Don't need to assume anything. He just read the liner credits on the albums lol.
Blizzard of Ozz literally says "All songs written by Ozzy Osbourne, Randy Rhoads and Bob Daisley, except where noted.". Same with Diary of a Madman. "All songs by Ozzy Osbourne, Randy Rhoads, Bob Daisley and Lee Kerslake, except where noted."
Bark at the Moon is where the controversy starts, where Ozzy is the sole writing credit, but Daisley and Jake Lee claim they wrote a good chunk of it and Sharon screwed them out of credits.
The Ultimate Sin lists Jake Lee with credits again. No Rest for the Wicked has everyone with writing credits. Same with Just Say Ozzy.
No More Tears explicitly credits everyone. "All tracks are written by Ozzy Osbourne, Zakk Wylde and Randy Castillo, except where noted." And Lemmy is credited on I Don't Want to Change the World, Mama I'm Coming Home, Desire, and Hellraiser.
Bark at the Moon is the only controversy. Every other album has credit given to other writers. And all of Black Sabbath's are credited to all four of them. It has been long known though that Geezer was the primary lyricist, and Ozzy only contributed some original lyrics, but did a lot more with the actual arrangement of the songs.