YES # parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023) Consider the left-linear TRS R: f(a(),f(a(),f(a(),f(a(),y)))) -> f(a(),f(a(),f(a(),g(y,f(a(),y))))) f(x,y) -> g(y,f(x,y)) Let C be the following subset of R: f(x,y) -> g(y,f(x,y)) The TRS R is left-linear and all parallel critical pairs are joinable by C. Therefore, the confluence of R is equivalent to that of C. # parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023) Consider the left-linear TRS R: f(x,y) -> g(y,f(x,y)) 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 is equivalent to 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. # Parallel rule labeling (Zankl et al. 2015). Consider the left-linear TRS R: (empty) All parallel critical peaks (except C's) are decreasing wrt rule labeling: