Proof using.
unfold acknowledged_op.
simpl. intuition congruence.
Qed.
Hint Constructors op_equivalent.

Lemma IR_equiv_AAOF_I :
forall k xs ys target,
op_equivalent xs ys ->
IR_equivalent (acknowledge_all_ops_func xs target)
(acknowledge_all_ops_func ys target) ->
IR_equivalent (acknowledge_all_ops_func (I k :: xs) target)
(acknowledge_all_ops_func (I k :: ys) target).