h$        ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~                                                                                                                                                                                                                                   ,:(c) Clark Gaebel 2012 (c) Johan Tibel 2012 BSD-stylelibraries@haskell.orgportableSafestrict-containers0Return a word where only the highest bit is set.   (c) David Feuer 2016 BSD-stylelibraries@haskell.orgportable Safe-Inferred strict-containersCreate an empty bit queue builder. This is represented as a single guard bit in the most significant position.strict-containersEnqueue a bit. This works by shifting the queue right one bit, then setting the most significant bit as requested.strict-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.strict-containers3Dequeue an element, or discover the queue is empty.strict-containersConvert a bit queue to a list of bits by unconsing. This is used to test that the queue functions properly.  None strict-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   8 9  Noneh strict-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. strict-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.  4 4! Safe-Inferred" Safe-Inferred Safe strict-containers'The same as a regular Haskell pair, but  (x :*: _|_) = (_|_ :*: y) = _|_ strict-containers)Convert a strict pair to a standard pair.1 Safe />?"strict-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   Safe-Inferred" !" "! Safe-Inferred# None(a )strict-containersCreate a new mutable array of specified size, in the specified state thread, with each element containing the specified initial value.6strict-containersUnsafely copy the elements of an array. Array bounds are not checked.7strict-containersUnsafely copy the elements of an array. Array bounds are not checked.8strict-containersCreate a new array of the n first elements of mary.9strict-containersO(n) Insert an element at the given position in this array, increasing its size by one.:strict-containersO(n) Insert an element at the given position in this array, increasing its size by one.;strict-containersO(n)8 Update the element at the given position in this array.<strict-containersO(n) Update the element at the given positio in this array, by applying a function to it. Evaluates the element to WHNF before inserting it into the array.=strict-containersO(1) Update the element at the given position in this array, without copying.Cstrict-containers=Verifies that a predicate holds for all elements of an array.Estrict-containersO(n) Delete an element at the given position in this array, decreasing its size by one.Gstrict-containersStrict version of F.)#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJK)$#)*+,-'(./021;<=9:E&834%567A>@?BCDFGJKIH None./3e4Sstrict-containersA map from keys to values. A map cannot contain duplicate keys; each key can map to at most one value.[strict-containers8A set of values. A set cannot contain duplicate values.Convenience function. Compute a hash value for the given value.^strict-containersO(1) Construct an empty map._strict-containersO(1)' Construct a map with a single element.`strict-containersO(1) Return   if this map is empty,   otherwise.astrict-containersO(n)5 Return the number of key-value mappings in this map.bstrict-containersO(log n) Return  . if the specified key is present in the map,   otherwise.cstrict-containersO(log n)< Return the value to which the specified key is mapped, or  - if this map contains no mapping for the key.dstrict-containerslookup' is a version of lookup that takes the hash separately. It is used to implement alterF.f strict-containersO(log n)< Return the value to which the specified key is mapped, or  - if this map contains no mapping for the key.This is a flipped version of c.g strict-containersO(log n) Return the value to which the specified key is mapped, or the default value if this map contains no mapping for the key.hstrict-containersO(log n) Return the value to which the specified key is mapped, or the default value if this map contains no mapping for the key.DEPRECATED: lookupDefault is deprecated as of version 0.2.11, replaced by g.istrict-containersO(log n)? Return the value to which the specified key is mapped. Calls  - if this map contains no mapping for the key.jstrict-containers Create a X value with two Y values.kstrict-containers Create a U or W node.lstrict-containersO(log n) Associate the specified value with the specified key in this map. If this map previously contained a mapping for the key, the old value is replaced.pstrict-containers!In-place update version of insertqstrict-containersCreate a map from two key-value pairs which hashes don't collide. To enhance sharing, the second key-value pair is represented by the hash of its key and a singleton HashMap pairing its key with its value.Note: to avoid silly thunks, this function must be strict in the key. See issue #232. We don't need to force the HashMap argument because it's already in WHNF (having just been matched) and we just put it directly in an array.rstrict-containersO(log n) Associate the value with the key in this map. If this map previously contained a mapping for the key, the old value is replaced by the result of applying the given function to the new and old value. Example: 2insertWith f k v map where f new old = new + oldsstrict-containersinsertModifying is a lot like insertWith; we use it to implement alterF. It takes a value to insert when the key is absent and a function to apply to calculate a new value when the key is present. Thanks to the unboxed unary tuple, we avoid introducing any unnecessary thunks in the tree.tstrict-containersO(log n) Remove the mapping for the specified key from this map if present.vstrict-containersDelete optimized for the case when we know the key is in the map.It is only valid to call this when the key exists in the map and you know the hash collision position if there was one. This information can be obtained from e0. If there is no collision pass (-1) as collPos.We can skip: - the key equality check on the leaf, if we reach a leaf it must be the keywstrict-containersO(log n) Adjust the value tied to a given key in this map only if it is present. Otherwise, leave the map alone.xstrict-containers Much like w, but not inherently leaky.ystrict-containersO(log n) The expression (y 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.zstrict-containersO(log n) The expression (z f k map) alters the value x at k, or absence thereof.z can be used to insert, delete, or update a value in a map. In short: c k (z f k m) = f (c k m) { strict-containersO(log n) The expression ({ f k map) alters the value x at k, or absence thereof.{; can be used to insert, delete, or update a value in a map.Note: { is a flipped version of the at combinator from  https://hackage.haskell.org/package/lens/docs/Control-Lens-At.html#v:atControl.Lens.At.| strict-containers O(n*log m) Inclusion of maps. A map is included in another map if the keys are subsets and the corresponding values are equal: isSubmapOf m1 m2 = keys m1 `isSubsetOf` keys m2 && and [ v1 == v2 | (k1,v1) <- toList m1; let v2 = m2 ! k1 ]Examples:fromList [(1,'a')] `isSubmapOf` fromList [(1,'a'),(2,'b')]True:fromList [(1,'a'),(2,'b')] `isSubmapOf` fromList [(1,'a')]False} strict-containers O(n*log m) Inclusion of maps with value comparison. A map is included in another map if the keys are subsets and if the comparison function is true for the corresponding values: isSubmapOfBy cmpV m1 m2 = keys m1 `isSubsetOf` keys m2 && and [ v1 `cmpV` v2 | (k1,v1) <- toList m1; let v2 = m2 ! k1 ]ExamplesisSubmapOfBy (<=) (fromList [(1,'a')]) (fromList [(1,'b'),(2,'c')])TrueisSubmapOfBy (<=) (fromList [(1,'b')]) (fromList [(1,'a'),(2,'c')])False~strict-containersO(n+m) The union of two maps. If a key occurs in both maps, the mapping from the first will be the mapping in the result.Examples?union (fromList [(1,'a'),(2,'b')]) (fromList [(2,'c'),(3,'d')])"fromList [(1,'a'),(2,'b'),(3,'d')]strict-containersO(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.strict-containersO(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.strict-containersStrict in the result of f.strict-containers fromList [(2,'c'),(3,'d')]"fromList [(1,'a'),(2,'b'),(3,'d')] strict-containersgstrict-containersDefault value to return.hstrict-containersDefault value to return.NOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~STUVWXYZ^_`abcfghilrptwyz{|}~RQkj[q]\eNOPmudnovsxi9 #2010-2012 Johan Tibell BSD-stylejohan.tibell@gmail.comportable Trustworthy|strict-containersO(1)' Construct a map with a single element.strict-containersO(log n) Associate the specified value with the specified key in this map. If this map previously contained a mapping for the key, the old value is replaced.strict-containersO(log n) Associate the value with the key in this map. If this map previously contained a mapping for the key, the old value is replaced by the result of applying the given function to the new and old value. Example: 2insertWith f k v map where f new old = new + oldstrict-containersO(log n) Adjust the value tied to a given key in this map only if it is present. Otherwise, leave the map alone.strict-containersO(log n) The expression ( f k map) updates the value x at k (if it is in the map). If (f x) is  $, the element is deleted. If it is (  y) , the key k is bound to the new value y.strict-containersO(log n) The expression ( f k map) alters the value x at k, or absence thereof. can be used to insert, delete, or update a value in a map. In short: c k ( f k m) = f (c k m)  strict-containersO(log n) The expression ( f k map) alters the value x at k, or absence thereof.; can be used to insert, delete, or update a value in a map.Note:  is a flipped version of the at combinator from  https://hackage.haskell.org/package/lens/docs/Control-Lens-At.html#v:atControl.Lens.At.strict-containersO(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.strict-containersO(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.strict-containersO(n): Transform this map by applying a function to every value.strict-containersO(n): Transform this map by applying a function to every value.strict-containersO(n) Transform this map by applying a function to every value and retaining only some of them.strict-containersO(n) Transform this map by applying a function to every value and retaining only some of them.strict-containersO(n) Perform an  & action for each key-value pair in a S and produce a S of all the results. Each S# will be strict in all its values. traverseWithKey f = fmap ( id) .  