Actually, (foldl.foldr) f z xs === foldr f z (concat $ reverse xs)
.
Even if f
is an associative operation, the correct sequence of applications matters, as it can have an impact on performance.
We begin with
(foldl.foldr) f z xs
foldl (foldr f) z xs
writing with g = foldr f
and [x1,x2,...,xn_1,xn] = xs
for a moment, this is
(...((z `g` x1) `g` x2) ... `g` xn)
(`g` xn) ((`g` xn_1) ... ((`g` x1) z) ... )
foldr f z $ concat [xn,xn_1, ..., x1]
foldr f z $ concat $ reverse xs
So in your case the correct reduction sequence is
(foldl.foldr) 1 [[1,2,3],[4,5,6]]
4+(5+(6+( 1+(2+(3+ 1)))))
22
To wit,
Prelude> (foldl.foldr) (:) [] [[1..3],[4..6],[7..8]]
[7,8,4,5,6,1,2,3]
Similarly, (foldl.foldl) f z xs == foldl f z $ concat xs
. With snoc a b = a++[b]
,
Prelude> (foldl.foldl) snoc [] [[1..3],[4..6],[7..8]]
[1,2,3,4,5,6,7,8]
Also, (foldl.foldl.foldl) f z xs == (foldl.foldl) (foldl f) z xs == foldl (foldl f) z $ concat xs == (foldl.foldl) f z $ concat xs == foldl f z $ concat (concat xs)
, etc.:
Prelude> (foldl.foldl.foldl) snoc [] [[[1..3],[4..6]],[[7..8]]]
[1,2,3,4,5,6,7,8]
Prelude> (foldl.foldr.foldl) snoc [] [[[1..3],[4..6]],[[7..8]]]
[7,8,1,2,3,4,5,6]
Prelude> (foldl.foldl.foldr) (:) [] [[[1..3],[4..6]],[[7..8]]]
[7,8,4,5,6,1,2,3]