YES

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

Consider the left-linear TRS R:

  g(f(f(h(x))),y) -> g(g(f(h(x)),f(f(h(x)))),y)
  f(x) -> g(x,f(x))
  h(x) -> g(f(x),x)
  g(x,y) -> h(g(f(x),f(y)))

Let C be the following subset of R:

  f(x) -> g(x,f(x))
  h(x) -> g(f(x),x)
  g(x,y) -> h(g(f(x),f(y)))

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(x) -> g(x,f(x))
  h(x) -> g(f(x),x)
  g(x,y) -> h(g(f(x),f(y)))

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.