YES # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: f(x1,g(x2)) -> f(x1,g(x1)) f(g(y1),y2) -> f(g(y1),g(y1)) g(a()) -> g(b()) b() -> a() Let C be the following subset of R: f(x1,g(x2)) -> f(x1,g(x1)) f(g(y1),y2) -> f(g(y1),g(y1)) b() -> a() 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. # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: f(x1,g(x2)) -> f(x1,g(x1)) f(g(y1),y2) -> f(g(y1),g(y1)) b() -> a() Let C be the following subset of R: f(g(y1),y2) -> f(g(y1),g(y1)) 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. # Compositional parallel critical pair system (Shintani and Hirokawa 2022). Consider the left-linear TRS R: f(g(y1),y2) -> f(g(y1),g(y1)) 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.