(* ---------------------------------------------------------------------------- *)

(* This internal tactic helps prove goals by examination of all cases. *)

Ltac just_do_it :=
  unfold subst, Subst_idx, subst_idx, lift, Lift_idx, var, Var_idx;
  intros; solve [ dblib_by_cases; eauto with f_equal omega ].

Lemma lift_idx_old: forall w k x, k <= x -> lift w k x = w + x.