YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

Let C be the following subset of R:

  f(b(),x) -> g(f(x,x))
  f(x,b()) -> g(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(b(),x) -> g(f(x,x))
  f(x,b()) -> g(f(x,x))

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.