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.

Lemma lookup_insert_old: forall A x y (o : option A) e, x > y -> lookup x (raw_insert y o e) = lookup (x - 1) e.