Proof using.
intros ir.
induction ir, good_trace using good_trace_ind; intros; simpl in *; intuition (auto; try congruence).
- subst. find_apply_hyp_hyp. intuition.
- subst. find_apply_hyp_hyp. intuition.
Qed.

Lemma acknowledge_all_ops_func_target_nil :
forall l,
(forall k, ~ In (O k) l) ->
acknowledge_all_ops_func l [] = [].