YES

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  b(a(b(b(x)))) -> b(b(b(a(b(x)))))
  b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x)))))))
  b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x)))))))))

Let C be the following subset of R:

  (empty)

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(b(a(b(b(x)))) -> b(b(b(a(b(x)))))) = 1
  phi(b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x)))))))) = 1
  phi(b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x)))))))))) = 1

  psi(b(a(b(b(x)))) -> b(b(b(a(b(x)))))) = 1
  psi(b(a(a(b(b(x))))) -> b(a(b(b(a(a(b(x)))))))) = 1
  psi(b(a(a(a(b(b(x)))))) -> b(a(a(b(b(a(a(a(b(x)))))))))) = 1


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.