Definition lub (x y: t) : t :=
    match x, y with
    | Bot, _ => y
    | _, Bot => x
    | State ae1 am1, State ae2 am2 => State (AE.lub ae1 ae2) (mlub am1 am2)
    end.

(* Specifications: *)
Lemma ge_lub_left: forall x y, ge (lub x y) x.
Lemma ge_lub_right: forall x y, ge (lub x y) y.