Definition indprinciple_3_aux (n : nat) :=
  forall P : term -> Prop,
  (forall (a : ad) (tl : term_list),
   (forall u : term, term_list_occur u tl -> P u) -> P (app a tl)) ->
  forall t : term, term_high t <= n -> P t.

Lemma indprinciple_term : forall P : term -> Prop, (forall (a : ad) (tl : term_list), (forall u : term, term_list_occur u tl -> P u) -> P (app a tl)) -> forall t : term, P t.