YES

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

Consider the left-linear TRS R:

  a() -> b()
  a() -> f(a())
  b() -> f(f(b()))
  f(f(f(b()))) -> b()

Let C be the following subset of R:

  a() -> b()
  a() -> f(a())
  b() -> f(f(b()))
  f(f(f(b()))) -> b()

The parallel critical pair system PCPS(R,C) is:

  (empty)

The TRS R is locally confluent 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() -> b()
  a() -> f(a())
  b() -> f(f(b()))
  f(f(f(b()))) -> b()

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

  phi(a() -> b()) = 2
  phi(a() -> f(a())) = 5
  phi(b() -> f(f(b()))) = 1
  phi(f(f(f(b()))) -> b()) = 1

  psi(a() -> b()) = 4
  psi(a() -> f(a())) = 3
  psi(b() -> f(f(b()))) = 1
  psi(f(f(f(b()))) -> b()) = 1