Data.Strict.HashMap.Autogen.Lazy.$% f Note: the order in which the actions occur is unspecified. In particular, when the map contains hash collisions, the order in which the actions associated with the keys involved will depend in an unspecified way on their insertion order.strict-containers O(n*log 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.strict-containersO(n+m) Intersection of two maps. If a key occurs in both maps the provided function is used to combine the values from the two maps.strict-containersO(n+m) Intersection of two maps. If a key occurs in both maps the provided function is used to combine the values from the two maps.strict-containers O(n*log n) Construct a map with the supplied mappings. If the list contains duplicate mappings, the later mappings take precedence.strict-containers O(n*log n) Construct a map from a list of elements. Uses the provided function f" to merge duplicate entries with (f newVal oldVal).Examples Given a list xs, create a map with the number of occurrences of each element in xs: let xs = ['a', 'b', 'a'] in fromListWith (+) [ (x, 1) | x <- xs ] = fromList [('a', 2), ('b', 1)] Given a list of key-value pairs xs :: [(k, v)]/, group all values by their keys and return a  HashMap k [v]. let xs = ('a', 1), ('b', 2), ('a', 3)] in fromListWith (++) [ (k, [v]) | (k, v) <- xs ] = fromList [('a', [3, 1]), ('b', [2])]Note that the lists in the resulting map contain elements in reverse order from their occurences in the original list.More generally, duplicate entries are accumulated as follows; this matters when f' is not commutative or not associative. fromListWith f [(k, a), (k, b), (k, c), (k, d)] = fromList [(k, f d (f c (f b a)))] strict-containers O(n*log n) Construct a map from a list of elements. Uses the provided function to merge duplicate entries.ExamplesGiven a list of key-value pairs where the keys are of different flavours, e.g: data Key = Div | Suband the values need to be combined differently when there are duplicates, depending on the key: #combine Div = div combine Sub = (-)then fromListWithKey can be used as follows: fromListWithKey combine [(Div, 2), (Div, 6), (Sub, 2), (Sub, 3)] = fromList [(Div, 3), (Sub, 1)]=More generally, duplicate entries are accumulated as follows; fromListWith f [(k, a), (k, b), (k, c), (k, d)] = fromList [(k, f k d (f k c (f k b a)))]4S^`abcfghit|}~4S^`abcfghit|}~ 2010-2012 Johan Tibell BSD-stylejohan.tibell@gmail.com provisionalportableSafe};4S^`abcfghit|}~4S^`abcfghit|}~ Safe-Inferred>~& Safe-Inferred~B4S^`abcfghit|}~' Safe-Inferred~ (c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 (c) wren romano 2016 BSD-stylelibraries@haskell.orgportable Trustworthy 234O strict-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. strict-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). strict-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. strict-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).strict-containersA map of integers to values a.strict-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' strict-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'strict-containersSame as .strict-containersO(1). Is the map empty? Data.Strict.IntMap.Autogen.null (empty) == True Data.Strict.IntMap.Autogen.null (singleton 1 'a') == Falsestrict-containersO(n) . Number of elements in the map. size empty == 0 size (singleton 1 'a') == 1 size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3strict-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')]) == Falsestrict-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')]) == Truestrict-containers O(min(n,W))1. Lookup the value at a key in the map. See also ().strict-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'strict-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')strict-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')]) == Nothingstrict-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')strict-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')]) == Nothingstrict-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)strict-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.Strict.IntMap.Autogen.Strict exposed a version of & that forced the values of the output ,. This version does not force these values.strict-containersO(1). The empty map. )empty == fromList [] size empty == 0strict-containersO(1). A map of one element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1strict-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'strict-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"strict-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"strict-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")])strict-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 == emptystrict-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 == emptystrict-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 == emptystrict-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"strict-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"strict-containers O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than (*>. 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")strict-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).strict-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.strict-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")]strict-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")]strict-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")]strict-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")]strict-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")]strict-containersO(n+m).. Difference between two maps (based on keys). difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"strict-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"strict-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"strict-containersO(n+m)0. Remove all the keys in a given set from a map. m `withoutKeys` s =  (k _ -> k   s) m strict-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"strict-containersO(n+m)0. The restriction of a map to the keys in a set. m `restrictKeys` s =  (k _ -> k   s) m strict-containersO(n+m)-. The intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"strict-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"strict-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. strict-containersMap covariantly over a  f x.strict-containersMap covariantly over a  f x', using only a 'Functor f' constraint.strict-containersMap covariantly over a  f k x', using only a 'Functor f' constraint. strict-containersMap contravariantly over a  f _ x. strict-containersMap contravariantly over a  f _ y z. strict-containersMap contravariantly over a  f x _ z. strict-containersAlong with zipWithMaybeAMatched, witnesses the isomorphism between WhenMatched f x y z and Key -> x -> y -> f (Maybe z). strict-containersAlong with traverseMaybeMissing, witnesses the isomorphism between WhenMissing f x y and Key -> x -> f (Maybe y). strict-containersMap covariantly over a  f x y. strict-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 strict-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. strict-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 strict-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. strict-containersDrop all the entries whose keys are missing from the other map. $dropMissing :: SimpleWhenMissing x y/dropMissing = mapMaybeMissing (\_ _ -> Nothing)but  dropMissing is much faster. strict-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. strict-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. strict-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. strict-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. strict-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. strict-containersTraverse over the entries whose keys are missing from the other map. strict-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.strict-containersO(n)'. Traverse keys/values and collect the   results. strict-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) strict-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.strict-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"strict-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"strict-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 == Nothingstrict-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 == Nothingstrict-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"strict-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"strict-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.strict-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.strict-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.strict-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.strict-containers O(min(n,W))&. The minimal key of the map. Returns   if the map is empty.strict-containers O(min(n,W))$. The minimal key of the map. Calls   if the map is empty. Use  if the map may be empty.strict-containers O(min(n,W))&. The maximal key of the map. Returns   if the map is empty.strict-containers O(min(n,W))$. The maximal key of the map. Calls   if the map is empty. Use  if the map may be empty.strict-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.strict-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.strict-containersO(n+m). Is this a proper submap? (ie. a submap but not equal). Defined as ( =  (==)).strict-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)])strict-containersO(n+m)!. Is this a submap? Defined as ( =  (==)).strict-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)])strict-containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]strict-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")]strict-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')]) == Nothingstrict-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")])strict-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")])strict-containersO(n). The function  threads an accumulating argument through the map in descending order of keys.strict-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"strict-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"strict-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")]strict-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")]) == emptystrict-containersO(n)5. Filter all keys/values that satisfy some predicate. filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"strict-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")])strict-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")])strict-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"strict-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"strict-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")])strict-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")])strict-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)strict-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)strict-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")]) == 4strict-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.strict-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")]) == 4strict-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.strict-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)"strict-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.strict-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)"strict-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.strict-containersO(n). Fold the keys and values in the map using the given monoid, such that  f = ,/ .  f*This can be an asymptotically faster than  or  for some monoids.strict-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 == []strict-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 == emptystrict-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 == []strict-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")]strict-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")]strict-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")]strict-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 (++) [] == emptystrict-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 [] == emptystrict-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")]strict-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")]strict-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")]strict-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")]strict-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.strict-containers Does the key i differ from the prefix p& before getting to the switching bit m?strict-containers Does the key i match the prefix p (up to but not including bit m)?strict-containersThe prefix of key i. up to (but not including) the switching bit m.strict-containersThe prefix of key i. up to (but not including) the switching bit m.strict-containers5Does the left switching bit specify a shorter prefix?strict-containers8The first switching bit where the two prefixes disagree.strict-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.strict-containersO(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format.strict-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. strict-containers strict-containers strict-containers strict-containersstrict-containersstrict-containers%Traverses in order of increasing key.strict-containers!Folds in order of increasing key.strict-containers strict-containersEquivalent to  ReaderT k (ReaderT x (MaybeT f)). strict-containersEquivalent to  ReaderT k (ReaderT x (MaybeT f)). strict-containers strict-containers strict-containersEquivalent to .ReaderT Key (ReaderT x (ReaderT y (MaybeT f))) strict-containersEquivalent to .ReaderT Key (ReaderT x (ReaderT y (MaybeT f))) strict-containers strict-containersstrict-containersWhat to do with keys in m1 but not m2strict-containersWhat to do with keys in m2 but not m1strict-containersWhat to do with keys in both m1 and m2strict-containersMap m1strict-containersMap m2strict-containersWhat to do with keys in m1 but not m2strict-containersWhat to do with keys in m2 but not m1strict-containersWhat to do with keys in both m1 and m2strict-containersMap m1strict-containersMap m29 9  Safe-Inferred7 Safe-Inferred/?7strict-containers has moved to 0strict-containers has moved to 1(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportableNone|-strict-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'strict-containersO(1). A map of one element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1strict-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'strict-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"strict-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.strict-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")])strict-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 == emptystrict-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 == emptystrict-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"strict-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"strict-containers O(min(n,W)). Lookup and update. The function returns original value, if it is updated. This is different behavior than (*>. 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")strict-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).strict-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.strict-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")]strict-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")]strict-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")]strict-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"strict-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"strict-containersO(n+m)-. The intersection with a combining function. intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"strict-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"strict-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.strict-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"strict-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"strict-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"strict-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"strict-containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]strict-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")]strict-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')]) == Nothingstrict-containersO(n)'. Traverse keys/values and collect the   results.strict-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")])strict-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")])strict-containersO(n). The function  threads an accumulating argument through the map in descending order of keys.strict-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"strict-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"strict-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"strict-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")])strict-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")])strict-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 == emptystrict-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")]strict-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 (++) [] == emptystrict-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 (++) [] == emptystrict-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")]strict-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")]strict-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")]strict-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")]2(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable Trustworthy~(c) wren romano 2016 BSD-stylelibraries@haskell.orgportable Trustworthy 23 strict-containersMap covariantly over a  f k x.strict-containersMap covariantly over a  f k x y.strict-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 strict-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.strict-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.strict-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 strict-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.strict-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.strict-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.strict-containersTraverse over the entries whose keys are missing from the other map. Safe-Inferred>U3 Safe-Inferred{4 Safe-Inferred2 (c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable Trustworthy 23h strict-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 . strict-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) . strict-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 . strict-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) .strict-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.strict-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' strict-containersO(log n)$. Find the value at a key. Returns  # when the element can not be found.