YES

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

Consider the left-linear TRS R:

  f(x) -> x
  f(x) -> f(f(x))

Let C be the following subset of R:

  f(x) -> x
  f(x) -> f(f(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:

  f(x) -> x
  f(x) -> f(f(x))

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

  phi(f(x) -> x) = 1
  phi(f(x) -> f(f(x))) = 2

  psi(f(x) -> x) = 1
  psi(f(x) -> f(f(x))) = 2