(** * Safe executions. *)

(** A state is safe according to the nondeterministic semantics
  if it cannot get stuck by doing silent transitions only. *)

Definition safe (s: Csem.state) : Prop :=
  forall s', star Csem.step ge s E0 s' ->
  (exists r, final_state s' r) \/ (exists t, exists s'', Csem.step ge s' t s'').

(* Specifications: *)
Lemma safe_steps: forall s s', safe s -> star Csem.step ge s E0 s' -> safe s'.
Lemma star_safe: forall s1 s2 t s3, safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> star Csem.step ge s2 t s3) -> star Csem.step ge s1 t s3.
Lemma plus_safe: forall s1 s2 t s3, safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> plus Csem.step ge s2 t s3) -> plus Csem.step ge s1 t s3.
Lemma safe_imm_safe: forall f C a k e m K, safe (ExprState f (C a) k e m) -> context K RV C -> imm_safe ge e K a m.
Lemma safe_steps: forall s s', safe s -> star Csem.step ge s E0 s' -> safe s'.
Lemma star_safe: forall s1 s2 t s3, safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> star Csem.step ge s2 t s3) -> star Csem.step ge s1 t s3.
Lemma plus_safe: forall s1 s2 t s3, safe s1 -> star Csem.step ge s1 E0 s2 -> (safe s2 -> plus Csem.step ge s2 t s3) -> plus Csem.step ge s1 t s3.
Lemma safe_imm_safe: forall f C a k e m K, safe (ExprState f (C a) k e m) -> context K RV C -> imm_safe ge e K a m.