Proof of the functor laws for the Maybe functor

fmap :: (a -> b) -> Maybe a -> Maybe b
fmap f (Just x) = Just (f x)
fmap f Nothing  = Nothing

Prove:
fmap id = id

Case 1: Just x

fmap id (Just x)
= Just (id x)
= Just x

Case 2: Nothing

fmap id Nothing
= Nothing


Prove:
fmap (g . f) = fmap g . fmap f

Case 1: Just a
fmap (g . f) (Just a)
= Just (g . f $ a)
= Just (g(f a))


(fmap g . fmap f) (Just a)
= fmap g (fmap f (Just a))
= fmap g (Just (f a))
= Just (g(f a))


Case 2: Nothing
(fmap g . fmap f) Nothing
= fmap g (fmap f Nothing)
= fmap g Nothing
= Nothing

fmap (g . f) Nothing
= Nothing