We revisit completion modulo equational theories for left-linear term rewrite systems where unification modulo the theory is avoided and the normal rewrite relation can be used in order to decide validity questions. To that end, we give a new correctness proof for finite runs and establish a simulation result between the two inference systems known from the literature. Given a concrete reduction order, novel canonicity results show that the resulting complete systems are unique up to the representation of their rules' right-hand sides. Furthermore, we show how left-linear AC completion can be simulated by general AC completion. In particular, this result allows us to switch from the former to the latter at any point during a completion process.

@article{paperbot3421,
  title = {Left-Linear Completion with AC Axioms},
  author = {Johannes Niederhauser and Nao Hirokawa and Aart Middeldorp},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 2},
  year = {2025},
  doi = {10.46298/lmcs-21(2:10)2025}
}