' empty == singleton 5 'x' containersO(log n)>. Insert with a function, combining new value and old value.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key, f new_value old_value). insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")] insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWith (++) 5 "xxx" empty == singleton 5 "xxx" containersO(log n). Insert with a function, combining key, new value and old value.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key,f key new_value old_value);. Note that the key passed to f is the same key passed to . let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")] insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWithKey f 5 "xxx" empty == singleton 5 "xxx" containersO(log n). Combines insert operation with old value retrieval. The expression ( f k x map2) is a pair where the first element is equal to ( k map$) and the second element equal to ( f k x map). let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")]) insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")]) insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")This is how to define  insertLookup using insertLookupWithKey: let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")]) insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")]) containersO(log n). Delete a key and its value from the map. When the key is not a member of the map, the original map is returned. delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] delete 5 empty == empty containersO(log n). Update a value at a specific key with the result of the provided function. When the key is not a member of the map, the original map is returned. adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjust ("new " ++) 7 empty == empty containersO(log n). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. let f key x = (show key) ++ ":new " ++ x adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjustWithKey f 7 empty == empty containersO(log n). The expression ( f k map) updates the value x at k (if it is in the map). If (f x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f x = if x == "a" then Just "new a" else Nothing update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n). The expression ( f k map) updates the value x at k (if it is in the map). If (f k x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n). Lookup and update. See also . The function returns changed value, if it is updated. Returns the original key value if the map entry is deleted. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")]) updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")]) updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a") containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof. 7 can be used to insert, delete, or update a value in a . In short :  k ( f k m) = f ( k m). let f _ = Nothing alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" let f _ = Just "c" alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")] alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")] Note that  = alter . fmap. containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof.  can be used to inspect, insert, delete, or update a value in a  . In short:  k <$>  f k m = f ( k m).Example: interactiveAlter :: Int -> Map Int String -> IO (Map Int String) interactiveAlter k m = alterF f k m where f Nothing = do putStrLn $ show k ++ " was not found in the map. Would you like to add it?" getUserResponse1 :: IO (Maybe String) f (Just old) = do putStrLn $ "The key is currently bound to " ++ show old ++ ". Would you like to change or delete it?" getUserResponse2 :: IO (Maybe String)  is the most general operation for working with an individual key that may or may not be in a given map. When used with trivial functors like  and , it is often slightly slower than more specialized combinators like  and . However, when the functor is non-trivial and key comparison is not particularly cheap, it is the fastest way.Note on rewrite rules:3This module includes GHC rewrite rules to optimize  for the  and  functors. In general, these rules improve performance. The sole exception is that when using , deleting a key that is already absent takes longer than it would without the rules. If you expect this to occur a very large fraction of the time, you might consider using a private copy of the  type.Note:  is a flipped version of the at combinator from Control.Lens.At. containersO(log n) . Return the index of a key, which is its zero-based index in the sequence sorted by keys. The index is a number from 0 up to, but not including, the  of the map. Calls  when the key is not a  of the map. findIndex 2 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0 findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1 findIndex 6 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map containersO(log n) . Lookup the index of a key, which is its zero-based index in the sequence sorted by keys. The index is a number from 0 up to, but not including, the  of the map. isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")])) == False fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0 fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1 isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")])) == False containersO(log n). Retrieve an element by its index, i.e. by its zero-based index in the sequence sorted by keys. If the index7 is out of range (less than zero, greater or equal to  of the map),  is called. elemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b") elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a") elemAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range containersTake a given number of entries in key order, beginning with the smallest keys.  take n =  . !$ n .   containersDrop a given number of entries in key order, beginning with the smallest keys.  drop n =  . !% n .   containersO(log n)$. Split a map at a particular index. splitAt !n !xs = ( n xs,  n xs)  containersO(log n). Update the element at index, i.e. by its zero-based index in the sequence sorted by keys. If the index7 is out of range (less than zero, greater or equal to  of the map),  is called. updateAt (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")] updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")] updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range containersO(log n). Delete the element at index, i.e. by its zero-based index in the sequence sorted by keys. If the index7 is out of range (less than zero, greater or equal to  of the map),  is called. deleteAt 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" deleteAt 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" deleteAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range deleteAt (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range  containersO(log n)&. The minimal key of the map. Returns  if the map is empty. lookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b") lookupMin empty = Nothing containersO(log n)$. The minimal key of the map. Calls  if the map is empty. findMin (fromList [(5,"a"), (3,"b")]) == (3,"b") findMin empty Error: empty map has no minimal element  containersO(log n)&. The maximal key of the map. Returns  if the map is empty. lookupMax (fromList [(5,"a"), (3,"b")]) == Just (5,"a") lookupMax empty = Nothing containersO(log n). Delete the minimal key. Returns an empty map if the map is empty. deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")] deleteMin empty == empty containersO(log n). Delete the maximal key. Returns an empty map if the map is empty. deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")] deleteMax empty == empty containersO(log n)&. Update the value at the minimal key. updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")] updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n)&. Update the value at the maximal key. updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")] updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containersO(log n)&. Update the value at the minimal key. updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")] updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n)&. Update the value at the maximal key. updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")] updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containersO(log n). Retrieves the minimal (key,value) pair of the map, and the map stripped of that element, or  if passed an empty map. minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a") minViewWithKey empty == Nothing containersO(log n). Retrieves the maximal (key,value) pair of the map, and the map stripped of that element, or  if passed an empty map. maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b") maxViewWithKey empty == Nothing containersO(log n). Retrieves the value associated with minimal key of the map, and the map stripped of that element, or  if passed an empty map. minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a") minView empty == Nothing containersO(log n). Retrieves the value associated with maximal key of the map, and the map stripped of that element, or  if passed an empty map. maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b") maxView empty == Nothing containers!The union of a list of maps: ( == !#  ). unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "b"), (5, "a"), (7, "C")] unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])] == fromList [(3, "B3"), (5, "A3"), (7, "C")] containers=The union of a list of maps, with a combining operation: ( f == !# ( f) ). unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")] containersO(m*log(n/m + 1)), m <= n. The expression ( t1 t2!) takes the left-biased union of t1 and t2. It prefers t1- when duplicate keys are encountered, i.e. ( ==  ). union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")] containersO(m*log(n/m + 1)), m <= n". Union with a combining function. unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")] containersO(m*log(n/m + 1)), m <= n#. Union with a combining function. let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")] containersO(m*log(n/m + 1)), m <= n. Difference of two maps. Return elements of the first map not existing in the second map. difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b" containersO(m*log(n/m + 1)), m <= n. Remove all keys in a  from a . m `withoutKeys` s =  (k _ -> k # s) m m `withoutKeys` s = m   (const ()) s  containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the values of these keys. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")]) == singleton 3 "b:B" containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")]) == singleton 3 "3:b|B" containersO(m*log(n/m + 1)), m <= n. Intersection of two maps. Return data in the first map for the keys existing in both maps. ( m1 m2 ==   m1 m2). intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a" containersO(m*log(n/m + 1)), m <= n . Restrict a  to only those keys found in a . m `restrictKeys` s =  (k _ -> k " s) m m `restrictKeys` s = m   (const ()) s  containersO(m*log(n/m + 1)), m <= n). Intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA" containersO(m*log(n/m + 1)), m <= n). Intersection with a combining function. let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A" containersO(m*log(n/m + 1)), m <= n. Check whether the key sets of two maps are disjoint (i.e., their  is empty). disjoint (fromList [(2,'a')]) (fromList [(1,()), (3,())]) == True disjoint (fromList [(2,'a')]) (fromList [(1,'a'), (2,'b')]) == False disjoint (fromList []) (fromList []) == True xs  ys = null (xs  ys)  containersRelate the keys of one map to the values of the other, by using the values of the former as keys for lookups in the latter. Complexity:  O (n * \log(m)) , where m" is the size of the first argument compose (fromList [('a', "A"), ('b', "B")]) (fromList [(1,'a'),(2,'b'),(3,'z')]) = fromList [(1,"A"),(2,"B")] ( bc ab ) = (bc  ) <=< (ab ) Note: Prior to v0.6.4, Data.Map.Strict exposed a version of & that forced the values of the output ,. This version does not force these values.  containersMap covariantly over a  f k x. containersMap covariantly over a  f k x', using only a 'Functor f' constraint. containersMap covariantly over a  f k x', using only a 'Functor f' constraint.  containersMap contravariantly over a  f k _ x.  containersMap contravariantly over a  f k _ y z.  containersMap contravariantly over a  f k x _ z.  containersAlong with zipWithMaybeAMatched, witnesses the isomorphism between WhenMatched f k x y z and k -> x -> y -> f (Maybe z).  containersAlong with traverseMaybeMissing, witnesses the isomorphism between WhenMissing f k x y and k -> x -> f (Maybe y).  containersMap covariantly over a  f k x y.  containersWhen a key is found in both maps, apply a function to the key and values and use the result in the merged map. zipWithMatched :: (k -> x -> y -> z) -> SimpleWhenMatched k x y z   containersWhen a key is found in both maps, apply a function to the key and values to produce an action and use its result in the merged map.  containersWhen a key is found in both maps, apply a function to the key and values and maybe use the result in the merged map. zipWithMaybeMatched :: (k -> x -> y -> Maybe z) -> SimpleWhenMatched k x y z   containersWhen a key is found in both maps, apply a function to the key and values, perform the resulting action, and maybe use the result in the merged map.This is the fundamental  tactic.  containersDrop all the entries whose keys are missing from the other map. 'dropMissing :: SimpleWhenMissing k x y /dropMissing = mapMaybeMissing (\_ _ -> Nothing)but  dropMissing is much faster.  