YES

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

Consider the left-linear TRS R:

  f(a(),b()) -> c()
  a() -> a'()
  b() -> b'()
  c() -> f(a'(),b())
  c() -> f(a(),b'())
  c() -> f(a(),b())

Let C be the following subset of R:

  (empty)

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

  phi(f(a(),b()) -> c()) = 1
  phi(a() -> a'()) = 3
  phi(b() -> b'()) = 7
  phi(c() -> f(a'(),b())) = 2
  phi(c() -> f(a(),b'())) = 8
  phi(c() -> f(a(),b())) = 10

  psi(f(a(),b()) -> c()) = 3
  psi(a() -> a'()) = 6
  psi(b() -> b'()) = 4
  psi(c() -> f(a'(),b())) = 5
  psi(c() -> f(a(),b'())) = 4
  psi(c() -> f(a(),b())) = 9


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.