[java] How to use @Nullable and @Nonnull annotations more effectively?

I agree that the annotations "don't propagate very far". However, I see the mistake on the programmer's side.

I understand the Nonnull annotation as documentation. The following method expresses that is requires (as a precondition) a non-null argument x.

    public void directPathToA(@Nonnull Integer x){
        x.toString(); // do stuff to x        
    }

The following code snippet then contains a bug. The method calls directPathToA() without enforcing that y is non-null (that is, it does not guarantee the precondition of the called method). One possibility is to add a Nonnull annotation as well to indirectPathToA() (propagating the precondition). Possibility two is to check for the nullity of y in indirectPathToA() and avoid the call to directPathToA() when y is null.

    public void indirectPathToA(Integer y){
        directPathToA(y);
    }

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How to use @Nullable and @Nonnull annotations more effectively?