containersPreserve, unchanged, the entries whose keys are missing from the other map. +preserveMissing :: SimpleWhenMissing k x x =preserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -> Just x)but preserveMissing is much faster.  containersForce the entries whose keys are missing from the other map and otherwise preserve them unchanged. ,preserveMissing' :: SimpleWhenMissing k x x preserveMissing' = Merge.Lazy.mapMaybeMissing (\_ x -> Just $! x)but preserveMissing' is quite a bit faster.  containers?Map over the entries whose keys are missing from the other map. 7mapMissing :: (k -> x -> y) -> SimpleWhenMissing k x y 5mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)but  mapMissing is somewhat faster.  containersMap over the entries whose keys are missing from the other map, optionally removing some. This is the most powerful 0 tactic, but others are usually more efficient. mapMaybeMissing :: (k -> x -> Maybe y) -> SimpleWhenMissing k x y ?mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))but mapMaybeMissing uses fewer unnecessary  operations.  containers=Filter the entries whose keys are missing from the other map. =filterMissing :: (k -> x -> Bool) -> SimpleWhenMissing k x x filterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -> guard (f k x) *> Just x#but this should be a little faster.  containersFilter the entries whose keys are missing from the other map using some  action. filterAMissing f = Merge.Lazy.traverseMaybeMissing $ k x -> (b -> guard b *> Just x)  $ f k x #but this should be a little faster.  containersTraverse over the entries whose keys are missing from the other map.  containersTraverse over the entries whose keys are missing from the other map, optionally producing values to put in the result. This is the most powerful 0 tactic, but others are usually more efficient.  containersMerge two maps. takes two  tactics, a  tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics,  and .Consider merge (mapMaybeMissing g1) (mapMaybeMissing g2) (zipWithMaybeMatched f) m1 m2 Take, for example, m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] & will first "align" these maps by key: m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate: maybes = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"] This produces a  for each key: keys = 0 1 2 3 4 results = [Nothing, Just True, Just False, Nothing, Just True]  Finally, the Just" results are collected into a map: 2return value = [(1, True), (2, False), (4, True)] The other tactics below are optimizations or simplifications of % for special cases. Most importantly, drops all the keys. leaves all the entries alone.When  is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should typically use , to define your custom combining functions. Examples:unionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)intersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)differenceWith f = merge preserveMissing dropMissing (zipWithMatched f)symmetricDifference = merge preserveMissing preserveMissing (zipWithMaybeMatched $ \ _ _ _ -> Nothing)mapEachPiece f g h = merge (mapMissing f) (mapMissing g) (zipWithMatched h)  containersAn applicative version of . takes two  tactics, a  tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics,  and .Consider mergeA (traverseMaybeMissing g1) (traverseMaybeMissing g2) (zipWithMaybeAMatched f) m1 m2 Take, for example, m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] mergeA& will first "align" these maps by key: m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate: actions = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"] )Next, it will perform the actions in the actions# list in order from left to right. keys = 0 1 2 3 4 results = [Nothing, Just True, Just False, Nothing, Just True]  Finally, the Just" results are collected into a map: 2return value = [(1, True), (2, False), (4, True)] The other tactics below are optimizations or simplifications of % for special cases. Most importantly, drops all the keys. leaves all the entries alone. does not use the  context.When  is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should generally only use & to define custom combining functions. containersO(n+m)'. An unsafe general combining function.WARNING: This function can produce corrupt maps and its results may depend on the internal structures of its inputs. Users should prefer  or .When  is given three arguments, it is inlined to the call site. You should therefore use  only to define custom combining functions. For example, you could define ,  and  as myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2 When calling  combine only1 only2, a function combining two s is created, such thatif a key is present in both maps, it is passed with both corresponding values to the combine function. Depending on the result, the key is either present in the result with specified value, or is left out;>a nonempty subtree present only in the first map is passed to only1* and the output is added to the result;?a nonempty subtree present only in the second map is passed to only2* and the output is added to the result.The only1 and only2 methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of only1 and only2 are  and  , but for example  f,  f, or  f could be used for any f. containersO(m*log(n/m + 1)), m <= n . This function is defined as ( =  (==)). containersO(m*log(n/m + 1)), m <= n. The expression ( f t1 t2 ) returns  if all keys in t1 are in tree t2 , and when f returns  when applied to their respective values. For example, the following expressions are all : isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)]) isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)]) isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])But the following are all : isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)]) isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)]) isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)]) Note that  isSubmapOfBy (_ _ -> True) m1 m2 tests whether all the keys in m1 are also keys in m2. containersO(m*log(n/m + 1)), m <= n. Is this a proper submap? (ie. a submap but not equal). Defined as ( =  (==)). containersO(m*log(n/m + 1)), m <= n. Is this a proper submap? (ie. a submap but not equal). The expression ( f m1 m2 ) returns  when keys m1 and keys m2 are not equal, all keys in m1 are in m2 , and when f returns  when applied to their respective values. For example, the following expressions are all : isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])But the following are all : isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)]) isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)]) isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) containersO(n)/. Filter all values that satisfy the predicate. filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty containersO(n)4. Filter all keys/values that satisfy the predicate. filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n). Take while a predicate on the keys holds. The user is responsible for ensuring that for all keys j and k in the map, j < k ==> p j >= p k. See note at . takeWhileAntitone p =  . &' (p . fst) .  takeWhileAntitone p =  (k _ -> p k)  containersO(log n). Drop while a predicate on the keys holds. The user is responsible for ensuring that for all keys j and k in the map, j < k ==> p j >= p k. See note at . dropWhileAntitone p =  . &( (p . fst) .  dropWhileAntitone p =  (k -> not (p k))  containersO(log n). Divide a map at the point where a predicate on the keys stops holding. The user is responsible for ensuring that for all keys j and k in the map, j < k ==> p j >= p k. spanAntitone p xs = ( p xs, < p xs) spanAntitone p xs = partitionWithKey (k _ -> p k) xs  Note: if p is not actually antitone, then  spanAntitone will split the map at some  unspecified point where the predicate switches from holding to not holding (where the predicate is seen to hold before the first key and to fail after the last key). containersO(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also . partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) containersO(n). Partition the map according to a predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also . partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b") partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) containersO(n). Map values and collect the  results. let f x = if x == "a" then Just "new a" else Nothing mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a" containersO(n)". Map keys/values and collect the  results. let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3" containersO(n)'. Traverse keys/values and collect the  results. containersO(n). Map values and separate the  and  results. let f a = if a < "c" then Left a else Right a mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")]) mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) containersO(n)#. Map keys/values and separate the  and  results. let f k a = if k < 5 then Left (k * 2) else Right (a ++ a) mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")]) mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")]) containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")] containersO(n),. Map a function over all values in the map. let f key x = (show key) ++ ":" ++ x mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")] containersO(n).  f m ==   $  ((k, v) -> (,) k  $ f k v) ( m)* That is, behaves exactly like a regular  except that the traversing function also has access to the key associated with a value. traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')]) traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a b = (a ++ b, b ++ "X") mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X") mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in descending order of keys. containers O(n*log n).  f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the greatest of the original keys is retained. mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")] mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c" mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c" containers O(n*log n).  c f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c. The value at the greater of the two original keys is used as the first argument to c. mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab" mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab" containersO(n).  f s ==  f s, but works only when f2 is strictly monotonic. That is, for any values x and y, if x < y then f x < f y.  The precondition is not checked. Semi-formally, we have: and [x < y ==> f x < f y | x <- ls, y <- ls] ==> mapKeysMonotonic f s == mapKeys f s where ls = keys sThis means that f maps distinct original keys to distinct resulting keys. This function has better performance than . mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")] valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False containersO(n). Fold the values in the map using the given right-associative binary operator, such that  f z == !" f z . . For example, elems map = foldr (:) [] map let f a len = len + (length a) foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4 containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the values in the map using the given left-associative binary operator, such that  f z == !# f z . . For example, %elems = reverse . foldl (flip (:)) [] let f len a = len + (length a) foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4 containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given right-associative binary operator, such that  f z == !" ( f) z . . For example, 0keys map = foldrWithKey (\k x ks -> k:ks) [] map let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")" foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)" containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given left-associative binary operator, such that  f z == !# (\z' (kx, x) -> f z' kx x) z . . For example, 2keys = reverse . foldlWithKey (\ks k x -> k:ks) [] let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")" foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)" containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given monoid, such that  f = !2 .  f*This can be an asymptotically faster than  or  for some monoids. containersO(n). Return all elements of the map in the ascending order of their keys. Subject to list fusion. elems (fromList [(5,"a"), (3,"b")]) == ["b","a"] elems empty == [] containersO(n). Return all keys of the map in ascending order. Subject to list fusion.  replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")] fromSet undefined Data.Set.empty == empty containers O(n*log n)7. Build a map from a list of key/value pairs. See also . If the list contains more than one value for the same key, the last value for the key is retained.If the keys of the list are ordered, linear-time implementation is used, with the performance equal to . fromList [] == empty fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")] fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")] containers O(n*log n). Build a map from a list of key/value pairs with a combining function. See also . fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")] fromListWith (++) [] == empty containers O(n*log n). Build a map from a list of key/value pairs with a combining function. See also . let f k a1 a2 = (show k) ++ a1 ++ a2 fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")] fromListWithKey f [] == empty containersO(n). Convert the map to a list of key/value pairs. Subject to list fusion. toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] toList empty == [] containersO(n). Convert the map to a list of key/value pairs where the keys are in ascending order. Subject to list fusion. =toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] containersO(n). Convert the map to a list of key/value pairs where the keys are in descending order. Subject to list fusion. >toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")] containersO(n)6. Build a map from an ascending list in linear time. :The precondition (input list is ascending) is not checked. fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")] valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n)6. Build a map from a descending list in linear time. ;The precondition (input list is descending) is not checked. fromDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")] fromDescList [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "b")] valid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True valid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list in linear time with a combining function for equal keys. :The precondition (input list is ascending) is not checked. fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from a descending list in linear time with a combining function for equal keys. ;The precondition (input list is descending) is not checked. fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")] valid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True valid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list in linear time with a combining function for equal keys. :The precondition (input list is ascending) is not checked. let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2 fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")] valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False containersO(n). Build a map from a descending list in linear time with a combining function for equal keys. ;The precondition (input list is descending) is not checked. let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2 fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")] valid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True valid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list of distinct elements in linear time.  The precondition is not checked. fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] valid (fromDistinctAscList [(3,"b"), (5,"a")]) == True valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False containersO(n). Build a map from a descending list of distinct elements in linear time.  The precondition is not checked. fromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")] valid (fromDistinctDescList [(5,"a"), (3,"b")]) == True valid (fromDistinctDescList [(5,"a"), (5,"b"), (3,"b")]) == False containersO(log n). The expression ( k map ) is a pair  (map1,map2) where the keys in map1 are smaller than k and the keys in map2 larger than k. Any key equal to k is found in neither map1 nor map2. split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")]) split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a") split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty) split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty) containersO(log n). The expression ( k map) splits a map just like  but also returns  k map. splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")]) splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a") splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a") splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty) splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty) containersO(log n)&. Delete and find the minimal element. deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")]) deleteFindMin empty Error: can not return the minimal element of an empty map containersO(log n)&. Delete and find the maximal element. deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")]) deleteFindMax empty Error: can not return the maximal element of an empty map containersO(1). Decompose a map into pieces based on the structure of the underlying tree. This function is useful for consuming a map in parallel.No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first submap less than all elements in the second, and so on). Examples: splitRoot (fromList (zip [1..6] ['a'..])) == [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d')],fromList [(5,'e'),(6,'f')]] splitRoot empty == []Note that the current implementation does not return more than three submaps, but you should not depend on this behaviour because it can change in the future without notice. containers containers!Folds in order of increasing key. containers%Traverses in order of increasing key.  containers  containers  containers  containers  containers  containers  containers containers  containersEquivalent to " ReaderT k (ReaderT x (MaybeT f)) .  containersEquivalent to " ReaderT k (ReaderT x (MaybeT f)) .  containers  containers  containersEquivalent to . ReaderT k (ReaderT x (ReaderT y (MaybeT f)))   containersEquivalent to . ReaderT k (ReaderT x (ReaderT y (MaybeT f)))   containers  containers containersWhat to do with keys in m1 but not m2 containersWhat to do with keys in m2 but not m1 containersWhat to do with keys in both m1 and m2 containersMap m1 containersMap m2 containersWhat to do with keys in m1 but not m2 containersWhat to do with keys in m2 but not m1 containersWhat to do with keys in both m1 and m2 containersMap m1 containersMap m29 9 9 (c) David Feuer 2016 BSD-stylelibraries@haskell.orgportableSafe 23 Safe-Inferred* containersO(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See . containersO(n). The expression ( showelem hang wide map) shows the tree that implements the map. Elements are shown using the showElem function. If hang is , a hanging6 tree is shown otherwise a rotated tree is shown. If wide is !, an extra wide version is shown.  Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]] Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t (4,()) +--(2,()) | +--(1,()) | +--(3,()) +--(5,()) Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t (4,()) | +--(2,()) | | | +--(1,()) | | | +--(3,()) | +--(5,()) Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t +--(5,()) | (4,()) | | +--(3,()) | | +--(2,()) | +--(1,()) containersO(n).. Test if the internal map structure is valid. valid (fromAscList [(3,"b"), (5,"a")]) == True valid (fromAscList [(5,"a"), (3,"b")]) == False containers'Test if the keys are ordered correctly. containers+Test if a map obeys the balance invariants. containers6Test if each node of a map reports its size correctly.  >(c) Daan Leijen 2002 (c) Joachim Breitner 2011 BSD-stylelibraries@haskell.orgportable Trustworthy 235 containersA set of integers. containersO(n+m). See . containersO(1). Is the set empty? containersO(n). Cardinality of the set. containers O(min(n,W))#. Is the value a member of the set? containers O(min(n,W)) . Is the element not in the set? containersO(log n)2. Find largest element smaller than the given one. lookupLT 3 (fromList [3, 5]) == Nothing lookupLT 5 (fromList [3, 5]) == Just 3 containersO(log n)3. Find smallest element greater than the given one. lookupGT 4 (fromList [3, 5]) == Just 5 lookupGT 5 (fromList [3, 5]) == Nothing containersO(log n)9. Find largest element smaller or equal to the given one. lookupLE 2 (fromList [3, 5]) == Nothing lookupLE 4 (fromList [3, 5]) == Just 3 lookupLE 5 (fromList [3, 5]) == Just 5 containersO(log n):. Find smallest element greater or equal to the given one. lookupGE 3 (fromList [3, 5]) == Just 3 lookupGE 4 (fromList [3, 5]) == Just 5 lookupGE 6 (fromList [3, 5]) == Nothing containersO(1). The empty set. containersO(1). A set of one element. containers O(min(n,W)). Add a value to the set. There is no left- or right bias for IntSets. containers O(min(n,W)). Delete a value in the set. Returns the original set when the value was not present. containers O(min(n,W)). ( f x s) can delete or insert x in s0 depending on whether it is already present in s. In short:  x <$>  f x s = f ( x s) Note:  is a variant of the at combinator from Control.Lens.At. containersThe union of a list of sets. containersO(n+m). The union of two sets. containersO(n+m). Difference between two sets. containersO(n+m). The intersection of two sets. containersO(n+m)8. Is this a proper subset? (ie. a subset but not equal). containersO(n+m). Is this a subset? (s1 `isSubsetOf` s2) tells whether s1 is a subset of s2.  containersO(n+m). Check whether two sets are disjoint (i.e. their intersection is empty). disjoint (fromList [2,4,6]) (fromList [1,3]) == True disjoint (fromList [2,4,6,8]) (fromList [2,3,5,7]) == False disjoint (fromList [1,2]) (fromList [1,2,3,4]) == False disjoint (fromList []) (fromList []) == True containersO(n)2. Filter all elements that satisfy some predicate. containersO(n)0. partition the set according to some predicate. containers O(min(n,W)). The expression ( x set ) is a pair  (set1,set2) where set1 comprises the elements of set less than x and set2 comprises the elements of set greater than x. =split 3 (fromList [1..5]) == (fromList [1,2], fromList [4,5]) containers O(min(n,W)) . Performs a  but also returns whether the pivot element was found in the original set. containers O(min(n,W)). Retrieves the maximal key of the set, and the set stripped of that element, or  if passed an empty set. containers O(min(n,W)). Retrieves the minimal key of the set, and the set stripped of that element, or  if passed an empty set. containers O(min(n,W))&. Delete and find the minimal element. 0deleteFindMin set = (findMin set, deleteMin set) containers O(min(n,W))&. Delete and find the maximal element. 0deleteFindMax set = (findMax set, deleteMax set) containers O(min(n,W))!. The minimal element of the set. containers O(min(n,W)). The maximal element of a set. containers O(min(n,W)). Delete the minimal element. Returns an empty set if the set is empty.=Note that this is a change of behaviour for consistency with 30 @ versions prior to 0.5 threw an error if the  was already empty. containers O(min(n,W)). Delete the maximal element. Returns an empty set if the set is empty.=Note that this is a change of behaviour for consistency with 30 @ versions prior to 0.5 threw an error if the  was already empty. containers O(n*min(n,W)).  f s! is the set obtained by applying f to each element of s.It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y containersO(n). The f s ==  f s, but works only when f is strictly increasing.  The precondition is not checked. Semi-formally, we have: and [x < y ==> f x < f y | x <- ls, y <- ls] ==> mapMonotonic f s == map f s where ls = toList s containersO(n). Fold the elements in the set using the given right-associative binary operator. This function is an equivalent of ( and is present for compatibility only.Please note that fold will be deprecated in the future and removed. containersO(n). Fold the elements in the set using the given right-associative binary operator, such that  f z == !" f z . . For example,  toAscList set = foldr (:) [] set containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the elements in the set using the given left-associative binary operator, such that  f z == !# f z . . For example, (toDescList set = foldl (flip (:)) [] set containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). An alias of . The elements of a set in ascending order. Subject to list fusion. containersO(n). Convert the set to a list of elements. Subject to list fusion. containersO(n). Convert the set to an ascending list of elements. Subject to list fusion. containersO(n). Convert the set to a descending list of elements. Subject to list fusion. containers O(n*min(n,W))'. Create a set from a list of integers. containersO(n)3. Build a set from an ascending list of elements. :The precondition (input list is ascending) is not checked. containersO(n)<. Build a set from an ascending list of distinct elements. The precondition (input list is strictly ascending) is not checked. containersO(n). Show the tree that implements the set. The tree is shown in a compressed, hanging format. containersO(n). The expression ( hang wide map.) shows the tree that implements the set. If hang is , a hanging6 tree is shown otherwise a rotated tree is shown. If wide is !, an extra wide version is shown. containersO(1). Decompose a set into pieces based on the structure of the underlying tree. This function is useful for consuming a set in parallel.No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first submap less than all elements in the second, and so on). Examples: splitRoot (fromList [1..120]) == [fromList [1..63],fromList [64..120]] splitRoot empty == []Note that the current implementation does not return more than two subsets, but you should not depend on this behaviour because it can change in the future without notice. Also, the current version does not continue splitting all the way to individual singleton sets -- it stops at some point. containers containers??9 >(c) Daan Leijen 2002 (c) Joachim Breitner 2011 BSD-stylelibraries@haskell.orgportableSafe֞44(c) Gershom Bazerman 2018 BSD-stylelibraries@haskell.orgportable Trustworthyު containers O(n \log d) . The nubOrd function removes duplicate elements from a list. In particular, it keeps only the first occurrence of each element. By using a 9 internally it has better asymptotics than the standard &4 function. StrictnessnubOrd' is strict in the elements of the list.Efficiency note3When applicable, it is almost always better to use  or  instead of this function, although it can be a little worse in certain pathological cases. For example, to nub a list of characters, use  nubIntOn fromEnum xs containersThe nubOrdOn function behaves just like  except it performs comparisons not on the original datatype, but a user-specified projection from that datatype. StrictnessnubOrdOn is strict in the values of the function applied to the elements of the list. containers O(n \min(d,W)) . The nubInt function removes duplicate  values from a list. In particular, it keeps only the first occurrence of each element. By using an > internally, it attains better asymptotics than the standard &4 function. See also *, a more widely applicable generalization. StrictnessnubInt' is strict in the elements of the list. containersThe nubIntOn function behaves just like  except it performs comparisons not on the original datatype, but a user-specified projection from that datatype. For example,  nubIntOn  can be used to nub characters and typical fixed-with numerical types efficiently. StrictnessnubIntOn is strict in the values of the function applied to the elements of the list.