Suppose you have something like:

data Tree a = Leaf a | Branch (Tree a) (Tree a)

instance Functor Tree where

  fmap f (Leaf a) = Leaf $ f a

  fmap f (Branch l r) = Branch (fmap f l) (fmap f r)

To check that fmap id = id, I can see that the case for Leaf is ok:

  fmap id (Leaf a) = Leaf $ id a = Leaf a

How would you prove it for the second case? Is some sort of inductive proof necessary? I found this: http://ssomayyajula.github.io/posts/2015-11-07-proofs-of-functor-laws-with-Haskell.html, which goes over an inductive (on the length of the list) proof for the list type, but I'm not sure how to do it for a Tree.

Thanks,

toz