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

Lemma first_activations_at_given_rank_rank rk p p' : In p' (first_activations_at_given_rank rk p) -> rk = activation_rank rank p'.