(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 (c) wren romano 2016 BSD-stylelibraries@haskell.orgportable Trustworthy 23C  containers7A tactic for dealing with keys present in both maps in .A tactic of type SimpleWhenMatched x y z6 is an abstract representation of a function of type Key -> x -> y -> Maybe z.  containers7A tactic for dealing with keys present in both maps in  or .A tactic of type WhenMatched f x y z6 is an abstract representation of a function of type Key -> x -> y -> f (Maybe z).  containersA tactic for dealing with keys present in one map but not the other in .A tactic of type SimpleWhenMissing x z6 is an abstract representation of a function of type Key -> x -> Maybe z.  containersA tactic for dealing with keys present in one map but not the other in  or .A tactic of type WhenMissing f k x z6 is an abstract representation of a function of type Key -> x -> f (Maybe z). containersA map of integers to values a. containers O(min(n,W))". Find the value at a key. Calls # when the element can not be found. fromList [(5,'a'), (3,'b')] ! 1 Error: element not in the map fromList [(5,'a'), (3,'b')] ! 5 == 'a'  containers O(min(n,W))$. Find the value at a key. Returns # when the element can not be found. fromList [(5,'a'), (3,'b')] !? 1 == Nothing fromList [(5,'a'), (3,'b')] !? 5 == Just 'a' containersSame as . containersO(1). Is the map empty? Data.IntMap.null (empty) == True Data.IntMap.null (singleton 1 'a') == False containersO(n) . Number of elements in the map. size empty == 0 size (singleton 1 'a') == 1 size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3 containers O(min(n,W))!. Is the key a member of the map? member 5 (fromList [(5,'a'), (3,'b')]) == True member 1 (fromList [(5,'a'), (3,'b')]) == False containers O(min(n,W))%. Is the key not a member of the map? notMember 5 (fromList [(5,'a'), (3,'b')]) == False notMember 1 (fromList [(5,'a'), (3,'b')]) == True containers O(min(n,W))1. Lookup the value at a key in the map. See also 5. containers O(min(n,W)). The expression ( def k map) returns the value at key k or returns def, when the key is not an element of the map. findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x' findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a' containersO(log n). Find largest key smaller than the given one and return the corresponding (key, value) pair. lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') containersO(log n). Find smallest key greater than the given one and return the corresponding (key, value) pair. lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing containersO(log n). Find largest key smaller or equal to the given one and return the corresponding (key, value) pair. lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') containersO(log n). Find smallest key greater or equal to the given one and return the corresponding (key, value) pair. lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a') lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b') lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing containersO(n+m). Check whether the key sets of two maps are disjoint (i.e. their  is empty). disjoint (fromList [(2,'a')]) (fromList [(1,()), (3,())]) == True disjoint (fromList [(2,'a')]) (fromList [(1,'a'), (2,'b')]) == False disjoint (fromList []) (fromList []) == True 'disjoint a b == null (intersection a b) containersRelate the keys of one map to the values of the other, by using the values of the former as keys for lookups in the latter. Complexity:  O(n * \min(m,W)) , where m" is the size of the first argument compose (fromList [('a', "A"), ('b', "B")]) (fromList [(1,'a'),(2,'b'),(3,'z')]) = fromList [(1,"A"),(2,"B")] ( bc ab ) = (bc  ) <=< (ab ) Note: Prior to v0.6.4, Data.IntMap.Strict exposed a version of & that forced the values of the output ,. This version does not force these values. containersO(1). The empty map. )empty == fromList [] size empty == 0 containersO(1). A map of one element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1 containers O(min(n,W)). Insert a new key/value pair in the map. If the key is already present in the map, the associated value is replaced with the supplied value, i.e.  is equivalent to  . insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')] insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')] insert 5 'x' empty == singleton 5 'x' containers O(min(n,W))%. Insert with a combining function.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert f new_value old_value. insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")] insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWith (++) 5 "xxx" empty == singleton 5 "xxx" containers O(min(n,W))%. Insert with a combining function.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert f key new_value old_value. let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")] insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWithKey f 5 "xxx" empty == singleton 5 "xxx" containers O(min(n,W)). The expression ( f k x map2) is a pair where the first element is equal to ( k map$) and the second element equal to ( f k x map). let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")]) insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")]) insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")This is how to define  insertLookup using insertLookupWithKey: let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")]) insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")]) containers O(min(n,W)). Delete a key and its value from the map. When the key is not a member of the map, the original map is returned. delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] delete 5 empty == empty containers O(min(n,W)). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjust ("new " ++) 7 empty == empty containers O(min(n,W)). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. let f key x = (show key) ++ ":new " ++ x adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjustWithKey f 7 empty == empty containers O(min(n,W)). The expression ( f k map) updates the value x at k (if it is in the map). If (f x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f x = if x == "a" then Just "new a" else Nothing update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W)). The expression ( f k map) updates the value x at k (if it is in the map). If (f k x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than 6>. Returns the original key value if the map entry is deleted. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")]) updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")]) updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a") containers O(min(n,W)). The expression ( f k map) alters the value x at k, or absence thereof. 8 can be used to insert, delete, or update a value in an . In short :  k ( f k m) = f ( k m). containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof.  can be used to inspect, insert, delete, or update a value in an . In short :  k  $  f k m = f ( k m).Example: interactiveAlter :: Int -> IntMap String -> IO (IntMap String) interactiveAlter k m = alterF f k m where f Nothing = do putStrLn $ show k ++ " was not found in the map. Would you like to add it?" getUserResponse1 :: IO (Maybe String) f (Just old) = do putStrLn $ "The key is currently bound to " ++ show old ++ ". Would you like to change or delete it?" getUserResponse2 :: IO (Maybe String)  is the most general operation for working with an individual key that may or may not be in a given map.Note:  is a flipped version of the at combinator from Control.Lens.At. containersThe union of a list of maps. unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "b"), (5, "a"), (7, "C")] unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])] == fromList [(3, "B3"), (5, "A3"), (7, "C")] containers8The union of a list of maps, with a combining operation. unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")] containersO(n+m). The (left-biased) union of two maps. It prefers the first map when duplicate keys are encountered, i.e. ( ==  ). union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")] containersO(n+m)&. The union with a combining function. unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")] containersO(n+m)&. The union with a combining function. let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")] containersO(n+m).. Difference between two maps (based on keys). difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b" containersO(n+m)'. Difference with a combining function. let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")]) == singleton 3 "b:B" containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")]) == singleton 3 "3:b|B" containersO(n+m)0. Remove all the keys in a given set from a map. m `withoutKeys` s =  (k _ -> k  s) m  containersO(n+m)=. The (left-biased) intersection of two maps (based on keys). intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a" containersO(n+m)0. The restriction of a map to the keys in a set. m `restrictKeys` s =  (k _ -> k  s) m  containersO(n+m)-. The intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA" containersO(n+m)-. The intersection with a combining function. let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A" containersO(n+m):. A high-performance universal combining function. Using , all combining functions can be defined without any loss of efficiency (with exception of ,  and ,, where sharing of some nodes is lost with ).6Please make sure you know what is going on when using , otherwise you can be surprised by unexpected code growth or even corruption of the data structure.When  is given three arguments, it is inlined to the call site. You should therefore use  only to define your custom combining functions. For example, you could define ,  and  as myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2 When calling  combine only1 only2, a function combining two s is created, such thatif a key is present in both maps, it is passed with both corresponding values to the combine function. Depending on the result, the key is either present in the result with specified value, or is left out;>a nonempty subtree present only in the first map is passed to only1* and the output is added to the result;?a nonempty subtree present only in the second map is passed to only2* and the output is added to the result.The only1 and only2 methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of only1 and only2 are  and  , but for example  f or  f could be used for any f.  containersMap covariantly over a  f x. containersMap covariantly over a  f x', using only a 'Functor f' constraint. containersMap covariantly over a  f k x', using only a 'Functor f' constraint.  containersMap contravariantly over a  f _ x.  containersMap contravariantly over a  f _ y z.  containersMap contravariantly over a  f x _ z.  containersAlong with zipWithMaybeAMatched, witnesses the isomorphism between WhenMatched f x y z and Key -> x -> y -> f (Maybe z).  containersAlong with traverseMaybeMissing, witnesses the isomorphism between WhenMissing f x y and Key -> x -> f (Maybe y).  containersMap covariantly over a  f x y.  containersWhen a key is found in both maps, apply a function to the key and values and use the result in the merged map. zipWithMatched :: (Key -> x -> y -> z) -> SimpleWhenMatched x y z  containersWhen a key is found in both maps, apply a function to the key and values to produce an action and use its result in the merged map.  containersWhen a key is found in both maps, apply a function to the key and values and maybe use the result in the merged map. zipWithMaybeMatched :: (Key -> x -> y -> Maybe z) -> SimpleWhenMatched x y z  containersWhen a key is found in both maps, apply a function to the key and values, perform the resulting action, and maybe use the result in the merged map.This is the fundamental  tactic.  containersDrop all the entries whose keys are missing from the other map. $dropMissing :: SimpleWhenMissing x y/dropMissing = mapMaybeMissing (\_ _ -> Nothing)but  dropMissing is much faster.  containersPreserve, unchanged, the entries whose keys are missing from the other map. (preserveMissing :: SimpleWhenMissing x x=preserveMissing = Merge.Lazy.mapMaybeMissing (\_ x -> Just x)but preserveMissing is much faster.  containers?Map over the entries whose keys are missing from the other map. 4mapMissing :: (k -> x -> y) -> SimpleWhenMissing x y5mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)but  mapMissing is somewhat faster.  