-fromList [(5, 'a'), (3, 'b')] !? 1 == Nothing.fromList [(5, 'a'), (3, 'b')] !? 5 == Just 'a'strict-containersSame as .strict-containersO(1). Is the map empty? Data.Strict.Map.Autogen.null (empty) == True Data.Strict.Map.Autogen.null (singleton 1 'a') == Falsestrict-containersO(1)$. The number of elements in the map. size empty == 0 size (singleton 1 'a') == 1 size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3strict-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.Strict.Map.Autogen 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: Nothingstrict-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')]) == Falsestrict-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')]) == Truestrict-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'strict-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')strict-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')]) == Nothingstrict-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')strict-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')]) == Nothingstrict-containersO(1). The empty map. )empty == fromList [] size empty == 0strict-containersO(1). A map with a single element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1strict-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'strict-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"strict-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"strict-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")])strict-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 == emptystrict-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 == emptystrict-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 == emptystrict-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"strict-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"strict-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")strict-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")]strict-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.strict-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 mapstrict-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")])) == Falsestrict-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 rangestrict-containersTake a given number of entries in key order, beginning with the smallest keys.  take n =  . ,5 n .  strict-containersDrop a given number of entries in key order, beginning with the smallest keys.  drop n =  . ,6 n .  strict-containersO(log n)$. Split a map at a particular index. splitAt !n !xs = ( n xs,  n xs) strict-containersO(log n). Update the element at index, i.e. by its zero-based index in the sequence sorted by keys. If the index7 is out of range (less than zero, greater or equal to  of the map),   is called. updateAt (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")] updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")] updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of rangestrict-containersO(log n). Delete the element at index, i.e. by its zero-based index in the sequence sorted by keys. If the index7 is out of range (less than zero, greater or equal to  of the map),   is called. deleteAt 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a" deleteAt 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b" deleteAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range deleteAt (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range strict-containersO(log n)&. The minimal key of the map. Returns   if the map is empty. lookupMin (fromList [(5,"a"), (3,"b")]) == Just (3,"b") lookupMin empty = Nothingstrict-containersO(log n)$. 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 strict-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 = Nothingstrict-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 == emptystrict-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 == emptystrict-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"strict-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"strict-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"strict-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"strict-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 == Nothingstrict-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 == Nothingstrict-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 == Nothingstrict-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 == Nothingstrict-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")]strict-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")]strict-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")]strict-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")]strict-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")]strict-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"strict-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 strict-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"strict-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"strict-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"strict-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 strict-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"strict-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"strict-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) strict-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.Strict.Map.Autogen.Strict exposed a version of & that forced the values of the output ,. This version does not force these values. strict-containersMap covariantly over a  f k x.strict-containersMap covariantly over a  f k x', using only a 'Functor f' constraint.strict-containersMap covariantly over a  f k x', using only a 'Functor f' constraint. strict-containersMap contravariantly over a  f k _ x. strict-containersMap contravariantly over a  f k _ y z. strict-containersMap contravariantly over a  f k x _ z. strict-containersAlong with zipWithMaybeAMatched, witnesses the isomorphism between WhenMatched f k x y z and k -> x -> y -> f (Maybe z). strict-containersAlong with traverseMaybeMissing, witnesses the isomorphism between WhenMissing f k x y and k -> x -> f (Maybe y). strict-containersMap covariantly over a  f k x y. strict-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  strict-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. strict-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  strict-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. strict-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. strict-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. strict-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. strict-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. strict-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. strict-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. strict-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. strict-containersTraverse over the entries whose keys are missing from the other map. strict-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. strict-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) strict-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.strict-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.strict-containersO(m*log(n/m + 1)), m <= n . This function is defined as ( =  (==)).strict-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.strict-containersO(m*log(n/m + 1)), m <= n. Is this a proper submap? (ie. a submap but not equal). Defined as ( =  (==)).strict-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)])strict-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")]) == emptystrict-containersO(n)4. Filter all keys/values that satisfy the predicate. filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"strict-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 =  . 78 (p . fst) .  takeWhileAntitone p =  (k _ -> p k) strict-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 =  . 79 (p . fst) .  