Inductive direct_son : Site -> Prop :=
    direct_son_intro :
      forall s : Site,
      s <> owner ->
      rt c0 s = true ->
      (forall s1 : Site, ~ inc_dec_in_queue s1 (bm c0 s owner)) ->
      direct_son s.Inductive direct_son : Site -> Prop :=
    direct_son_intro :
      forall s : Site,
      s <> owner ->
      rt c0 s = true ->
      (forall s1 : Site, ~ inc_dec_in_queue s1 (bm c0 s owner)) ->
      direct_son s.Inductive indirect_son1 : Site -> Prop :=
    indirect_son1_intro :
      forall s : Site,
      s <> owner ->
      rt c0 s = true ->
      ~ (forall s1 : Site, ~ inc_dec_in_queue s1 (bm c0 s owner)) ->
      indirect_son1 s.Inductive indirect_son1 : Site -> Prop :=
    indirect_son1_intro :
      forall s : Site,
      s <> owner ->
      rt c0 s = true ->
      ~ (forall s1 : Site, ~ inc_dec_in_queue s1 (bm c0 s owner)) ->
      indirect_son1 s.(*  {s1:Site | (inc_dec_in_queue s1 (bm c0 s owner))}  *)

Inductive indirect_son2 : Site -> Prop :=
    indirect_son2_intro :
      forall s s1 : Site,
      s <> owner /\ rt c0 s = true /\ direct_son s1 /\ ancestor s1 s ->
      indirect_son2 s.(*  {s1:Site | (inc_dec_in_queue s1 (bm c0 s owner))}  *)

Inductive indirect_son2 : Site -> Prop :=
    indirect_son2_intro :
      forall s s1 : Site,
      s <> owner /\ rt c0 s = true /\ direct_son s1 /\ ancestor s1 s ->
      indirect_son2 s.

Lemma inc_dec_not_in_first_out : forall (q : queue Message) (m : Message) (s3 : Site), first Message q = value Message m -> m <> inc_dec s3 -> ~ inc_dec_in_queue s3 q -> ~ inc_dec_in_queue s3 (first_out Message q).