YES

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

Consider the left-linear TRS R:

  a(x) -> x
  a(a(x)) -> b(c(x))
  b(x) -> x
  c(x) -> x
  c(b(x)) -> b(a(c(x)))

Let C be the following subset of R:

  a(x) -> x
  a(a(x)) -> b(c(x))
  b(x) -> x
  c(x) -> x
  c(b(x)) -> b(a(c(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:

  a(x) -> x
  a(a(x)) -> b(c(x))
  b(x) -> x
  c(x) -> x
  c(b(x)) -> b(a(c(x)))

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

  phi(a(x) -> x) = 2
  phi(a(a(x)) -> b(c(x))) = 3
  phi(b(x) -> x) = 2
  phi(c(x) -> x) = 2
  phi(c(b(x)) -> b(a(c(x)))) = 3

  psi(a(x) -> x) = 1
  psi(a(a(x)) -> b(c(x))) = 3
  psi(b(x) -> x) = 1
  psi(c(x) -> x) = 2
  psi(c(b(x)) -> b(a(c(x)))) = 3