dropWhileAntitone p =  (k -> not (p k)) strict-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).strict-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")])strict-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")])strict-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"strict-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"strict-containersO(n)'. Traverse keys/values and collect the   results.strict-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")])strict-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")])strict-containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]strict-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")]strict-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')]) == Nothingstrict-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")])strict-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")])strict-containersO(n). The function  threads an accumulating argument through the map in descending order of keys.strict-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"strict-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"strict-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")])) == Falsestrict-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")]) == 4strict-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.strict-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")]) == 4strict-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.strict-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)"strict-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.strict-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)"strict-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.strict-containersO(n). Fold the keys and values in the map using the given monoid, such that  f = ,/ .  f*This can be an asymptotically faster than  or  for some monoids.strict-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 == []strict-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 == emptystrict-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")]strict-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 (++) [] == emptystrict-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 [] == emptystrict-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 == []strict-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")]strict-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")]strict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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)strict-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)strict-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 mapstrict-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 mapstrict-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.strict-containersstrict-containers!Folds in order of increasing key.strict-containers%Traverses in order of increasing key. strict-containers strict-containers strict-containers strict-containers strict-containers strict-containers strict-containersstrict-containers strict-containersEquivalent to " ReaderT k (ReaderT x (MaybeT f)) . strict-containersEquivalent to " ReaderT k (ReaderT x (MaybeT f)) . strict-containers strict-containers strict-containersEquivalent to . ReaderT k (ReaderT x (ReaderT y (MaybeT f)))  strict-containersEquivalent to . ReaderT k (ReaderT x (ReaderT y (MaybeT f)))  strict-containers strict-containersstrict-containersWhat to do with keys in m1 but not m2strict-containersWhat to do with keys in m2 but not m1strict-containersWhat to do with keys in both m1 and m2strict-containersMap m1strict-containersMap m2strict-containersWhat to do with keys in m1 but not m2strict-containersWhat to do with keys in m2 but not m1strict-containersWhat to do with keys in both m1 and m2strict-containersMap m1strict-containersMap m29 9 9  Safe-InferredpLstrict-containersO(n). Show the tree that implements the map. The tree is shown in a compressed, hanging format. See .strict-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,())strict-containersO(n).. Test if the internal map structure is valid. valid (fromAscList [(3,"b"), (5,"a")]) == True valid (fromAscList [(5,"a"), (3,"b")]) == Falsestrict-containers'Test if the keys are ordered correctly.strict-containers+Test if a map obeys the balance invariants.strict-containers6Test if each node of a map reports its size correctly.   Safe-Inferred/?q&strict-containersThis function has moved to 0.strict-containersThis function has moved to 1.(c) Daan Leijen 2002 (c) Andriy Palamarchuk 2008 BSD-stylelibraries@haskell.orgportable TrustworthyЧ<strict-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'strict-containersO(1). A map with a single element. singleton 1 'a' == fromList [(1, 'a')] size (singleton 1 'a') == 1strict-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'strict-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"strict-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"strict-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")])strict-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 == emptystrict-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 == emptystrict-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"strict-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"strict-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")strict-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")]strict-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.strict-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 rangestrict-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"strict-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"strict-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"strict-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"strict-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")]strict-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")]strict-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")]strict-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"strict-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"strict-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"strict-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"strict-containersMap covariantly over a  f k x.strict-containersMap covariantly over a  f k x y.strict-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 strict-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.strict-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.strict-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 strict-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.strict-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.strict-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.strict-containersTraverse over the entries whose keys are missing from the other map.strict-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.strict-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"strict-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"strict-containersO(n)'. Traverse keys/values and collect the   results.strict-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")])strict-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")])strict-containersO(n),. Map a function over all values in the map. map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]strict-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")]strict-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')]) == Nothingstrict-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")])strict-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")])strict-containersO(n). The function  threads an accumulating argument through the map in descending order of keys.strict-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"strict-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 == emptystrict-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")]strict-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 (++) [] == emptystrict-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 [] == emptystrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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")]) == Falsestrict-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.orgportableSafeӬ(c) David Feuer 2016 BSD-stylelibraries@haskell.orgportableSafe 23 Safe-Inferred>֜< Safe-Inferred(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>strict-containers$View of the right end of a sequence.strict-containersempty sequencestrict-containersthe sequence minus the rightmost element, and the rightmost elementstrict-containers#View of the left end of a sequence.strict-containersempty sequencestrict-containers-leftmost element and the rest of the sequencestrict-containers!General-purpose finite sequences.strict-containersA bidirectional pattern synonym viewing the rear of a non-empty sequence.strict-containersA bidirectional pattern synonym viewing the front of a non-empty sequence.strict-containers;A bidirectional pattern synonym matching an empty sequence.strict-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] strict-containers O(1) . The empty sequence.strict-containers O(1) . A singleton sequence.strict-containers O(\log n) .  