Ltac inv_safe H :=
  inv H;
  try solve[match goal with
    | H : semantics.at_external _ _ _ = _ |- _ =>
      simpl in H; congruence
    | H : j_at_external _ _ _ = _ |- _ =>
      simpl in H; congruence
    | H : semantics.halted _ _ _ |- _ =>
      simpl in H; unfold cl_halted in H; contradiction
  end].

Lemma control_as_safe_refl psi n k : control_as_safe psi n k k.