Definition equivalent (l : list op) (ir : list IR) : Prop :=
    good_trace ir /\
    exists ir',
      acknowledge_all_ops l ir' /\
      IR_equivalent ir' ir.

Lemma acknowledge_all_ops_func_IRU_In : forall l ir k, In (IRU k) (acknowledge_all_ops_func l ir) -> In (I k) l.