Proof using.
induction 1; eauto.
Qed.

(* Specifications: *)
Lemma IR_equiv_app_head :
forall l xs ys,
IR_equivalent xs ys ->
IR_equivalent (l ++ xs) (l ++ ys).
Lemma op_equiv_ack_op_lr :
forall xs ys,
op_equivalent xs ys ->
forall k,
acknowledged_op k xs ->
acknowledged_op k ys.