**map** +containers

*O(n*min(n,W))*. map f s is the set obtained by applying f to each element of s.
It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y
*O(n)*. Map a function over all values in the map.
> map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

*O(n)*. Map a function over all values in the map.
> map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]

*O(n*log n)*. map f s is the set obtained by applying f to each element of s.
It's worth noting that the size of the result may be smaller if, for some (x,y), x /= y && f x == f y
*O(n)*. The function mapAccum threads an accumulating argument through the map in ascending order of keys.
> let f a b = (a ++ b, b ++ "X")
> mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccum threads an accumulating argument through the map in ascending order of keys.
> let f a b = (a ++ b, b ++ "X")
> mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccumR threads an accumulating argument through the map in descending order of keys.

*O(n)*. The function mapAccumR threads an accumulating argument through the map in descending order of keys.

*O(n)*. The function mapAccumWithKey threads an accumulating argument through the map in ascending order of keys.
> let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
> mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
*O(n)*. The function mapAccumWithKey threads an accumulating argument through the map in ascending order of keys.
> let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")
> mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])
*O(n)*. Map values and separate the Left and Right results.
> let f a = if a < "c" then Left a else Right a
> mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
>
> mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
*O(n)*. Map values and separate the Left and Right results.
> let f a = if a < "c" then Left a else Right a
> mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])
>
> mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
*O(n)*. Map keys/values and separate the Left and Right results.
> let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
> mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
>
> mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
*O(n)*. Map keys/values and separate the Left and Right results.
> let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)
> mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])
>
> mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])
> == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])
*O(n*min(n,W))*. mapKeys f s is the map obtained by applying f to each key of s.
The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the greatest of the original keys is retained.
> mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]
> mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
> mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
*O(n*log n)*. mapKeys f s is the map obtained by applying f to each key of s.
The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the value at the greatest of the original keys is retained.
> mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]
> mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"
> mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"
*O(n*min(n,W))*. mapKeysMonotonic f s == mapKeys f s, but works only when f is strictly monotonic. That is, for any values x and y, if x < y then f x < f y. *The precondition is not checked.* Semi-formally, we have:
> and [x < y ==> f x < f y | x <- ls, y <- ls]
> ==> mapKeysMonotonic f s == mapKeys f s
>
This means that f maps distinct original keys to distinct resulting keys. This function has slightly better performance than mapKeys.
> mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
*O(n)*. mapKeysMonotonic f s == mapKeys f s, but works only when f is strictly monotonic. That is, for any values x and y, if x < y then f x < f y. *The precondition is not checked.* Semi-formally, we have:
> and [x < y ==> f x < f y | x <- ls, y <- ls]
> ==> mapKeysMonotonic f s == mapKeys f s
>
This means that f maps distinct original keys to distinct resulting keys. This function has better performance than mapKeys.
> mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]
> valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True
> valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False
*O(n*log n)*. mapKeysWith c f s is the map obtained by applying f to each key of s.
The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c.
> mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
> mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
*O(n*min(n,W))*. mapKeysWith c f s is the map obtained by applying f to each key of s.
The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case the associated values will be combined using c.
> mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"
> mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"
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