YES # parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023) Consider the left-linear TRS R: from(x) -> :(x,from(s(x))) sel(0(),:(y,z)) -> y sel(s(x),:(y,z)) -> sel(x,z) Let C be the following subset of R: sel(0(),:(y,z)) -> y sel(s(x),:(y,z)) -> sel(x,z) 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: sel(0(),:(y,z)) -> y sel(s(x),:(y,z)) -> sel(x,z) 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 rule labeling (Shintani and Hirokawa 2022). Consider the left-linear TRS R: (empty) Let C be the following subset of R: (empty) All parallel critical peaks (except C's) are decreasing wrt rule labeling: 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.