containersMap over the entries whose keys are missing from the other map, optionally removing some. This is the most powerful / tactic, but others are usually more efficient. mapMaybeMissing :: (Key -> x -> Maybe y) -> SimpleWhenMissing x y?mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))but mapMaybeMissing uses fewer unnecessary  operations.  containers=Filter the entries whose keys are missing from the other map. :filterMissing :: (k -> x -> Bool) -> SimpleWhenMissing x xfilterMissing f = Merge.Lazy.mapMaybeMissing $ \k x -> guard (f k x) *> Just x#but this should be a little faster.  containersFilter the entries whose keys are missing from the other map using some  action. filterAMissing f = Merge.Lazy.traverseMaybeMissing $ \k x -> (\b -> guard b *> Just x) <$> f k x#but this should be a little faster.  containersTraverse over the entries whose keys are missing from the other map.  containersTraverse over the entries whose keys are missing from the other map, optionally producing values to put in the result. This is the most powerful 0 tactic, but others are usually more efficient. containersO(n)'. Traverse keys/values and collect the  results.  containersMerge two maps. takes two  tactics, a  tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics,  and .Consider merge (mapMaybeMissing g1) (mapMaybeMissing g2) (zipWithMaybeMatched f) m1 m2 Take, for example, m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] & will first "align" these maps by key: m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate: maybes = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"] This produces a  for each key: keys = 0 1 2 3 4 results = [Nothing, Just True, Just False, Nothing, Just True]  Finally, the Just" results are collected into a map: 2return value = [(1, True), (2, False), (4, True)] The other tactics below are optimizations or simplifications of % for special cases. Most importantly, drops all the keys. leaves all the entries alone.When  is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should typically use + to define your custom combining functions. Examples:unionWithKey f = merge preserveMissing preserveMissing (zipWithMatched f)intersectionWithKey f = merge dropMissing dropMissing (zipWithMatched f)0differenceWith f = merge diffPreserve diffDrop fsymmetricDifference = merge diffPreserve diffPreserve (\ _ _ _ -> Nothing)mapEachPiece f g h = merge (diffMapWithKey f) (diffMapWithKey g)  containersAn applicative version of . takes two  tactics, a  tactic and two maps. It uses the tactics to merge the maps. Its behavior is best understood via its fundamental tactics,  and .Consider mergeA (traverseMaybeMissing g1) (traverseMaybeMissing g2) (zipWithMaybeAMatched f) m1 m2 Take, for example, m1 = [(0, 'a'), (1, 'b'), (3,'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] & will first "align" these maps by key: m1 = [(0, 'a'), (1, 'b'), (3, 'c'), (4, 'd')] m2 = [(1, "one"), (2, "two"), (4, "three")] It will then pass the individual entries and pairs of entries to g1, g2, or f as appropriate: actions = [g1 0 'a', f 1 'b' "one", g2 2 "two", g1 3 'c', f 4 'd' "three"] )Next, it will perform the actions in the actions# list in order from left to right. keys = 0 1 2 3 4 results = [Nothing, Just True, Just False, Nothing, Just True]  Finally, the Just" results are collected into a map: 2return value = [(1, True), (2, False), (4, True)] The other tactics below are optimizations or simplifications of % for special cases. Most importantly, drops all the keys. leaves all the entries alone. does not use the  context.When  is given three arguments, it is inlined at the call site. To prevent excessive inlining, you should generally only use & to define custom combining functions. containers O(min(n,W))&. Update the value at the minimal key. updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")] updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W))&. Update the value at the maximal key. updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")] updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containers O(min(n,W)). Retrieves the maximal (key,value) pair of the map, and the map stripped of that element, or  if passed an empty map. maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b") maxViewWithKey empty == Nothing containers O(min(n,W)). Retrieves the minimal (key,value) pair of the map, and the map stripped of that element, or  if passed an empty map. minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a") minViewWithKey empty == Nothing containers O(min(n,W))&. Update the value at the maximal key. updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")] updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containers O(min(n,W))&. Update the value at the minimal key. updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")] updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W)). Retrieves the maximal key of the map, and the map stripped of that element, or  if passed an empty map. containers O(min(n,W)). Retrieves the minimal key of the map, and the map stripped of that element, or  if passed an empty map. containers O(min(n,W)). Delete and find the maximal element. This function throws an error if the map is empty. Use  if the map may be empty. containers O(min(n,W)). Delete and find the minimal element. This function throws an error if the map is empty. Use  if the map may be empty. containers O(min(n,W))&. The minimal key of the map. Returns  if the map is empty. containers O(min(n,W))$. The minimal key of the map. Calls  if the map is empty. Use  if the map may be empty. containers O(min(n,W))&. The maximal key of the map. Returns  if the map is empty. containers O(min(n,W))$. The maximal key of the map. Calls  if the map is empty. Use  if the map may be empty. containers O(min(n,W)). Delete the minimal key. Returns an empty map if the map is empty.=Note that this is a change of behaviour for consistency with +0 @ versions prior to 0.5 threw an error if the  was already empty. containers O(min(n,W)). Delete the maximal key. Returns an empty map if the map is empty.=Note that this is a change of behaviour for consistency with +0 @ versions prior to 0.5 threw an error if the  was already empty. containersO(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as ( =  (==)). containersO(n+m). Is this a proper submap? (ie. a submap but not equal). The expression ( f m1 m2 ) returns  when keys m1 and keys m2 are not equal, all keys in m1 are in m2 , and when f returns  when applied to their respective values. For example, the following expressions are all : isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])But the following are all : isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)]) isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)]) isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) containersO(n+m)!. Is this a submap? Defined as ( =  (==)). containersO(n+m). The expression ( f m1 m2 ) returns  if all keys in m1 are in m2 , and when f returns  when applied to their respective values. For example, the following expressions are all : isSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) isSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])But the following are all : isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)]) isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)]) isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)]) containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")] containersO(n),. Map a function over all values in the map. let f key x = (show key) ++ ":" ++ x mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")] containersO(n).  f s ==   $  ((k, v) -> (,) k  $ f k v) ( m)* That is, behaves exactly like a regular  except that the traversing function also has access to the key associated with a value. traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')]) traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a b = (a ++ b, b ++ "X") mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X") mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in descending order of keys. containers O(n*min(n,W)).  f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the greatest of the original keys is retained. mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")] mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c" mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c" containers O(n*min(n,W)).  c f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c. mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab" mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab" containers O(n*min(n,W)).  f s ==  f s, but works only when f2 is strictly monotonic. That is, for any values x and y, if x < y then f x < f y.  The precondition is not checked. Semi-formally, we have: and [x < y ==> f x < f y | x <- ls, y <- ls] ==> mapKeysMonotonic f s == mapKeys f s where ls = keys sThis means that f maps distinct original keys to distinct resulting keys. This function has slightly better performance than . mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")] containersO(n)0. Filter all values that satisfy some predicate. filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty containersO(n)5. Filter all keys/values that satisfy some predicate. filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(n). Partition the map according to some predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also . partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) containersO(n). Partition the map according to some predicate. The first map contains all elements that satisfy the predicate, the second all elements that fail the predicate. See also . partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b") partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty) partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")]) containersO(n). Map values and collect the  results. let f x = if x == "a" then Just "new a" else Nothing mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a" containersO(n)". Map keys/values and collect the  results. let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3" containersO(n). Map values and separate the  and  results. let f a = if a < "c" then Left a else Right a mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")]) mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) containersO(n)#. Map keys/values and separate the  and  results. let f k a = if k < 5 then Left (k * 2) else Right (a ++ a) mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")]) mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")]) containers O(min(n,W)). The expression ( k map ) is a pair  (map1,map2) where all keys in map1 are lower than k and all keys in map2 larger than k. Any key equal to k is found in neither map1 nor map2. split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")]) split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a") split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a") split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty) split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty) containers O(min(n,W)) . Performs a  but also returns whether the pivot key was found in the original map. splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")]) splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a") splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a") splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty) splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty) containersO(n). Fold the values in the map using the given right-associative binary operator, such that  f z == !" f z . . For example, elems map = foldr (:) [] map let f a len = len + (length a) foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4 containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the values in the map using the given left-associative binary operator, such that  f z == !# f z . . For example, %elems = reverse . foldl (flip (:)) [] let f len a = len + (length a) foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4 containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given right-associative binary operator, such that  f z == !" ( f) z . . For example, 0keys map = foldrWithKey (\k x ks -> k:ks) [] map let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")" foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)" containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given left-associative binary operator, such that  f z == !# (\z' (kx, x) -> f z' kx x) z . . For example, 2keys = reverse . foldlWithKey (\ks k x -> k:ks) [] let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")" foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)" containersO(n). A strict version of . Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value. containersO(n). Fold the keys and values in the map using the given monoid, such that  f = !2 .  f*This can be an asymptotically faster than  or  for some monoids. containersO(n). Return all elements of the map in the ascending order of their keys. Subject to list fusion. elems (fromList [(5,"a"), (3,"b")]) == ["b","a"] elems empty == [] containersO(n). Return all keys of the map in ascending order. Subject to list fusion.  replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")] fromSet undefined Data.IntSet.empty == empty containersO(n). Convert the map to a list of key/value pairs. Subject to list fusion. toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] toList empty == [] containersO(n). Convert the map to a list of key/value pairs where the keys are in ascending order. Subject to list fusion. =toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")] containersO(n). Convert the map to a list of key/value pairs where the keys are in descending order. Subject to list fusion. >toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")] containers O(n*min(n,W)).. Create a map from a list of key/value pairs. fromList [] == empty fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")] fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")] containers O(n*min(n,W)). Create a map from a list of key/value pairs with a combining function. See also . fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "ab"), (5, "cba")] fromListWith (++) [] == empty containers O(n*min(n,W)). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey'. let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"c")] == fromList [(3, "3:a|b"), (5, "5:c|5:b|a")] fromListWithKey f [] == empty containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order. fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. :The precondition (input list is ascending) is not checked. fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. :The precondition (input list is ascending) is not checked. let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "5:b|a")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order and all distinct. The precondition (input list is strictly ascending) is not checked. fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] containers-Should this key follow the left subtree of a  with switching bit m&? N.B., the answer is only valid when  match i p m is true. containers Does the key i differ from the prefix p& before getting to the switching bit m? containers Does the key i match the prefix p (up to but not including bit m)? containersThe prefix of key i. up to (but not including) the switching bit m. containersThe prefix of key i. up to (but not including) the switching bit m. containers5Does the left switching bit specify a shorter prefix? containers8The first switching bit where the two prefixes disagree. containersO(1). Decompose a map into pieces based on the structure of the underlying tree. This function is useful for consuming a map in parallel.No guarantee is made as to the sizes of the pieces; an internal, but deterministic process determines this. However, it is guaranteed that the pieces returned will be in ascending order (all elements in the first submap less than all elements in the second, and so on). Examples: splitRoot (fromList (zip [1..6::Int] ['a'..])) == [fromList [(1,'a'),(2,'b'),(3,'c')],fromList [(4,'d'),(5,'e'),(6,'f')]] splitRoot empty == []Note that the current implementation does not return more than two submaps, but you should not depend on this behaviour because it can change in the future without notice. containersO(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. containersO(n). The expression ( hang wide map.) shows the tree that implements the map. If hang is , a hanging6 tree is shown otherwise a rotated tree is shown. If wide is !, an extra wide version is shown.  containers  containers  containers  containers containers containers%Traverses in order of increasing key. containers!Folds in order of increasing key. containers  containersEquivalent to  ReaderT k (ReaderT x (MaybeT f)).  containersEquivalent to  ReaderT k (ReaderT x (MaybeT f)).  containers  containers  containersEquivalent to .ReaderT Key (ReaderT x (ReaderT y (MaybeT f)))  containersEquivalent to .ReaderT Key (ReaderT x (ReaderT y (MaybeT f)))  containers  containers containersWhat to do with keys in m1 but not m2 containersWhat to do with keys in m2 but not m1 containersWhat to do with keys in both m1 and m2 containersMap m1 containersMap m2 containersWhat to do with keys in m1 but not m2 containersWhat to do with keys in m2 but not m1 containersWhat to do with keys in both m1 and m2 containersMap m1 containersMap m29 9 (c) wren romano 2016 BSD-stylelibraries@haskell.orgportableSafe 23S Safe-Inferred7Safe />? containersThe constraint Whoops s is unsatisfiable for every  s<. Under GHC 8.0 and above, trying to use a function with a Whoops s constraint will lead to a pretty type error explaining how to fix the problem. Under earlier GHC versions, it will produce an extremely ugly type error within which the desired message is buried.Example oldFunction :: Whoops "oldFunction is gone now. Use newFunction." => Int -> IntMap a -> IntMap a 8 Safe-Inferred/? containersThis function has moved to 9. containersThis function has moved to :. (c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable Trustworthy< containersO(log n). The expression ( def k map) returns the value at key k or returns default value def! when the key is not in the map. findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x' findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a' containersO(1). A map with a single element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1 containersO(log n). Insert a new key and value in the map. If the key is already present in the map, the associated value is replaced with the supplied value.  is equivalent to  . insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')] insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')] insert 5 'x' empty == singleton 5 'x' containersO(log n)>. Insert with a function, combining new value and old value.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key, f new_value old_value). insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")] insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWith (++) 5 "xxx" empty == singleton 5 "xxx" containersO(log n). Insert with a function, combining key, new value and old value.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert the pair (key,f key new_value old_value);. Note that the key passed to f is the same key passed to . let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")] insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWithKey f 5 "xxx" empty == singleton 5 "xxx" containersO(log n). Combines insert operation with old value retrieval. The expression ( f k x map2) is a pair where the first element is equal to ( k map$) and the second element equal to ( f k x map). let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")]) insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")]) insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")This is how to define  insertLookup using insertLookupWithKey: let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")]) insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")]) containersO(log n). Update a value at a specific key with the result of the provided function. When the key is not a member of the map, the original map is returned. adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjust ("new " ++) 7 empty == empty containersO(log n). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. let f key x = (show key) ++ ":new " ++ x adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjustWithKey f 7 empty == empty containersO(log n). The expression ( f k map) updates the value x at k (if it is in the map). If (f x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f x = if x == "a" then Just "new a" else Nothing update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n). The expression ( f k map) updates the value x at k (if it is in the map). If (f k x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n). Lookup and update. See also . The function returns changed value, if it is updated. Returns the original key value if the map entry is deleted. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")]) updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")]) updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a") containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof. 7 can be used to insert, delete, or update a value in a . In short :  k ( f k m) = f ( k m). let f _ = Nothing alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" let f _ = Just "c" alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")] alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")] Note that  = alter . fmap. containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof.  can be used to inspect, insert, delete, or update a value in a  . In short:  k <$>  f k m = f ( k m).Example: interactiveAlter :: Int -> Map Int String -> IO (Map Int String) interactiveAlter k m = alterF f k m where f Nothing = do putStrLn $ show k ++ " was not found in the map. Would you like to add it?" getUserResponse1 :: IO (Maybe String) f (Just old) = do putStrLn $ "The key is currently bound to " ++ show old ++ ". Would you like to change or delete it?" getUserResponse2 :: IO (Maybe String)  is the most general operation for working with an individual key that may or may not be in a given map. When used with trivial functors like  and , it is often slightly slower than more specialized combinators like  and . However, when the functor is non-trivial and key comparison is not particularly cheap, it is the fastest way.Note on rewrite rules:3This module includes GHC rewrite rules to optimize  for the  and  functors. In general, these rules improve performance. The sole exception is that when using , deleting a key that is already absent takes longer than it would without the rules. If you expect this to occur a very large fraction of the time, you might consider using a private copy of the  type.Note:  is a flipped version of the at combinator from Control.Lens.At. containersO(log n). Update the element at index. Calls  when an invalid index is used. updateAt (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")] updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")] updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range containersO(log n)&. Update the value at the minimal key. updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")] updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n)&. Update the value at the maximal key. updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")] updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containersO(log n)&. Update the value at the minimal key. updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")] updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n)&. Update the value at the maximal key. updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")] updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containers=The union of a list of maps, with a combining operation: ( f == !# ( f) ). unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")] containersO(m*log(n/m + 1)), m <= n". Union with a combining function. unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")] containersO(m*log(n/m + 1)), m <= n#. Union with a combining function. let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")] containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the values of these keys. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")]) == singleton 3 "b:B" containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")]) == singleton 3 "3:b|B" containersO(m*log(n/m + 1)), m <= n). Intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA" containersO(m*log(n/m + 1)), m <= n). Intersection with a combining function. let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A" containersMap covariantly over a  f k x. containersMap covariantly over a  f k x y. containersWhen a key is found in both maps, apply a function to the key and values and maybe use the result in the merged map. zipWithMaybeMatched :: (k -> x -> y -> Maybe z) -> SimpleWhenMatched k x y z  containersWhen a key is found in both maps, apply a function to the key and values, perform the resulting action, and maybe use the result in the merged map.This is the fundamental  tactic. containersWhen a key is found in both maps, apply a function to the key and values to produce an action and use its result in the merged map. containersWhen a key is found in both maps, apply a function to the key and values and use the result in the merged map. zipWithMatched :: (k -> x -> y -> z) -> SimpleWhenMatched k x y z  containersMap over the entries whose keys are missing from the other map, optionally removing some. This is the most powerful 0 tactic, but others are usually more efficient. mapMaybeMissing :: (k -> x -> Maybe y) -> SimpleWhenMissing k x y ?mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))but mapMaybeMissing uses fewer unnecessary  operations. containers?Map over the entries whose keys are missing from the other map. 7mapMissing :: (k -> x -> y) -> SimpleWhenMissing k x y 5mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)but  mapMissing is somewhat faster. containersTraverse over the entries whose keys are missing from the other map, optionally producing values to put in the result. This is the most powerful 0 tactic, but others are usually more efficient. containersTraverse over the entries whose keys are missing from the other map. containersO(n+m)). An unsafe universal combining function.WARNING: This function can produce corrupt maps and its results may depend on the internal structures of its inputs. Users should prefer ; or <.When  is given three arguments, it is inlined to the call site. You should therefore use  only to define custom combining functions. For example, you could define ,  and  as myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2 When calling  combine only1 only2, a function combining two s is created, such thatif a key is present in both maps, it is passed with both corresponding values to the combine function. Depending on the result, the key is either present in the result with specified value, or is left out;>a nonempty subtree present only in the first map is passed to only1* and the output is added to the result;?a nonempty subtree present only in the second map is passed to only2* and the output is added to the result.The only1 and only2 methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of only1 and only2 are  and  , but for example  f or  f could be used for any f. containersO(n). Map values and collect the  results. let f x = if x == "a" then Just "new a" else Nothing mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a" containersO(n)". Map keys/values and collect the  results. let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3" containersO(n)'. Traverse keys/values and collect the  results. containersO(n). Map values and separate the  and  results. let f a = if a < "c" then Left a else Right a mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")]) mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) containersO(n)#. Map keys/values and separate the  and  results. let f k a = if k < 5 then Left (k * 2) else Right (a ++ a) mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")]) mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")]) containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")] containersO(n),. Map a function over all values in the map. let f key x = (show key) ++ ":" ++ x mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")] containersO(n).  