YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

  phi(a(x) -> b(x)) = 2
  phi(a(b(x)) -> b(a(c(a(x))))) = 5
  phi(b(x) -> c(x)) = 2
  phi(c(c(x)) -> x) = 1

  psi(a(x) -> b(x)) = 4
  psi(a(b(x)) -> b(a(c(a(x))))) = 5
  psi(b(x) -> c(x)) = 3
  psi(c(c(x)) -> x) = 2