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.
(* by specifying the lemma as follows, I don't need to know that there is no cycle in the relation *)

Lemma aux :
forall s0 s1 s2 : Site,
parent s0 s2 -> direct_son s0 -> ancestor s1 s2 -> s1 = s0.