Ltac induction2 :=
    simple induction s;
     [ simpl; auto; try solve [ intros; inv ]
     | intros x l Hrec; simple induction s';
        [ simpl; auto; try solve [ intros; inv ]
        | intros x' l' Hrec'; simpl; elim_compare x x'; intros; inv; auto ]].

Lemma union_inf : forall (s s' : t) (a : elt) (Hs : Ok s) (Hs' : Ok s'), Inf a s -> Inf a s' -> Inf a (union s s').