foldr and foldl on ∫ A : A^{op} ⊗ A → Set

composition.hs

-- post-composition 

foldl_foldr f z l = foldr (.>) id (foldr (\x -> (flip f x :)) [] l) z

-- f x .> (f y .> (f z .> id)) rs = f z (f y (f x rs)) -- .> quite not lazy

-- pre-composition = mieux que remplacement des ctors

foldr_foldr f z l = foldr (.) id (foldr (\x -> (f x :)) [] l) z

foldl_foldl f z l = foldl (.) id (foldl (\rl x -> flip f x : rl) [] l) z

foldr_foldl f z l = foldl (.>) id (foldl (\rl x -> f x : rl) [] l) z

(.>) :: (a -> b) -> (b -> c) -> (a -> c)
f .> g = g . f

-- >>> foldl (+) 0 [1, 2, 3]
-- >>> foldr (+) 0 [1, 2, 3]
-- >>> foldl_foldr (+) 0 [1, 2, 3]
-- >>> foldr_foldr (+) 0 [1, 2, 3]
-- >>> foldl_foldl (+) 0 [1, 2, 3]
-- >>> foldr_foldl (+) 0 [1, 2, 3]
-- 6
-- 6
-- 6
-- 6
-- 6
-- 6

-- >>>  foldl (flip (:)) [] [1, 2, 3]
-- >>>  foldl_foldr (flip (:)) [] [1, 2, 3]
-- >>>  foldl_foldl (flip (:)) [] [1, 2, 3]
-- [3,2,1]
-- [3,2,1]
-- [3,2,1]

-- >>> foldr (:) [] [1, 2, 3]
-- >>> foldr_foldr (:) [] [1, 2, 3]
-- >>> foldr_foldl (:) [] [1, 2, 3]
-- [1,2,3]
-- [1,2,3]
-- [1,2,3]

-- >>> take 5 (map (+1) [1..])
-- >>> take 5 (foldr ((:) . (+1)) [] [1..])
-- >>> take 5 (foldr_foldr ((:) . (+1)) [] [1..])
-- [2,3,4,5,6]
-- [2,3,4,5,6]
-- [2,3,4,5,6]

-- >>> take 5 (foldr_foldl ((:) . (+1)) [] [1..]) -- might take some time
-- ProgressCancelledException

-- >>> take 5 (foldl_foldr (flip ((:) . (+ 1))) [] [1 ..]) -- better but still slow
-- ProgressCancelledException