Definition activation_rank (p : cbv) : nat :=
  match p with
  | cbv_update _ _ _ _ (fapply f _) _ _ => rank f
  | _                                   => 0
  end.

Lemma PPO_first_activations i s p c1 t c2 v p' : let proof_tree := cbv_update i s p c1 t c2 v in PPO_prog -> wf proof_tree -> In p' (first_activations proof_tree) -> PPO (proj_left p') (proj_left proof_tree).