YES

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

Consider the left-linear TRS R:

  from(x) -> :(x,from(s(x)))
  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

Let C be the following subset of R:

  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

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:

  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

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.