replicate n x is a sequence consisting of n copies of x.strict-containers is an   version of  , and makes  O(\log n)  calls to   and  . *replicateA n x = sequenceA (replicate n x)strict-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 .strict-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 . toListstrict-containers O(1) . Add an element to the left end of a sequence. Mnemonic: a triangle with the single element at the pointy end.strict-containers O(1) . Add an element to the right end of a sequence. Mnemonic: a triangle with the single element at the pointy end.strict-containers O(\log(\min(n_1,n_2))) . Concatenate two sequences.strict-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.strict-containers f x is equivalent to  ( (  swap . f) x).strict-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))strict-containers O(1) . Is this the empty sequence?strict-containers O(1) ). The number of elements in the sequence.strict-containers O(1) %. Analyse the left end of a sequence.strict-containers O(1) &. Analyse the right end of a sequence.strict-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, ...]strict-containers is a variant of % that has no starting value argument: scanl1 f (fromList [x1, x2, ...]) = fromList [x1, x1 `f` x2, ...]strict-containers is the right-to-left dual of .strict-containers is a variant of % that has no starting value argument.strict-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 .strict-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 strict-containers O(\log(\min(i,n-i))) . A flipped, infix version of .strict-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.strict-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 strict-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 xsstrict-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] strict-containersA generalization of  ,  takes a mapping function that also depends on the element's index, and applies it to every element in the sequence.strict-containers is a version of  7 that also offers access to the index of each element.strict-containers O(n) . Convert a given sequence length and a function representing that sequence into a sequence.strict-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.strict-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.strict-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.strict-containers O(\log(\min(i,n-i))) ). Split a sequence at a given position.  i s = ( i s,  i s).strict-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) strict-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.strict-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.strict-containers is a version of  9 that also provides access to the index of each element.strict-containers is a version of  9 that also provides access to the index of each element.strict-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.strict-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)).strict-containers O(i)  where  i  is the prefix length.  p xs% returns the suffix remaining after  p xs.strict-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)).strict-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.strict-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.strict-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).strict-containers p is equivalent to  (not . p).strict-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.strict-containers O(n) . The  function takes a predicate p and a sequence xs and returns a sequence of those elements which satisfy the predicate.strict-containers finds the leftmost index of the specified element, if it is present, and otherwise  .strict-containers finds the rightmost index of the specified element, if it is present, and otherwise  .strict-containers finds the indices of the specified element, from left to right (i.e. in ascending order).strict-containers finds the indices of the specified element, from right to left (i.e. in descending order).strict-containers p xs9 finds the index of the leftmost element that satisfies p, if any exist.strict-containers p xs: finds the index of the rightmost element that satisfies p, if any exist.strict-containers p, finds all indices of elements that satisfy p, in ascending order.strict-containers p, finds all indices of elements that satisfy p, in descending order.strict-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 .strict-containers O(n) . The reverse of a sequence. strict-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 . strict-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.strict-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.strict-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.strict-containers O(\min(n_1,n_2,n_3)) .  takes three sequences and returns a sequence of triples, analogous to .strict-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 .strict-containers O(\min(n_1,n_2,n_3,n_4)) .  takes four sequences and returns a sequence of quadruples, analogous to .strict-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 .strict-containers    =     = strict-containersstrict-containers strict-containers strict-containers strict-containers strict-containersstrict-containersstrict-containers strict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containersstrict-containers 5555 6 05 55 55 5 Safe-Inferred(strict-containersA pairing heap tagged with both a key and the original position of its elements, for use in .strict-containersA pairing heap tagged with some key for sorting elements, for use in .strict-containersA pairing heap tagged with the original position of elements, to allow for stable sorting.strict-containersA simple pairing heap.strict-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.strict-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. strict-containers O(n \log n) .  sorts the specified  by comparing the results of a key function applied to each element.  f is equivalent to  (  AB 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  ),  (  AB f) will be faster than  f.strict-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 .strict-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 . strict-containers O(n \log n) .  sorts the specified  by comparing the results of a key function applied to each element.  f is equivalent to  (  AB 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  ),  (  AB f) will be faster than  f.strict-containers merges two s.strict-containers merges two s, based on the tag value.strict-containers merges two >s, taking into account the original position of the elements.strict-containers merges two s, based on the tag value, taking into account the original position of the elements.strict-containersPop the smallest element from the queue, using the supplied comparator.strict-containersPop the smallest element from the queue, using the supplied comparator, deferring to the item's original position when the comparator returns  .strict-containersPop the smallest element from the queue, using the supplied comparator on the tag.strict-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  .strict-containersA  $-like function, specialized to the CD= monoid, which takes advantage of the internal structure of  to avoid wrapping in   at certain points.strict-containersA E$-like function, specialized to the CD= 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-Inferred* Safe-Inferred>,estrict-containersThe position in the  is available as the index.F Safe-Inferred,G Safe-Inferred-\  (c) Roman Leshchinskiy 2008-2010 BSD-style'Roman Leshchinskiy  experimental non-portableNone 3>M8strict-containers7Mutable boxed vectors keyed on the monad they live in (  or ST s).strict-containersLength of the mutable vector.strict-containers!Check whether the vector is emptystrict-containersYield a part of the mutable vector without copying it. The vector must contain at least i+n elements.strict-containersYield a part of the mutable vector without copying it. No bounds checks are performed.strict-containers"Check whether two vectors overlap.strict-containers,Create a mutable vector of the given length.strict-containersCreate a mutable vector of the given length. The vector elements are set to bottom so accessing them will cause an exception.strict-containersCreate a mutable vector of the given length (0 if the length is negative) and fill it with an initial value.strict-containersCreate a mutable vector of the given length (0 if the length is negative) and fill it with values produced by repeatedly executing the monadic action. strict-containersO(n) Create a mutable vector of the given length (0 if the length is negative) and fill it with the results of applying the function to each index. strict-containersO(n) Create a mutable vector of the given length (0 if the length is negative) and fill it with the results of applying the monadic function to each index. Iteration starts at index 0.strict-containers"Create a copy of a mutable vector.strict-containersGrow a boxed vector by the given number of elements. The number must be non-negative. Same semantics as in  & for generic vector. It differs from grow functions for unpacked vectors, however, in that only pointers to values are copied over, therefore values themselves will be shared between two vectors. This is an important distinction to know about during memory usage analysis and in case when values themselves are of a mutable type, eg. HI or another mutable vector.Examples0import qualified Data.Strict.Vector.Autogen as V9import qualified Data.Strict.Vector.Autogen.Mutable as MV5mv <- V.thaw $ V.fromList ([10, 20, 30] :: [Integer])mv' <- MV.grow mv 2The two extra elements at the end of the newly allocated vector will be uninitialized and will result in an error if evaluated, so me must overwrite them with new values first:MV.write mv' 3 999MV.write mv' 4 777V.unsafeFreeze mv'[10,20,30,999,777]It is important to note that the source mutable vector is not affected when the newly allocated one is mutated.MV.write mv' 2 888V.unsafeFreeze mv'[10,20,888,999,777]V.unsafeFreeze mv [10,20,30]strict-containersGrow a vector by the given number of elements. The number must be non-negative but this is not checked. Same semantics as in   for generic vector.strict-containersReset all elements of the vector to some undefined value, clearing all references to external objects. This is usually a noop for unboxed