YES

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

Consider the left-linear TRS R:

  f(g(g(x))) -> a()
  f(g(h(x))) -> b()
  f(h(g(x))) -> b()
  f(h(h(x))) -> c()
  g(x) -> h(x)
  a() -> b()
  b() -> c()

Let C be the following subset of R:

  (empty)

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

  phi(f(g(g(x))) -> a()) = 4
  phi(f(g(h(x))) -> b()) = 2
  phi(f(h(g(x))) -> b()) = 2
  phi(f(h(h(x))) -> c()) = 1
  phi(g(x) -> h(x)) = 3
  phi(a() -> b()) = 1
  phi(b() -> c()) = 1

  psi(f(g(g(x))) -> a()) = 5
  psi(f(g(h(x))) -> b()) = 2
  psi(f(h(g(x))) -> b()) = 2
  psi(f(h(h(x))) -> c()) = 1
  psi(g(x) -> h(x)) = 4
  psi(a() -> b()) = 1
  psi(b() -> c()) = 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.