Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion; subst.
- find_apply_lem_hyp null_sound. find_rewrite. simpl.
intuition. left. eexists. intuition.
- simpl. find_apply_lem_hyp null_false_neq_nil.
intuition. left. eexists. intuition.
- simpl. intuition (auto; congruence).
- simpl. destruct st; simpl in *; subst; intuition (auto; congruence).
- simpl in *. intuition.
- simpl in *. intuition eauto.
- simpl. intuition (auto; congruence).
Qed.
Definition at_head_of_queue sigma c := (exists t, queue (sigma Server) = c :: t).

Lemma at_head_of_queue_intro :
forall sigma c t,
queue (sigma Server) = c :: t ->
at_head_of_queue sigma c.