(* s1 is parent of s0 *)
Inductive mul_parent : Site -> Site -> Prop :=
    mul_parent_intro :
      forall s1 s0 : Site,
      rt c0 s0 = true ->
      In_queue Message (inc_dec s1) (bm c0 s0 owner) -> mul_parent s1 s0.(* s1 is parent of s0 *)
Inductive mul_parent : Site -> Site -> Prop :=
    mul_parent_intro :
      forall s1 s0 : Site,
      rt c0 s0 = true ->
      In_queue Message (inc_dec s1) (bm c0 s0 owner) -> mul_parent s1 s0.(* s1 is ancestor of s0 *)
Inductive mul_ancestor : Site -> Site -> Prop :=
  | mul_short : forall s1 s0 : Site, mul_parent s1 s0 -> mul_ancestor s1 s0
  | mul_long :
      forall s2 s1 s0 : Site,
      mul_ancestor s2 s1 -> mul_parent s1 s0 -> mul_ancestor s2 s0.(* s1 is ancestor of s0 *)
Inductive mul_ancestor : Site -> Site -> Prop :=
  | mul_short : forall s1 s0 : Site, mul_parent s1 s0 -> mul_ancestor s1 s0
  | mul_long :
      forall s2 s1 s0 : Site,
      mul_ancestor s2 s1 -> mul_parent s1 s0 -> mul_ancestor s2 s0.(* s1 is ancestor of s0 *)
Inductive mul_ancestor : Site -> Site -> Prop :=
  | mul_short : forall s1 s0 : Site, mul_parent s1 s0 -> mul_ancestor s1 s0
  | mul_long :
      forall s2 s1 s0 : Site,
      mul_ancestor s2 s1 -> mul_parent s1 s0 -> mul_ancestor s2 s0.

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