YES

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

Consider the left-linear TRS R:

  f(g(x,a(),b())) -> x
  g(f(h(c(),d())),x,y) -> h(k1(x),k2(y))
  k1(a()) -> c()
  k2(b()) -> d()
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))
  f(h(c(),k2(b()))) -> f(h(c(),d()))
  f(h(k1(a()),d())) -> f(h(c(),d()))

Let C be the following subset of R:

  f(g(x,a(),b())) -> x
  g(f(h(c(),d())),x,y) -> h(k1(x),k2(y))
  k2(b()) -> d()
  k1(a()) -> c()
  f(h(k1(a()),d())) -> f(h(c(),d()))
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))

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(g(x,a(),b())) -> x
  g(f(h(c(),d())),x,y) -> h(k1(x),k2(y))
  k2(b()) -> d()
  k1(a()) -> c()
  f(h(k1(a()),d())) -> f(h(c(),d()))
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))

Let C be the following subset of R:

  k1(a()) -> c()
  f(h(k1(a()),d())) -> f(h(c(),d()))
  k2(b()) -> d()
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))

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:

  k1(a()) -> c()
  f(h(k1(a()),d())) -> f(h(c(),d()))
  k2(b()) -> d()
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))

Let C be the following subset of R:

  k1(a()) -> c()
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))
  k2(b()) -> d()

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:

  k1(a()) -> c()
  f(h(k1(a()),k2(b()))) -> f(h(c(),d()))
  k2(b()) -> d()

Let C be the following subset of R:

  k2(b()) -> d()
  k1(a()) -> c()

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:

  k2(b()) -> d()
  k1(a()) -> c()

Let C be the following subset of R:

  k1(a()) -> c()

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:

  k1(a()) -> c()

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.