# (a -> b -> c) -> f a -> f b -> f c

Promote a function to a monad, scanning the monadic arguments from left to right. For example,
> liftM2 (+) [0,1] [0,2] = [0,2,1,3]
> liftM2 (+) (Just 1) Nothing = Nothing

Lift a binary function to actions.

*O(n+m)*. The intersection with a combining function.
> intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"

*O(min(n1,n2))*. zipWith generalizes zip 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.
zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function. For example, zipWith (+) is applied to two lists to produce the list of corresponding sums.
*O(n+m)*. The union with a combining function.
> unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]

scanl is similar to foldl, 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, ...]
scanl is similar to foldl, but returns a list of successive reduced values from the left:
> scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]
Note that
> last (scanl f z xs) == foldl f z xs.
scanr is the right-to-left dual of scanl. Note that
> head (scanr f z xs) == foldr f z xs.
*O(n+m)*. Is this a proper submap? (ie. a submap but not equal). The expression (isProperSubmapOfBy f m1 m2) returns True when m1 and m2 are not equal, all keys in m1 are in m2, and when f returns True when applied to their respective values. For example, the following expressions are all True:
> isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
But the following are all False:
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
> isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
> isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
*O(n+m)*. The expression (isSubmapOfBy f m1 m2) returns True if all keys in m1 are in m2, and when f returns True when applied to their respective values. For example, the following expressions are all True:
> isSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])
But the following are all False:
> isSubmapOfBy (==) (fromList [(1,2)]) (fromList [(1,1),(2,2)])
> isSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])
> isSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])
flip f takes its (first) two arguments in the reverse order of f.
Fold over the elements of a structure, associating to the left, but strictly.

The deleteFirstsBy function takes a predicate and two lists and returns the first list with the first occurrence of each element of the second list removed.
The unionBy function is the non-overloaded version of union.
*O(n)*. Fold the values in the map using the given left-associative binary operator, such that foldl f z == foldl f z . elems.
For example,
> elems = reverse . foldl (flip (:)) []
> let f len a = len + (length a)
> foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4
*O(n)*. A strict version of foldl. Each application of the operator is evaluated before using the result in the next application. This function is strict in the starting value.
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