YES

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

Consider the left-linear TRS R:

  max(x,0()) -> x
  max(0(),y) -> y
  max(s(x),s(y)) -> s(max(y,x))
  max(x,y) -> max(y,x)

Let C be the following subset of R:

  (empty)

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

  phi(max(x,0()) -> x) = 3
  phi(max(0(),y) -> y) = 2
  phi(max(s(x),s(y)) -> s(max(y,x))) = 1
  phi(max(x,y) -> max(y,x)) = 1

  psi(max(x,0()) -> x) = 5
  psi(max(0(),y) -> y) = 5
  psi(max(s(x),s(y)) -> s(max(y,x))) = 1
  psi(max(x,y) -> max(y,x)) = 4


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.