Definition activation_function (function_default : function) (p: cbv) :=
  match p with
    | cbv_update _ _ _ _ (fapply f _) _ _ => f
    | _                                   => function_default
  end.

Lemma activations_at_rank_activation_rank proof_tree p: In p (activations proof_tree) -> In p (activations_at_rank (activation_rank rank p) proof_tree).