Corollary SsafeN_ksafe: forall n st,
      SsafeN n (@P_init st) -> forall U, valid st U -> ksafe st U n.

Lemma finite_Ssafe_safe: (forall P : Prop, P \/ ~ P) -> (forall P : ST -> Prop, finite P -> finite (SST_step P)) -> forall st, ( forall n : nat, safeN (ST -> Prop) SST_step n (@P_init st)) -> Ssafe (@P_init st).