YES

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

Consider the left-linear TRS R:

  b(w(x)) -> w(w(w(b(x))))
  w(b(x)) -> b(x)
  b(b(x)) -> w(w(w(w(x))))
  w(w(x)) -> w(x)

Let C be the following subset of R:

  (empty)

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

  phi(b(w(x)) -> w(w(w(b(x))))) = 6
  phi(w(b(x)) -> b(x)) = 4
  phi(b(b(x)) -> w(w(w(w(x))))) = 6
  phi(w(w(x)) -> w(x)) = 1

  psi(b(w(x)) -> w(w(w(b(x))))) = 5
  psi(w(b(x)) -> b(x)) = 5
  psi(b(b(x)) -> w(w(w(w(x))))) = 2
  psi(w(w(x)) -> w(x)) = 3


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.