The missing folds

Question

If you want to fold a list, I see four ways to do it.

Fold from the right of the list, with the recursive term on the right

foldrr (-) 100 [1..10] = 1 - (2 - (3 - (4 - (5 - (6 - (7 - (8 - (9 - (10 - (100)))))))))) = 95

foldrr :: (a -> b -> b) -> b -> [a] -> b
foldrr step zero (x:xs) = step x (foldrr step zero xs)
foldrr _    zero []     = zero

Fold from the right of the list, with the recursive term on the left

foldrl (-) 100 [1..10] = ((((((((((100) - 10) - 9) - 8) - 7) - 6) - 5) - 4) - 3) - 2) - 1 = 45

foldrl :: (a -> b -> a) -> a -> [b] -> a
foldrl step zero (x:xs) = step (foldrl step zero xs) x
foldrl _    zero []     = zero

Fold from the left of the list with the recursive term on the right

foldlr (-) 100 [1..10] = 10 - (9 - (8 - (7 - (6 - (5 - (4 - (3 - (2 - (1 - (100)))))))))) = 105

foldlr :: (a -> b -> b) -> b -> [a] -> b
foldlr step zero (x:xs) = foldlr step (step x zero) xs
foldlr _    zero []     = zero

Fold from the left of the list with the recursive term on the left

foldll (-) 100 [1..10] = ((((((((((100) - 1) - 2) - 3) - 4) - 5) - 6) - 7) - 8) - 9) - 10 = 45

foldll :: (a -> b -> a) -> a -> [b] -> a
foldll step zero (x:xs) = foldll step (step zero x) xs
foldll _    zero []     = zero

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Only two of these folds made it into Prelude as foldr and foldl. Was there any reason to just include two folds, and why those two?

Answer 1

foldrl and foldlr don't add any expressive power: they are just the same as the other two folds but with the folding function flipped.

foldrl f = foldr (flip f)
foldlr f = foldl (flip f)

-- Or this, if you prefer
foldrl = foldr . flip
foldlr = foldl . flip

But it is not so easy to define foldl in terms of foldr, so providing them both is useful.