YES

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

Consider the left-linear TRS R:

  f(a(),a()) -> g(f(a(),a()))
  a() -> b()
  f(b(),x) -> g(f(x,x))
  f(x,b()) -> g(f(x,x))

Let C be the following subset of R:

  (empty)

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

  phi(f(a(),a()) -> g(f(a(),a()))) = 4
  phi(a() -> b()) = 2
  phi(f(b(),x) -> g(f(x,x))) = 1
  phi(f(x,b()) -> g(f(x,x))) = 1

  psi(f(a(),a()) -> g(f(a(),a()))) = 4
  psi(a() -> b()) = 3
  psi(f(b(),x) -> g(f(x,x))) = 1
  psi(f(x,b()) -> g(f(x,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.