YES

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

Consider the left-linear TRS R:

  Ap(Ap(Ap(S(),x),y),z) -> Ap(Ap(x,z),Ap(y,z))
  Ap(Ap(K(),x),y) -> x
  Ap(I(),x) -> x

Let C be the following subset of R:

  Ap(Ap(K(),x),y) -> x
  Ap(I(),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:

  Ap(Ap(K(),x),y) -> x
  Ap(I(),x) -> x

Let C be the following subset of R:

  Ap(I(),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:

  Ap(I(),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.