Fixpoint activations (proof_tree : cbv) : list cbv :=
  match proof_tree with
  | cbv_constr lp _ _ _ _ => flat_map activations lp
  | cbv_split lp p _ _ _ _ => activations p ++ flat_map activations lp
  | cbv_update _ _ p _ _ _ _ as p' => p' :: activations p
  | cbv_read _ _ _ => []
  end.Fixpoint activations_cache_order (proof_tree : cbv) : list cbv :=
  match proof_tree with
  | cbv_constr lp _ _ _ _ => revflatten (map activations_cache_order lp)
  | cbv_split lp p _ _ _ _ => activations_cache_order p ++ revflatten (map activations_cache_order lp)
  | cbv_update _ _ p _ _ _ _ as p' => p' :: activations_cache_order p
  | cbv_read _ _ _ => []
  end.

Lemma activations_are_subtrees: forall p pi:cbv, In p (activations pi) -> In p (sub_trees pi).