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 parent_does_not_join : forall (c0 : Config) (s2 s3 s4 : Site), legal c0 -> legal (rec_copy2_trans c0 s2) -> parent (rec_copy2_trans c0 s2) s3 s4 -> rt c0 s2 = false -> first Message (bm c0 owner s2) = value Message copy -> s3 <> s2.