Definition safe s k := forall s2 k2,
  star s k s2 k2 -> exists s', exists k', step s2 k2 s' k'.Definition safeN s k := forall s2 k2 n,
  stepN n s k s2 k2 -> exists s', exists k', step s2 k2 s' k'.

(* Specifications: *)
Lemma safe_safeN s k : safe s k <-> safeN s k.
Lemma safety_induction R s k : (forall s k, R s k -> exists s', exists k', step s k s' k') -> (forall s k s' k', R s k -> step s k s' k' -> R s' k') -> R s k -> safe s k.