zip package:semialign

Combines two structures by taking the intersection of their shapes and using pair to hold the elements.

Functors supporting a zip operation that takes the intersection of non-uniform shapes. Minimal definition: either zip or zipWith. Idempotency

join zip   ≡ fmap (join (,))

Commutativity

zip x y ≡ swap <$> zip y x

Associativity

zip x (zip y z) ≡ assoc <$> zip (zip x y) z

Absorption

fst    <$> zip xs (align xs ys) ≡ xs
toThis <$> align xs (zip xs ys) ≡ This <$> xs
where
toThis (This a)    = This a
toThis (These a _) = This a
toThis (That b)    = That b

With

zipWith f a b ≡ f <$> zip a b

Functoriality

zip (f <$> x) (g <$> y) ≡ bimap f g <$> zip x y

Zippyness

fmap fst (zip x x) ≡ x
fmap snd (zip x x) ≡ x
zip (fmap fst x) (fmap snd x) ≡ x

Distributivity

align (zip xs ys) zs ≡ undistrThesePair <$> zip (align xs zs) (align ys zs)
distrPairThese <$> zip (align xs ys) zs ≡                      align (zip xs zs) (zip ys zs)
zip (align xs ys) zs ≡ undistrPairThese <$> align (zip xs zs) (zip ys zs)

Note, the following doesn't hold:

distrThesePair <$> align (zip xs ys) zs ≢ zip (align xs zs) (align ys zs)

when xs = [] and ys = zs = [0], then the left hand side is "only" [(That 0, That 0)], but the right hand side is [(That 0, These 0 0)].

Zipping and unzipping of functors with non-uniform shapes.

Combines two structures by taking the intersection of their shapes and combining the elements with the given function.

Indexed version of Zip.

Left-padded zip.

Left-padded zipWith.

Align two structures as in zip, but filling in blanks with Nothing.

Align two structures as in zipWith, but filling in blanks with Nothing.

Right-padded zip.

Right-padded zipWith.

Right inverse of zip. This class is definable for every Functor. See unzipDefault.

Laws

uncurry zip (unzip xs) ≡ xs
unzip (zip xs xs) ≡ (xs, xs)

Note:

unzip (zip xs ys) ≢ (xs, _) or (_, ys)

For sequence-like types this holds, but for Map-like it doesn't.

Analogous to zipWith, but also provides an index.