YES

# Compositional parallel critical pair system (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(x,y))
  max(x,y) -> max(y,x)

Let C be the following subset of R:

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

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.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

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

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

  phi(max(x,0()) -> x) = 5
  phi(max(0(),y) -> y) = 4
  phi(max(s(x),s(y)) -> s(max(x,y))) = 1
  phi(max(x,y) -> max(y,x)) = 3

  psi(max(x,0()) -> x) = 2
  psi(max(0(),y) -> y) = 1
  psi(max(s(x),s(y)) -> s(max(x,y))) = 1
  psi(max(x,y) -> max(y,x)) = 6