YES # parallel critical pair closing system (Shintani and Hirokawa 2022) Consider the left-linear TRS R: p(x) -> q(x) p(x) -> r(x) q(x) -> s(p(x)) r(x) -> s(p(x)) s(x) -> f(p(x)) Let C be the following subset of R: q(x) -> s(p(x)) r(x) -> s(p(x)) The TRS R is left-linear and all parallel critical pairs are joinable by C. Therefore, the confluence of R follows from that of C. # parallel critical pair closing system (Shintani and Hirokawa 2022) Consider the left-linear TRS R: q(x) -> s(p(x)) r(x) -> s(p(x)) Let C be the following subset of R: (empty) The TRS R is left-linear and all parallel critical pairs are joinable by C. Therefore, the confluence of R follows from that of C. # emptiness The empty TRS is confluent.