Теоретическое доказательство:
Maybe:
1) pure id <*> v = v
   pure id <*> v -> Just id <*> v -> fmap id v -> либо Just (id x) = Just x = v, либо Nothing = Nothing

2)  pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
    Пусть u, v или w = Nothing. Тогда, и справа, и слева будет Nothing
    Если же нет:
    pure (.) <*> Just x <*> Just y <*> Just z | Just x <*> ((Just y) <*> (Just z))
    Just (.) <*> Just x <*> Just y <*> Just z | Just x <*> (Just (y z)) = Just (x (y z))
    Just ((.) x) <*> Just y <*> Just z        |
    Just (x . y) <*> Just z                   |
    Just ((x y (z)))                          |

3) pure g <*> pure x = pure (g x)
   Just g <*> Just x | Just (g x)
   Just (g x)

4) g <*> pure x = pure ($ x) <*> g
   Just gx <*> Just x | Just ($ x) <*> (Just gx)
   Just (gx x)        | Just (gx x)

List:
1) pure id <*> v = v
   [id] <*> [a, b, ..] -> [id x | x <- v] -> [id a, id b, ..] -> [a, b, ..]

2) pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
   [(.)] <*> u <*> v <*> w            | u <*> (v <*> w)
   [x . y | x <- u, y <- v] <*> w     | u <*> [y z | y <- v, z <- w]
   [x (y z) | x <- u, y <- v, z <- w] | [x (y z) | x <- u, y <- v, z <- w]

3) pure g <*> pure x = pure (g x)
   [g] <*> [x] = [g x]

4) g <*> pure x = pure ($ x) <*> g
   [..] <*> [x]     | [$ x] <*> g
   [gx x | gx <- g] | [($ x) gx | gx <- g] = [gx x | gx <- g]