YES

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

Consider the left-linear TRS R:

  f(a(),a()) -> b()
  a() -> a'()
  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())

Let C be the following subset of R:

  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())
  a() -> a'()
  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)

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.

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

Consider the left-linear TRS R:

  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())
  a() -> a'()
  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)

Let C be the following subset of R:

  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> 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(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())

Let C be the following subset of R:

  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> 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(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())

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

  phi(f(a'(),x) -> f(x,x)) = 1
  phi(f(x,a'()) -> f(x,x)) = 1
  phi(f(a'(),a'()) -> b()) = 1
  phi(b() -> f(a'(),a'())) = 1

  psi(f(a'(),x) -> f(x,x)) = 1
  psi(f(x,a'()) -> f(x,x)) = 1
  psi(f(a'(),a'()) -> b()) = 1
  psi(b() -> f(a'(),a'())) = 1