Data.Zip

join align ≡ fmap (join These)

align x y ≡ swap <$> align y x

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

alignWith f a b ≡ f <$> align a b

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

toList x ≡ toListOf (folded . here) (align x y)
         ≡ mapMaybe justHere (toList (align x y))

join zip   ≡ fmap (join (,))

zip x y ≡ swap <$> zip y x

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

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

zipWith f a b ≡ f <$> zip a b

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

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

                   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)

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

fst <$> zip xs (repeat y) ≡ xs
snd <$> zip (repeat x) ys ≡ ys

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

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