f m ==   $ $ ((k, v) -> (v' -> v' `seq` (k,v'))  $ f k v) ( m)* That is, it behaves much like a regular  except that the traversing function also has access to the key associated with a value and the values are forced before they are installed in the result map. traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')]) traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a b = (a ++ b, b ++ "X") mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X") mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in descending order of keys. containers O(n*log n).  c f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c. The value at the greater of the two original keys is used as the first argument to c. mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab" mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab" containersO(n). Build a map from a set of keys and a function which for each key computes its value. fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")] fromSet undefined Data.Set.empty == empty containers O(n*log n)7. Build a map from a list of key/value pairs. See also . If the list contains more than one value for the same key, the last value for the key is retained.If the keys of the list are ordered, linear-time implementation is used, with the performance equal to . fromList [] == empty fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")] fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")] containers O(n*log n). Build a map from a list of key/value pairs with a combining function. See also . fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")] fromListWith (++) [] == empty containers O(n*log n). Build a map from a list of key/value pairs with a combining function. See also . let f k a1 a2 = (show k) ++ a1 ++ a2 fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")] fromListWithKey f [] == empty containersO(n)6. Build a map from an ascending list in linear time. :The precondition (input list is ascending) is not checked. fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")] valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n)6. Build a map from a descending list in linear time. ;The precondition (input list is descending) is not checked. fromDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")] fromDescList [(5,"a"), (5,"b"), (3,"a")] == fromList [(3, "b"), (5, "b")] valid (fromDescList [(5,"a"), (5,"b"), (3,"b")]) == True valid (fromDescList [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list in linear time with a combining function for equal keys. :The precondition (input list is ascending) is not checked. fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from a descending list in linear time with a combining function for equal keys. ;The precondition (input list is descending) is not checked. fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "ba")] valid (fromDescListWith (++) [(5,"a"), (5,"b"), (3,"b")]) == True valid (fromDescListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list in linear time with a combining function for equal keys. :The precondition (input list is ascending) is not checked. let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2 fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")] valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False containersO(n). Build a map from a descending list in linear time with a combining function for equal keys. ;The precondition (input list is descending) is not checked. let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2 fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")] == fromList [(3, "b"), (5, "5:b5:ba")] valid (fromDescListWithKey f [(5,"a"), (5,"b"), (5,"b"), (3,"b")]) == True valid (fromDescListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False containersO(n). Build a map from an ascending list of distinct elements in linear time.  The precondition is not checked. fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] valid (fromDistinctAscList [(3,"b"), (5,"a")]) == True valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False containersO(n). Build a map from a descending list of distinct elements in linear time.  The precondition is not checked. fromDistinctDescList [(5,"a"), (3,"b")] == fromList [(3, "b"), (5, "a")] valid (fromDistinctDescList [(5,"a"), (3,"b")]) == True valid (fromDistinctDescList [(5,"a"), (3,"b"), (3,"a")]) == False(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableSafe(c) David Feuer 2016 BSD-stylelibraries@haskell.orgportableSafe 23(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableSafe(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableSafe/?3 containers=This function is being removed and is no longer usable. Use =. containers=This function is being removed and is no longer usable. Use >. containers=This function is being removed and is no longer usable. Use ?. containers=This function is being removed and is no longer usable. Use ". containers=This function is being removed and is no longer usable. Use .@ Safe-Inferred/? containers has moved to 9 containers has moved to :(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableNoneC}- containers O(min(n,W)). The expression ( def k map) returns the value at key k or returns def, when the key is not an element of the map. findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x' findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a' containersO(1). A map of one element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1 containers O(min(n,W)). Insert a new key/value pair in the map. If the key is already present in the map, the associated value is replaced with the supplied value, i.e.  is equivalent to  . insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')] insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')] insert 5 'x' empty == singleton 5 'x' containers O(min(n,W))%. Insert with a combining function.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert f new_value old_value. insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")] insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWith (++) 5 "xxx" empty == singleton 5 "xxx" containers O(min(n,W))%. Insert with a combining function.  f key value mp) will insert the pair (key, value) into mp if key does not exist in the map. If the key does exist, the function will insert f key new_value old_value. let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")] insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")] insertWithKey f 5 "xxx" empty == singleton 5 "xxx"7If the key exists in the map, this function is lazy in value but strict in the result of f. containers O(min(n,W)). The expression ( f k x map2) is a pair where the first element is equal to ( k map$) and the second element equal to ( f k x map). let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")]) insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")]) insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")This is how to define  insertLookup using insertLookupWithKey: let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")]) insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")]) containers O(min(n,W)). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjust ("new " ++) 7 empty == empty containers O(min(n,W)). Adjust a value at a specific key. When the key is not a member of the map, the original map is returned. let f key x = (show key) ++ ":new " ++ x adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] adjustWithKey f 7 empty == empty containers O(min(n,W)). The expression ( f k map) updates the value x at k (if it is in the map). If (f x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f x = if x == "a" then Just "new a" else Nothing update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")] update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W)). The expression ( f k map) updates the value x at k (if it is in the map). If (f k x) is %, the element is deleted. If it is ( y ), the key k is bound to the new value y. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")] updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")] updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containers O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than 6>. Returns the original key value if the map entry is deleted. let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:new a")]) updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")]) updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a") containers O(min(n,W)). The expression ( f k map) alters the value x at k, or absence thereof. 8 can be used to insert, delete, or update a value in an . In short :  k ( f k m) = f ( k m). containersO(log n). The expression ( f k map) alters the value x at k, or absence thereof.  can be used to inspect, insert, delete, or update a value in an . In short :  k  $  f k m = f ( k m).Example: interactiveAlter :: Int -> IntMap String -> IO (IntMap String) interactiveAlter k m = alterF f k m where f Nothing = do putStrLn $ show k ++ " was not found in the map. Would you like to add it?" getUserResponse1 :: IO (Maybe String) f (Just old) = do putStrLn $ "The key is currently bound to " ++ show old ++ ". Would you like to change or delete it?" getUserResponse2 :: IO (Maybe String)  is the most general operation for working with an individual key that may or may not be in a given map. containers8The union of a list of maps, with a combining operation. unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])] == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")] containersO(n+m)&. The union with a combining function. unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")] containersO(n+m)&. The union with a combining function. let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")] containersO(n+m)'. Difference with a combining function. let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")]) == singleton 3 "b:B" containersO(n+m). Difference with a combining function. When two equal keys are encountered, the combining function is applied to the key and both values. If it returns , the element is discarded (proper set difference). If it returns ( y+), the element is updated with a new value y. let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")]) == singleton 3 "3:b|B" containersO(n+m)-. The intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA" containersO(n+m)-. The intersection with a combining function. let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A" containersO(n+m):. A high-performance universal combining function. Using , all combining functions can be defined without any loss of efficiency (with exception of ,  and ,, where sharing of some nodes is lost with ).6Please make sure you know what is going on when using , otherwise you can be surprised by unexpected code growth or even corruption of the data structure.When  is given three arguments, it is inlined to the call site. You should therefore use  only to define your custom combining functions. For example, you could define ,  and  as myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2 myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2 myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2 When calling  combine only1 only2, a function combining two s is created, such thatif a key is present in both maps, it is passed with both corresponding values to the combine function. Depending on the result, the key is either present in the result with specified value, or is left out;>a nonempty subtree present only in the first map is passed to only1* and the output is added to the result;?a nonempty subtree present only in the second map is passed to only2* and the output is added to the result.The only1 and only2 methods must return a map with a subset (possibly empty) of the keys of the given map. The values can be modified arbitrarily. Most common variants of only1 and only2 are  and  , but for example  f or  f could be used for any f. containersO(log n)&. Update the value at the minimal key. updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")] updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(log n)&. Update the value at the maximal key. updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")] updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containersO(log n)&. Update the value at the maximal key. updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")] updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" containersO(log n)&. Update the value at the minimal key. updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")] updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")] containersO(n),. Map a function over all values in the map. let f key x = (show key) ++ ":" ++ x mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")] containersO(n).  