YES # Compositional parallel rule labeling (Shintani and Hirokawa 2022). Consider the left-linear TRS R: g(f(f(h(x))),y) -> g(g(f(h(x)),f(f(h(x)))),y) f(x) -> g(x,f(x)) h(x) -> g(f(x),x) g(x,y) -> h(g(f(x),f(y))) Let C be the following subset of R: (empty) All parallel critical peaks (except C's) are decreasing wrt rule labeling: phi(g(f(f(h(x))),y) -> g(g(f(h(x)),f(f(h(x)))),y)) = 1 phi(f(x) -> g(x,f(x))) = 2 phi(h(x) -> g(f(x),x)) = 4 phi(g(x,y) -> h(g(f(x),f(y)))) = 4 psi(g(f(f(h(x))),y) -> g(g(f(h(x)),f(f(h(x)))),y)) = 4 psi(f(x) -> g(x,f(x))) = 3 psi(h(x) -> g(f(x),x)) = 3 psi(g(x,y) -> h(g(f(x),f(y)))) = 3 Therefore, the confluence of R follows from that of C. # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: (empty) Let C be the following subset of R: (empty) 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. # emptiness The empty TRS is confluent.