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.

# parallel critical pair closing 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:

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

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

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  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.

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

Consider the left-linear TRS R:

  f(a()) -> a()

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.