Definition contained (l: list ident) (g1 g2: generator) : Prop :=
  forall id, In id l -> within id g1 g2.

(* Specifications: *)
Lemma contained_nil: forall g1 g2, contained nil g1 g2.
Lemma contained_widen: forall l g1 g2 g1' g2', contained l g1 g2 -> Ple (gen_next g1') (gen_next g1) -> Ple (gen_next g2) (gen_next g2') -> contained l g1' g2'.
Lemma contained_cons: forall id l g1 g2, within id g1 g2 -> contained l g1 g2 -> contained (id :: l) g1 g2.
Lemma contained_app: forall l1 l2 g1 g2, contained l1 g1 g2 -> contained l2 g1 g2 -> contained (l1 ++ l2) g1 g2.
Lemma contained_disjoint: forall g1 l1 g2 l2 g3, contained l1 g1 g2 -> contained l2 g2 g3 -> list_disjoint l1 l2.
Lemma contained_notin: forall g1 l g2 id g3, contained l g1 g2 -> within id g2 g3 -> ~In id l.