YES

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

Consider the left-linear TRS R:

  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  d(0()) -> 0()
  d(s(x)) -> s(s(d(x)))
  f(0()) -> 0()
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  f(g(0())) -> +(+(g(0()),g(0())),g(0()))
  g(x) -> s(d(x))

Let C be the following subset of R:

  +(x,s(y)) -> s(+(x,y))
  d(s(x)) -> s(s(d(x)))
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  g(x) -> s(d(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:

  +(x,s(y)) -> s(+(x,y))
  d(s(x)) -> s(s(d(x)))
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  g(x) -> s(d(x))

Let C be the following subset of R:

  +(x,s(y)) -> s(+(x,y))
  f(s(x)) -> +(+(s(x),s(x)),s(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:

  +(x,s(y)) -> s(+(x,y))
  f(s(x)) -> +(+(s(x),s(x)),s(x))

Let C be the following subset of R:

  +(x,s(y)) -> s(+(x,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:

  +(x,s(y)) -> s(+(x,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.