YES # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: b(w(x)) -> w(w(w(b(x)))) w(b(x)) -> b(x) b(b(x)) -> w(w(w(w(x)))) w(w(x)) -> w(x) Let C be the following subset of R: (empty) The parallel critical pair system PCPS(R,C) is: b(w(b(x1_1))) -> b(b(x1_1)) b(w(b(x1_1))) -> w(w(w(b(b(x1_1))))) b(w(w(x1_1))) -> b(w(x1_1)) b(w(w(x1_1))) -> w(w(w(b(w(x1_1))))) w(b(w(x1_1))) -> w(w(w(w(b(x1_1))))) w(b(w(x1_1))) -> b(w(x1_1)) w(b(b(x1_1))) -> w(w(w(w(w(x1_1))))) w(b(b(x1_1))) -> b(b(x1_1)) b(b(w(x1_1))) -> b(w(w(w(b(x1_1))))) b(b(w(x1_1))) -> w(w(w(w(w(x1_1))))) b(b(b(x1_1))) -> b(w(w(w(w(x1_1))))) b(b(b(x1_1))) -> w(w(w(w(b(x1_1))))) All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating. Therefore, the confluence of R follows from that of C. # emptiness The empty TRS is confluent.