Inductive D_queue : queue Message -> Prop :=
  | D_empty : D_queue (empty Message)
  | D_dec :
      forall qm : queue Message,
      alternate qm -> In_queue_before Message dec is_inc_dec qm -> D_queue qm.Inductive D_queue : queue Message -> Prop :=
  | D_empty : D_queue (empty Message)
  | D_dec :
      forall qm : queue Message,
      alternate qm -> In_queue_before Message dec is_inc_dec qm -> D_queue qm.Inductive D_queue : queue Message -> Prop :=
  | D_empty : D_queue (empty Message)
  | D_dec :
      forall qm : queue Message,
      alternate qm -> In_queue_before Message dec is_inc_dec qm -> D_queue qm.

Lemma alt_first_out : forall q0 : queue Message, alternate q0 -> alternate (first_out Message q0).