YES

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

Consider the left-linear TRS R:

  f(x) -> f(f(x))
  g(x) -> f(x)
  f(x) -> g(x)

Let C be the following subset of R:

  (empty)

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

  phi(f(x) -> f(f(x))) = 2
  phi(g(x) -> f(x)) = 1
  phi(f(x) -> g(x)) = 3

  psi(f(x) -> f(f(x))) = 2
  psi(g(x) -> f(x)) = 1
  psi(f(x) -> g(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.