YES

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

Consider the left-linear TRS R:

  f(0(),0()) -> f(0(),1())
  f(1(),0()) -> f(0(),0())
  f(x,y) -> f(y,x)

Let C be the following subset of R:

  (empty)

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

  phi(f(0(),0()) -> f(0(),1())) = 5
  phi(f(1(),0()) -> f(0(),0())) = 5
  phi(f(x,y) -> f(y,x)) = 4

  psi(f(0(),0()) -> f(0(),1())) = 1
  psi(f(1(),0()) -> f(0(),0())) = 3
  psi(f(x,y) -> f(y,x)) = 2


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.