0.00/0.03 YES 0.00/0.03 0.00/0.03 # Compositional parallel rule labeling (Shintani and Hirokawa 2022). 0.00/0.03 0.00/0.03 Consider the left-linear TRS R: 0.00/0.03 0.00/0.03 f(a()) -> f(f(a())) 0.00/0.03 f(x) -> f(a()) 0.00/0.03 0.00/0.03 Let C be the following subset of R: 0.00/0.03 0.00/0.03 (empty) 0.00/0.03 0.00/0.03 All parallel critical peaks (except C's) are decreasing wrt rule labeling: 0.00/0.03 0.00/0.03 phi(f(a()) -> f(f(a()))) = 1 0.00/0.03 phi(f(x) -> f(a())) = 1 0.00/0.03 0.00/0.03 psi(f(a()) -> f(f(a()))) = 1 0.00/0.03 psi(f(x) -> f(a())) = 1 0.00/0.03 0.00/0.03 0.00/0.03 Therefore, the confluence of R follows from that of C. 0.00/0.03 0.00/0.03 # Compositional parallel critical pair system (Shintani and Hirokawa 2022). 0.00/0.03 0.00/0.03 Consider the left-linear TRS R: 0.00/0.03 0.00/0.03 (empty) 0.00/0.03 0.00/0.03 Let C be the following subset of R: 0.00/0.03 0.00/0.03 (empty) 0.00/0.03 0.00/0.03 The parallel critical pair system PCPS(R,C) is: 0.00/0.03 0.00/0.03 (empty) 0.00/0.03 0.00/0.03 The TRS R is locally confluent and PCPS(R,C)/R is terminating. 0.00/0.03 Therefore, the confluence of R follows from that of C. 0.00/0.03 0.00/0.03 # Emptiness. 0.00/0.03 0.00/0.03 The empty TRS is confluent. 0.00/0.03 0.00/0.03 EOF