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

(* Specifications: *)
Lemma not_D_queue : forall (q0 : queue Message) (m : Message), m <> dec -> ~ D_queue (input Message m q0).
Lemma D_queue_is_alternate : forall q0 : queue Message, D_queue q0 -> alternate q0.
Lemma alt_first_out : forall q0 : queue Message, alternate q0 -> alternate (first_out Message q0).
Lemma D_first_out : forall q0 : queue Message, D_queue q0 -> D_queue (first_out Message q0).
Lemma D_collect : forall s1 s2 : Site, D_queue (b0 s0 owner) -> D_queue (Collect_message Message b0 s1 s2 s0 owner).
Lemma D_post_elsewhere : forall (s1 s2 : Site) (m : Message), s1 <> s0 \/ s2 <> owner -> D_queue (b0 s0 owner) -> D_queue (Post_message Message m b0 s1 s2 s0 owner).
Lemma D_post_dec : alternate (b0 s0 owner) -> D_queue (Post_message Message dec b0 s0 owner s0 owner).
Lemma alt_collect : forall s1 s2 : Site, alternate (b0 s0 owner) -> alternate (Collect_message Message b0 s1 s2 s0 owner).
Lemma alt_post_elsewhere : forall (s1 s2 : Site) (m : Message), s1 <> s0 \/ s2 <> owner -> alternate (b0 s0 owner) -> alternate (Post_message Message m b0 s1 s2 s0 owner).
Lemma alt_post_any : forall m : Message, (forall s : Site, m <> inc_dec s) -> alternate (b0 s0 owner) -> alternate (Post_message Message m b0 s0 owner s0 owner).
Lemma alt_post_inc : forall s1 : Site, alternate (b0 s0 owner) -> b0 s0 owner = empty Message \/ last Message (b0 s0 owner) = value Message dec -> alternate (Post_message Message (inc_dec s1) b0 s0 owner s0 owner).