YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  p(x) -> q(x)
  p(x) -> r(x)
  q(x) -> s(p(x))
  r(x) -> s(p(x))
  s(x) -> f(p(x))

Let C be the following subset of R:

  p(x) -> q(x)
  p(x) -> r(x)
  q(x) -> s(p(x))
  r(x) -> s(p(x))
  s(x) -> f(p(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:

  p(x) -> q(x)
  p(x) -> r(x)
  q(x) -> s(p(x))
  r(x) -> s(p(x))
  s(x) -> f(p(x))

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

  phi(p(x) -> q(x)) = 2
  phi(p(x) -> r(x)) = 4
  phi(q(x) -> s(p(x))) = 2
  phi(r(x) -> s(p(x))) = 1
  phi(s(x) -> f(p(x))) = 1

  psi(p(x) -> q(x)) = 3
  psi(p(x) -> r(x)) = 3
  psi(q(x) -> s(p(x))) = 1
  psi(r(x) -> s(p(x))) = 1
  psi(s(x) -> f(p(x))) = 1