I am studying modified gravity theories, these days, and I have learned from many papers that, among various modified gravity theories, f(R) theory can avoid the Ostrogradski instability. Most of these papers cited Woodard's paper in 2007 (Lect. Notes. Phys. 720, 403 (2007)), which just gives a brief explanation of the reason. Is there any rigorous explanation of this issue in a mathematical way? Thx a lot!