YES # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: b(b(c(a(b(c(x)))))) -> a(b(b(c(b(c(a(x))))))) Let C be the following subset of R: b(b(c(a(b(c(x)))))) -> a(b(b(c(b(c(a(x))))))) The parallel critical pair system PCPS(R,C) is: (empty) All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating. Therefore, the confluence of R follows from that of C. # Parallel rule labeling (Zankl et al. 2015). Consider the left-linear TRS R: b(b(c(a(b(c(x)))))) -> a(b(b(c(b(c(a(x))))))) All parallel critical peaks (except C's) are decreasing wrt rule labeling: phi(b(b(c(a(b(c(x)))))) -> a(b(b(c(b(c(a(x)))))))) = 1 psi(b(b(c(a(b(c(x)))))) -> a(b(b(c(b(c(a(x)))))))) = 1