Ltac one_plus_x_minus_one :=
repeat rewrite one_plus_x_minus_one_left in *;
repeat rewrite one_plus_x_minus_one_right in * by omega.
(* I tried [autorewrite with ... using omega]; it does not work. *)
(* ---------------------------------------------------------------------------- *)
(* Trivial facts. *)

Lemma empty_eq_insert:
forall A x o (e : env A),
empty _ = insert x o e ->
False.