Definition Above x s := forall y, In y s -> E.lt y x.Definition Below x s := forall y, In y s -> E.lt x y.

Lemma set_induction_max : forall P : t -> Type, (forall s : t, Empty s -> P s) -> (forall s s', P s -> forall x, Above x s -> Add x s s' -> P s') -> forall s : t, P s.