YES

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  f(f(f(x))) -> a()
  f(f(a())) -> a()
  f(a()) -> a()
  f(f(g(g(x)))) -> f(a())
  g(f(a())) -> a()
  g(a()) -> a()

Let C be the following subset of R:

  f(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

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

Consider the left-linear TRS R:

  f(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

Let C be the following subset of R:

  f(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

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(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

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

  phi(f(f(g(g(x)))) -> f(a())) = 5
  phi(g(a()) -> a()) = 6
  phi(f(a()) -> a()) = 2
  phi(f(f(f(x))) -> a()) = 1
  phi(f(f(a())) -> a()) = 3

  psi(f(f(g(g(x)))) -> f(a())) = 7
  psi(g(a()) -> a()) = 7
  psi(f(a()) -> a()) = 4
  psi(f(f(f(x))) -> a()) = 4
  psi(f(f(a())) -> a()) = 4