Lemma parent_invariant_inductive : forall (c : Config) (t : class_trans c), legal c -> (legal c -> forall s : Site, indirect_son1 c s -> exists s1 : Site, direct_son c s1 /\ ancestor c s1 s) -> legal (transition c t) -> forall s : Site, indirect_son1 (transition c t) s -> exists s1 : Site, direct_son (transition c t) s1 /\ ancestor (transition c t) s1 s.