Lucidity might not be required. I once solved a math problem in a non-lucid dream, and after waking up I could confirm the solution was correct.
I've had enough lucid dreams when I was kid to know the difference. These days lucid dreaming for me is rare and not deliberate--i.e. if I have the presence of mind to wake up in a dream, it's mostly by chance, not because I orchestrated it through habituation. It came mostly naturally as a kid, after I stumbled upon a reproducible technique. Adults usually have to practice at it, and I'm just too lazy (and too tired). The vast majority of my dreams now are non-lucid, or quasi-lucid (i.e. I'm not in deep sleep) such that I know it's not like a proper lucid dream. The experiences are quite different.