Definition bad_seq_in_lbl (t : Tree) : Prop := 
forall vs l ts, t = node (vs,l) ts -> bad embeds vs.

(* Specifications: *)
Lemma bad_seqs_in_forest : forall ws, bad embeds ws -> forall f, is_forest ws f -> forall t, tree_in_forest t f -> bad_seq_in_lbl t.