vectors.strict-containers(Yield the element at the given position.strict-containers*Replace the element at the given position.strict-containers)Modify the element at the given position. strict-containersModify the element at the given position using a monadic function.strict-containers)Swap the elements at the given positions.strict-containersReplace the element at the given position and return the old element.strict-containersYield the element at the given position. No bounds checks are performed.strict-containersReplace the element at the given position. No bounds checks are performed.strict-containersModify the element at the given position. No bounds checks are performed. strict-containersModify the element at the given position using a monadic function. No bounds checks are performed.strict-containersSwap the elements at the given positions. No bounds checks are performed.strict-containersReplace the element at the given position and return the old element. No bounds checks are performed.strict-containers2Set all elements of the vector to the given value.strict-containersCopy a vector. The two vectors must have the same length and may not overlap.strict-containersCopy a vector. The two vectors must have the same length and may not overlap. This is not checked.strict-containersMove the contents of a vector. The two vectors must have the same length.:If the vectors do not overlap, then this is equivalent to . Otherwise, the copying is performed as if the source vector were copied to a temporary vector and then the temporary vector was copied to the target vector.strict-containersMove the contents of a vector. The two vectors must have the same length, but this is not checked.:If the vectors do not overlap, then this is equivalent to . Otherwise, the copying is performed as if the source vector were copied to a temporary vector and then the temporary vector was copied to the target vector.strict-containersCompute the next (lexicographically) permutation of given vector in-place. Returns False when input is the last permutation strict-containersO(n) Apply the monadic action to every element of the vector, discarding the results. strict-containersO(n) Apply the monadic action to every element of the vector and its index, discarding the results. strict-containersO(n) Apply the monadic action to every element of the vector, discarding the results. It's same as the  flip mapM_. strict-containersO(n) Apply the monadic action to every element of the vector and its index, discarding the results. It's same as the  flip imapM_. strict-containersO(n) Pure left fold. strict-containersO(n)( Pure left fold with strict accumulator. strict-containersO(n) Pure left fold (function applied to each element and its index). strict-containersO(n) Pure left fold with strict accumulator (function applied to each element and its index). strict-containersO(n) Pure right fold. strict-containersO(n)) Pure right fold with strict accumulator. strict-containersO(n) Pure right fold (function applied to each element and its index). strict-containersO(n) Pure right fold with strict accumulator (function applied to each element and its index). strict-containersO(n) Monadic fold. strict-containersO(n)& Monadic fold with strict accumulator. strict-containersO(n)= Monadic fold (action applied to each element and its index). strict-containersO(n) Monadic fold with strict accumulator (action applied to each element and its index). strict-containersO(n) Monadic right fold. strict-containersO(n), Monadic right fold with strict accumulator. strict-containersO(n) Monadic right fold (action applied to each element and its index). strict-containersO(n) Monadic right fold with strict accumulator (action applied to each element and its index). strict-containersO(n)8 Make a copy of a mutable array to a new mutable vector. strict-containersO(n): Make a copy of a mutable vector into a new mutable array.strict-containersi starting indexstrict-containersn lengthstrict-containersstarting indexstrict-containerslength of the slicestrict-containerstargetstrict-containerssourcestrict-containerstargetstrict-containerssourcestrict-containerstargetstrict-containerssourcestrict-containerstargetstrict-containerssource (c) Roman Leshchinskiy 2008-2010 BSD-style'Roman Leshchinskiy  experimental non-portableNone 3>strict-containers,Boxed vectors, supporting efficient slicing.strict-containersO(1) Yield the length of the vectorstrict-containersO(1) Test whether a vector is emptystrict-containers O(1) Indexingstrict-containersO(1) Safe indexingstrict-containersO(1) First elementstrict-containersO(1) Last elementstrict-containersO(1)( Unsafe indexing without bounds checkingstrict-containersO(1)6 First element without checking if the vector is emptystrict-containersO(1)5 Last element without checking if the vector is emptystrict-containersO(1) Indexing in a monad.The monad allows operations to be strict in the vector when necessary. Suppose vector copying is implemented like this: © mv v = ... write mv i (v ! i) ...For lazy vectors, v ! i) would not be evaluated which means that mv, would unnecessarily retain a reference to v in each element written.With /, copying can be implemented like this instead: copy mv v = ... do x <- indexM v i write mv i xHere, no references to v$ are retained because indexing (but not% the elements) is evaluated eagerly.strict-containersO(1)+ First element of a vector in a monad. See + for an explanation of why this is useful.strict-containersO(1)* Last element of a vector in a monad. See + for an explanation of why this is useful.strict-containersO(1)0 Indexing in a monad without bounds checks. See + for an explanation of why this is useful.strict-containersO(1) First element in a monad without checking for empty vectors. See * for an explanation of why this is useful.strict-containersO(1) Last element in a monad without checking for empty vectors. See * for an explanation of why this is useful.strict-containersO(1) Yield a slice of the vector without copying it. The vector must contain at least i+n elements.strict-containersO(1) Yield all but the last element without copying. The vector may not be empty.strict-containersO(1) Yield all but the first element without copying. The vector may not be empty.strict-containersO(1) Yield at the first n= elements without copying. The vector may contain less than n1 elements in which case it is returned unchanged.strict-containersO(1) Yield all but the first n= elements without copying. The vector may contain less than n4 elements in which case an empty vector is returned.strict-containersO(1) Yield the first n4 elements paired with the remainder without copying. Note that  n v is equivalent to ( n v,  n v) but slightly more efficient. strict-containersO(1) Yield the  and  of the vector, or   if empty. strict-containersO(1) Yield the  and  of the vector, or   if empty.strict-containersO(1) Yield a slice of the vector without copying. The vector must contain at least i+n" elements but this is not checked.strict-containersO(1) Yield all but the last element without copying. The vector may not be empty but this is not checked.strict-containersO(1) Yield all but the first element without copying. The vector may not be empty but this is not checked.strict-containersO(1) Yield the first n= elements without copying. The vector must contain at least n" elements but this is not checked.strict-containersO(1) Yield all but the first n= elements without copying. The vector must contain at least n" elements but this is not checked.strict-containersO(1) Empty vectorstrict-containersO(1) Vector with exactly one elementstrict-containersO(n) Vector of the given length with the same value in each positionstrict-containersO(n) Construct a vector of the given length by applying the function to each indexstrict-containersO(n) Apply function \max(n - 1, 0): times to an initial value, producing a vector of length  \max(n, 0). Zeroth element will contain the initial value, that's why there is one less function application than the number of elements in the produced vector. \underbrace{x, f (x), f (f (x)), \ldots}_{\max(0,n)\rm{~elements}} Examples0import qualified Data.Strict.Vector.Autogen as V3V.iterateN 0 undefined undefined :: V.Vector String[] V.iterateN 4 (\x -> x <> x) "Hi"+["Hi","HiHi","HiHiHiHi","HiHiHiHiHiHiHiHi"]strict-containersO(n) Construct a vector by repeatedly applying the generator function to a seed. The generator function yields  ' the next element and the new seed or   if there are no more elements. unfoldr (\n -> if n == 0 then Nothing else Just (n,n-1)) 10 = <10,9,8,7,6,5,4,3,2,1>strict-containersO(n)! Construct a vector with at most n elements by repeatedly applying the generator function to a seed. The generator function yields  ' the next element and the new seed or   if there are no more elements. -unfoldrN 3 (\n -> Just (n,n-1)) 10 = <10,9,8> strict-containersO(n)! Construct a vector with exactly n elements by repeatedly applying the generator function to a seed. The generator function yields the next element and the new seed. -unfoldrExactN 3 (\n -> (n,n-1)) 10 = <10,9,8>strict-containersO(n) Construct a vector by repeatedly applying the monadic generator function to a seed. The generator function yields  ' the next element and the new seed or   if there are no more elements.strict-containersO(n) Construct a vector by repeatedly applying the monadic generator function to a seed. The generator function yields  ' the