f s ==   $  ((k, v) -> (,) k  $ f k v) ( m)* That is, behaves exactly like a regular  except that the traversing function also has access to the key associated with a value. traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')]) traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing containersO(n)'. Traverse keys/values and collect the  results. containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a b = (a ++ b, b ++ "X") mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in ascending order of keys. let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X") mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")]) containersO(n). The function  threads an accumulating argument through the map in descending order of keys. containers O(n*log n).  c f s! is the map obtained by applying f to each key of s.)The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c. mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab" mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab" containersO(n). Map values and collect the  results. let f x = if x == "a" then Just "new a" else Nothing mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a" containersO(n)". Map keys/values and collect the  results. let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3" containersO(n). Map values and separate the  and  results. let f a = if a < "c" then Left a else Right a mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")]) mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) containersO(n)#. Map keys/values and separate the  and  results. let f k a = if k < 5 then Left (k * 2) else Right (a ++ a) mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")]) mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")]) == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")]) containersO(n). Build a map from a set of keys and a function which for each key computes its value. fromSet (\k -> replicate k 'a') (Data.IntSet.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")] fromSet undefined Data.IntSet.empty == empty containers O(n*min(n,W)).. Create a map from a list of key/value pairs. fromList [] == empty fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")] fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")] containers O(n*min(n,W)). Create a map from a list of key/value pairs with a combining function. See also . fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")] fromListWith (++) [] == empty containers O(n*min(n,W)). Build a map from a list of key/value pairs with a combining function. See also fromAscListWithKey'. fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")] fromListWith (++) [] == empty containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order. fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")] fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. :The precondition (input list is ascending) is not checked. fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order, with a combining function on equal keys. :The precondition (input list is ascending) is not checked. fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")] containersO(n). Build a map from a list of key/value pairs where the keys are in ascending order and all distinct. The precondition (input list is strictly ascending) is not checked. fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]A(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable TrustworthyE(c) wren romano 2016 BSD-stylelibraries@haskell.orgportable Trustworthy 23Of  containersMap covariantly over a  f k x. containersMap covariantly over a  f k x y. containersWhen a key is found in both maps, apply a function to the key and values and maybe use the result in the merged map. zipWithMaybeMatched :: (k -> x -> y -> Maybe z) -> SimpleWhenMatched k x y z  containersWhen a key is found in both maps, apply a function to the key and values, perform the resulting action, and maybe use the result in the merged map.This is the fundamental  tactic. containersWhen a key is found in both maps, apply a function to the key and values to produce an action and use its result in the merged map. containersWhen a key is found in both maps, apply a function to the key and values and use the result in the merged map. zipWithMatched :: (k -> x -> y -> z) -> SimpleWhenMatched k x y z  containersMap over the entries whose keys are missing from the other map, optionally removing some. This is the most powerful 0 tactic, but others are usually more efficient. mapMaybeMissing :: (k -> x -> Maybe y) -> SimpleWhenMissing k x y ?mapMaybeMissing f = traverseMaybeMissing (\k x -> pure (f k x))but mapMaybeMissing uses fewer unnecessary  operations. containers?Map over the entries whose keys are missing from the other map. 7mapMissing :: (k -> x -> y) -> SimpleWhenMissing k x y 5mapMissing f = mapMaybeMissing (\k x -> Just $ f k x)but  mapMissing is somewhat faster. containersTraverse over the entries whose keys are missing from the other map, optionally producing values to put in the result. This is the most powerful 0 tactic, but others are usually more efficient. containersTraverse over the entries whose keys are missing from the other map.(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableSafePG(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableSafe/?S containers=This function is being removed and is no longer usable. Use A= containers=This function is being removed and is no longer usable. Use A>. containers=This function is being removed and is no longer usable. Use . containers=This function is being removed and is no longer usable. Use .BCDBCEBCEFGHIJKLMNOPQRSTTUV W X YZ[\]3^_`abcdefghij,klmnopqrstuvwxyz{|}2"~#$%;9:bij*a51$%{z                                   O                                                                                     ]+^_`ab5cdefghij,=>?k6l$%pqrstuvwxyo;<z{|"~#9:^_`abcdefghij,klvwxymnoz{qstu|}2"~#9:^_`abcd5efghoij,=>?k6lvwxy;<pqrstu|z{"~#9:898:  j , = > ?     6  l                             |                  2@9@:j,=>?6l|2B#   BBBBBBBBB"BBBBBBBBBBBBBBBBBB7)containers-0.6.5.1-DtpF2niqxyY5fgKWwj67yiData.Map.Internal!Utils.Containers.Internal.BitUtil"Utils.Containers.Internal.BitQueueData.Sequence.Internal$Utils.Containers.Internal.StrictPairData.Set.InternalData.Set Data.SequenceData.Sequence.Internal.Sorting Data.Tree Data.GraphData.Map.Merge.LazyData.Map.Strict.InternalData.Map.Strict Data.Map.LazyData.Map.Merge.StrictData.Map.Internal.Debug Data.IntSetData.IntSet.InternalData.Containers.ListUtilsData.IntMap.Merge.LazyData.IntMap.InternalData.IntMap.Strict.InternalData.IntMap.LazyData.IntMap.Internal.DebugData.MapData.IntMap.Merge.Strict Data.IntMap#Utils.Containers.Internal.Coercions%Utils.Containers.Internal.PtrEqualityUtils.Containers.Internal.State%Utils.Containers.Internal.StrictMaybePreludefoldrfoldltakedrop Data.List takeWhile dropWhile Control.Monad replicateMMapinsert Data.FunctiononData.SemigroupOptionfoldMapWithIndexfoldSetnublookupupdateLookupWithKey#Utils.Containers.Internal.TypeError$Data.Map.Internal.DeprecatedShowTreeshowTree showTreeWithmergemergeA insertWith insertWithKeyinsertLookupWithKey$Data.IntMap.Internal.DeprecatedDebugData.IntMap.StrictbaseData.Functor.Identity runIdentityIdentitybitcounthighestBitMaskshiftRLshiftLLwordSizeBitQueue BitQueueBemptyQBsnocQBbuildQunconsQtoListQ$fShowBitQueueB$fShowBitQueueStaterunState execStateMaybeSNothingSJustS StrictPair:*:toPairSizeBinTip\\nullsizemember notMemberlookupLTlookupGTlookupLElookupGEempty singletondeletealterFisProperSubsetOf isSubsetOfdisjoint lookupMinfindMin lookupMaxfindMax deleteMin deleteMaxunionsunion difference intersectionfilter partitionmap mapMonotonicfoldr'foldl'elemstoList toAscList toDescListfromList fromAscList fromDescListfromDistinctAscListfromDistinctDescListsplit splitMember findIndex lookupIndexelemAtdeleteAtsplitAttakeWhileAntitonedropWhileAntitone spanAntitonelink deleteFindMin deleteFindMaxminViewmaxViewbin splitRootpowerSetcartesianProduct disjointUnionvalidbalanced $fNFDataSet $fReadSet $fShow1Set $fOrd1Set$fEq1Set $fShowSet$fOrdSet$fEqSet $fIsListSet $fDataSet $fFoldableSet$fSemigroupSet $fMonoidSet$fMonoidMergeSet$fSemigroupMergeSetViewREmptyR:>ViewLEmptyL:<ElemgetElemNodeNode2Node3DigitOneTwoThreeFour FingerTreeEmptyTSingleDeepSeq MaybeForceSized:|>:<|Empty liftA2Seq intersperse foldDigitfoldNode replicate replicateA 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$fShowSeq$fOrdSeq$fEqSeq$fAlternativeSeq$fMonadPlusSeq$fApplicativeSeq $fMonadFixSeq $fMonadSeq $fNFDataSeq$fTraversableSeq $fFoldableSeq $fFunctorSeq$fTraversableViewL$fFoldableViewL$fFunctorViewL $fDataSeq$fTraversableViewR$fFoldableViewR$fFunctorViewR$fUnzipWithSeq$fUnzipWithFingerTree$fUnzipWithDigit$fUnzipWithNode$fUnzipWithElem $fEqViewR $fOrdViewR $fShowViewR $fReadViewR $fEqViewL $fOrdViewL $fShowViewL $fReadViewL$fGenericViewR$fGeneric1TYPEViewR $fDataViewR$fGenericViewL$fGeneric1TYPEViewL $fDataViewL $fGenericElem$fGeneric1TYPEElem $fGenericNode$fGeneric1TYPENode$fGenericDigit$fGeneric1TYPEDigit$fGenericFingerTree$fGeneric1TYPEFingerTreeITQListITQNilITQConsIndexedTaggedQueueITQTQListTQNilTQCons TaggedQueueTQIQListIQNilIQCons IndexedQueueIQQListNilQConsQueueQsortsortBysortOn unstableSortunstableSortByunstableSortOnmergeQmergeTQmergeIQmergeITQpopMinQpopMinIQpopMinTQ popMinITQbuildIQbuildTQbuildITQfoldToMaybeTreefoldToMaybeWithIndexTreeForestTree rootLabel subForestdrawTree drawForestflattenlevelsfoldTree unfoldTree unfoldForest unfoldTreeM unfoldForestMunfoldTreeM_BFunfoldForestM_BF$fMonadZipTree $fNFDataTree$fFoldableTree$fTraversableTree$fMonadFixTree $fMonadTree$fApplicativeTree $fFunctorTree $fRead1Tree $fShow1Tree $fOrd1Tree $fEq1Tree$fEqTree $fOrdTree $fReadTree $fShowTree $fDataTree $fGenericTree$fGeneric1TYPETreeEdgeBoundsGraphTableVertexSCC AcyclicSCC CyclicSCC flattenSCCs flattenSCCstronglyConnCompstronglyConnCompRverticesedgesbuildG transposeG outdegreeindegreegraphFromEdges'graphFromEdgesdffdfstopSortreverseTopSort componentsscc reachablepathbcc $fFunctorSCC $fNFDataSCC$fTraversableSCC $fFoldableSCC $fRead1SCC $fShow1SCC$fEq1SCC$fApplicativeSetM $fFunctorSetM $fMonadSetM$fEqSCC $fShowSCC $fReadSCC $fGenericSCC$fGeneric1TYPESCC $fDataSCCSimpleWhenMatched WhenMatched matchedKeySimpleWhenMissing WhenMissingmissingSubtree missingKey AreWeStrictStrictLazy!findWithDefault adjustWithKey updateWithKeyalter atKeyImpl atKeyPlainupdateAt updateMin updateMaxupdateMinWithKeyupdateMaxWithKeyminViewWithKeymaxViewWithKey unionsWith unionWith unionWithKey withoutKeysdifferenceWithdifferenceWithKey restrictKeysintersectionWithintersectionWithKeycomposemapWhenMissingmapGentlyWhenMissingmapGentlyWhenMatchedlmapWhenMissingcontramapFirstWhenMatchedcontramapSecondWhenMatchedrunWhenMatchedrunWhenMissingmapWhenMatchedzipWithMatchedzipWithAMatchedzipWithMaybeMatchedzipWithMaybeAMatched dropMissingpreserveMissingpreserveMissing' mapMissingmapMaybeMissing filterMissingfilterAMissingtraverseMissingtraverseMaybeMissing mergeWithKey isSubmapOf isSubmapOfByisProperSubmapOfisProperSubmapOfBy filterWithKeypartitionWithKeymapMaybemapMaybeWithKeytraverseMaybeWithKey mapEithermapEitherWithKey mapWithKeytraverseWithKeymapAccummapAccumWithKeymapAccumRWithKeymapKeys mapKeysWithmapKeysMonotonic foldrWithKey foldrWithKey' foldlWithKey foldlWithKey'foldMapWithKeykeysassocskeysSetfromSet fromListWithfromListWithKeyfromAscListWithfromDescListWithfromAscListWithKeyfromDescListWithKey splitLookup insertMaxlink2gluedeltabalancebalanceLbalanceR $fShowMap $fReadMap $fNFDataMap$fBifoldableMap $fFoldableMap$fTraversableMap $fFunctorMap $fRead1Map $fShow1Map $fShow2Map $fOrd1Map $fOrd2Map$fEq1Map$fEq2Map$fOrdMap$fEqMap $fIsListMap $fDataMap$fSemigroupMap $fMonoidMap$fMonadWhenMissing$fApplicativeWhenMissing$fCategoryTYPEWhenMissing$fFunctorWhenMissing$fMonadWhenMatched$fApplicativeWhenMatched$fCategoryTYPEWhenMatched$fFunctorWhenMatched showsTree showsTreeHangshowWide showsBarsnodewithBar withEmptyordered validsizeKeyBitMapMaskPrefixIntSet suffixBitMask prefixBitMaskbitmapOfzeromatch$fNFDataIntSet $fReadIntSet $fShowIntSet $fOrdIntSet $fEqIntSet $fDataIntSet$fSemigroupIntSet$fMonoidIntSet$fIsListIntSetnubOrdnubOrdOnnubIntnubIntOnIntMapNat natFromInt intFromNat mergeWithKey' linkWithMask binCheckLeft binCheckRightnomatchmaskmaskWshorter branchMask $fRead1IntMap $fReadIntMap $fShow1IntMap $fShowIntMap$fFunctorIntMap $fOrd1IntMap $fOrdIntMap $fEq1IntMap $fEqIntMap$fIsListIntMap $fDataIntMap$fNFDataIntMap$fTraversableIntMap$fFoldableIntMap$fSemigroupIntMap$fMonoidIntMap insertWith'insertWithKey'insertLookupWithKey' foldWithKey.^# Data.Foldable.#ptrEqhetPtrEqmaybeStoMaybetoMaybeSGHC.Errerror GHC.MaybeNothingghc-prim GHC.TypesTrueGHC.Base ApplicativeliftA2purefmapData.TraversabletraverseGHC.ArrArrayFoldableGHC.Primseq Data.TuplefstsndControl.Monad.ZipmzipWithmunzip<>$!?consTreesnocTree appendTree0 GHC.ClassescompareEQfoldMapMaybe SemigroupJustconstData.Functor.ConstConstidFalse Data.EitherLeftRightuncurryIntGHC.EnumfromEnumWhoopsSymbol