0.00/0.05 YES 0.00/0.05 0.00/0.05 # Compositional parallel rule labeling (Shintani and Hirokawa 2022). 0.00/0.05 0.00/0.05 Consider the left-linear TRS R: 0.00/0.05 0.00/0.05 b(a(b(b(x)))) -> b(b(b(a(b(x))))) 0.00/0.05 b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x))))))) 0.00/0.05 b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x))))))))) 0.00/0.05 0.00/0.05 Let C be the following subset of R: 0.00/0.05 0.00/0.05 (empty) 0.00/0.05 0.00/0.05 All parallel critical peaks (except C's) are decreasing wrt rule labeling: 0.00/0.05 0.00/0.05 phi(b(a(b(b(x)))) -> b(b(b(a(b(x)))))) = 1 0.00/0.05 phi(b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x)))))))) = 1 0.00/0.05 phi(b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x)))))))))) = 1 0.00/0.05 0.00/0.05 psi(b(a(b(b(x)))) -> b(b(b(a(b(x)))))) = 1 0.00/0.05 psi(b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x)))))))) = 1 0.00/0.05 psi(b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x)))))))))) = 1 0.00/0.05 0.00/0.05 0.00/0.05 Therefore, the confluence of R follows from that of C. 0.00/0.05 0.00/0.05 # Compositional parallel critical pair system (Shintani and Hirokawa 2022). 0.00/0.05 0.00/0.05 Consider the left-linear TRS R: 0.00/0.05 0.00/0.05 (empty) 0.00/0.05 0.00/0.05 Let C be the following subset of R: 0.00/0.05 0.00/0.05 (empty) 0.00/0.05 0.00/0.05 The parallel critical pair system PCPS(R,C) is: 0.00/0.05 0.00/0.05 (empty) 0.00/0.05 0.00/0.05 The TRS R is locally confluent and PCPS(R,C)/R is terminating. 0.00/0.05 Therefore, the confluence of R follows from that of C. 0.00/0.05 0.00/0.05 # Emptiness. 0.00/0.05 0.00/0.05 The empty TRS is confluent. 0.00/0.05 0.00/0.05 EOF