Proving Composition Law for Maybe Applicative

Question

So, I wanted to manually prove the Composition law for Maybe applicative which is:

u <*> (v <*> w) = pure (.) <*> u <*> v <*> w         


        

Answer 1

The rules that are generated by the definition of Maybe are

x :: a
---------------
Just x :: Maybe a

and

a type
-----------------
Nothing :: Maybe a

Along with

a type
------------------
bottom :: a

If these are the only rules which result in Maybe A then we can always invert them (run from bottom to top) in proofs so long as we're exhaustive. This is argument by case examination of a value of type Maybe A.

You did two cases analyses, but weren't exhaustive. It might be that u or v are actually Nothing or bottom.