YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  (empty)

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

  phi(f(a(),f(a(),f(a(),f(a(),y)))) -> f(a(),f(a(),f(a(),g(y,f(a(),y)))))) = 3
  phi(f(x,y) -> g(y,f(x,y))) = 2

  psi(f(a(),f(a(),f(a(),f(a(),y)))) -> f(a(),f(a(),f(a(),g(y,f(a(),y)))))) = 3
  psi(f(x,y) -> g(y,f(x,y))) = 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.