next element and the new seed or   if there are no more elements. strict-containersO(n)! Construct a vector with exactly n elements by repeatedly applying the monadic generator function to a seed. The generator function yields the next element and the new seed.strict-containersO(n) Construct a vector with n elements by repeatedly applying the generator function to the already constructed part of the vector. constructN 3 f = let a = f <> ; b = f ; c = f in strict-containersO(n) Construct a vector with n elements from right to left by repeatedly applying the generator function to the already constructed part of the vector. constructrN 3 f = let a = f <> ; b = f ; c = f in strict-containersO(n): Yield a vector of the given length containing the values x, x+15 etc. This operation is usually more efficient than . enumFromN 5 3 = <5,6,7>strict-containersO(n): Yield a vector of the given length containing the values x, x+y, x+y+y5 etc. This operations is usually more efficient than . +enumFromStepN 1 0.1 5 = <1,1.1,1.2,1.3,1.4>strict-containersO(n) Enumerate values from x to y.WARNING: This operation can be very inefficient. If at all possible, use  instead.strict-containersO(n) Enumerate values from x to y with a specific step z.WARNING: This operation can be very inefficient. If at all possible, use  instead.strict-containersO(n) Prepend an elementstrict-containersO(n) Append an elementstrict-containersO(m+n) Concatenate two vectorsstrict-containersO(n)$ Concatenate all vectors in the liststrict-containersO(n) Execute the monadic action the given number of times and store the results in a vector.strict-containersO(n) Construct a vector of the given length by applying the monadic action to each index strict-containersO(n) Apply monadic function \max(n - 1, 0): times to an initial value, producing a vector of length  \max(n, 0). Zeroth element will contain the initial value, that's why there is one less function application than the number of elements in the produced vector.For non-monadic version see strict-containers;Execute the monadic action and freeze the resulting vector. create (do { v <- new 2; write v 0 'a'; write v 1 'b'; return v }) = <a,b> strict-containers)Here, the slice retains a reference to the huge vector. Forcing it creates a copy of just the elements that belong to the slice and allows the huge vector to be garbage collected.strict-containersO(m+n) For each pair (i,a)8 from the list, replace the vector element at position i by a. ,<5,9,2,7> // [(2,1),(0,3),(2,8)] = <3,9,8,7>strict-containersO(m+n) For each pair (i,a) from the vector of index/value pairs, replace the vector element at position i by a. 0update <5,9,2,7> <(2,1),(0,3),(2,8)> = <3,9,8,7>strict-containersO(m+min(n1,n2)) For each index i4 from the index vector and the corresponding value a from the value vector, replace the element of the initial vector at position i by a. .update_ <5,9,2,7> <2,0,2> <1,3,8> = <3,9,8,7> The function  provides the same functionality and is usually more convenient. update_ xs is ys =  xs ( is ys) strict-containers Same as () but without bounds checking.strict-containersSame as  but without bounds checking.strict-containersSame as  but without bounds checking.strict-containersO(m+n) For each pair (i,b), from the list, replace the vector element a at position i by f a b.Examples0import qualified Data.Strict.Vector.Autogen as VV.accum (+) (V.fromList [1000.0,2000.0,3000.0]) [(2,4),(1,6),(0,3),(1,10)][1003.0,2016.0,3004.0]strict-containersO(m+n) For each pair (i,b)7 from the vector of pairs, replace the vector element a at position i by f a b.Examples0import qualified Data.Strict.Vector.Autogen as VV.accumulate (+) (V.fromList [1000.0,2000.0,3000.0]) (V.fromList [(2,4),(1,6),(0,3),(1,10)])[1003.0,2016.0,3004.0]strict-containersO(m+min(n1,n2)) For each index i4 from the index vector and the corresponding value b from the the value vector, replace the element of the initial vector at position i by f a b. ?accumulate_ (+) <5,9,2> <2,1,0,1> <4,6,3,7> = <5+3, 9+6+7, 2+4> The function  provides the same functionality and is usually more convenient. accumulate_ f as is bs =  f as ( is bs) strict-containersSame as  but without bounds checking.strict-containersSame as  but without bounds checking.strict-containersSame as  but without bounds checking.strict-containersO(n) Reverse a vectorstrict-containersO(n)5 Yield the vector obtained by replacing each element i of the index vector by xsi. This is equivalent to  (xs) is# but is often much more efficient. 3backpermute <0,3,2,3,1,0> = strict-containersSame as  but without bounds checking.strict-containersApply a destructive operation to a vector. The operation will be performed in place if it is safe to do so and will modify a copy of the vector otherwise. modify (\v -> write v 0 'x') ( 3 'a') = <'x','a','a'> strict-containersO(n)- Pair each element in a vector with its indexstrict-containersO(n) Map a function over a vectorstrict-containersO(n)< Apply a function to every element of a vector and its indexstrict-containers9Map a function over a vector and concatenate the results.strict-containersO(n) Apply the monadic action to all elements of the vector, yielding a vector of resultsstrict-containersO(n) Apply the monadic action to every element of a vector and its index, yielding a vector of resultsstrict-containersO(n) Apply the monadic action to all elements of a vector and ignore the resultsstrict-containersO(n) Apply the monadic action to every element of a vector and its index, ignoring the resultsstrict-containersO(n) Apply the monadic action to all elements of the vector, yielding a vector of results. Equivalent to flip .strict-containersO(n) Apply the monadic action to all elements of a vector and ignore the results. Equivalent to flip . strict-containersO(n) Apply the monadic action to all elements of the vector and their indices, yielding a vector of results. Equivalent to   . strict-containersO(n) Apply the monadic action to all elements of the vector and their indices and ignore the results. Equivalent to   .strict-containers O(min(m,n))) Zip two vectors with the given function.strict-containers*Zip three vectors with the given function.strict-containers O(min(m,n)) Zip two vectors with a function that also takes the elements' indices.strict-containers Monadic fold with strict accumulator that discards the result strict-containersO(n) Monadic fold with strict accumulator that discards the result (action applied to each element and its index) strict-containersO(n) Monadic fold over non-empty vectors with strict accumulator that discards the result strict-containers,Evaluate each action and collect the results strict-containers,Evaluate each action and discard the results strict-containersO(n) Prescan prescanl f z =  .   f z  Example: $prescanl (+) 0 <1,2,3,4> = <0,1,3,6> strict-containersO(n) Prescan with strict accumulator strict-containersO(n) Scan postscanl f z =  .   f z  Example: &postscanl (+) 0 <1,2,3,4> = <1,3,6,10> strict-containersO(n) Scan with strict accumulator strict-containersO(n) Haskell-style scan scanl f z = where y1 = z yi = f y(i-1) x(i-1) Example: $scanl (+) 0 <1,2,3,4> = <0,1,3,6,10> strict-containersO(n)+ Haskell-style scan with strict accumulator  strict-containersO(n)" Scan over a vector with its index  strict-containersO(n)- Scan over a vector (strictly) with its index strict-containersO(n) Scan over a non-empty vector scanl f = where y1 = x1 yi = f y(i-1) xi strict-containersO(n)7 Scan over a non-empty vector with a strict accumulator strict-containersO(n) Right-to-left prescan prescanr f z =  .   (flip f) z .  strict-containersO(n). Right-to-left prescan with strict accumulator strict-containersO(n) Right-to-left scan strict-containersO(n)+ Right-to-left scan with strict accumulator strict-containersO(n)! Right-to-left Haskell-style scan strict-containersO(n)9 Right-to-left Haskell-style scan with strict accumulator  strict-containersO(n)0 Right-to-left scan over a vector with its index  strict-containersO(n); Right-to-left scan over a vector (strictly) with its index strict-containersO(n)+ Right-to-left scan over a non-empty vector strict-containersO(n) Right-to-left scan over a non-empty vector with a strict accumulator  strict-containersO(n) Check if two vectors are equal using supplied equality predicate.  strict-containersO(n) Compare two vectors using supplied comparison function for vector elements. Comparison works same as for lists. cmpBy compare == compare strict-containersO(n) Convert a vector to a list strict-containersO(n) Convert a list to a vector strict-containersO(n) Convert the first n elements of a list to a vector fromListN n xs =   ( n xs)  strict-containersO(1) Convert an array to a vector.  strict-containersO(n) Convert a vector to an array. strict-containersO(1) Unsafe convert a mutable vector to an immutable one without copying. The mutable vector may not be used after this operation. strict-containersO(1) Unsafely convert an immutable vector to a mutable one without copying. The immutable vector may not be used after this operation. strict-containersO(n). Yield a mutable copy of the immutable vector. strict-containersO(n)/ Yield an immutable copy of the mutable vector. strict-containersO(n) Copy an immutable vector into a mutable one. The two vectors must have the same length. This is not checked. strict-containersO(n) Copy an immutable vector into a mutable one. 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