YES

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

Consider the left-linear TRS R:

  +(x,y) -> +(y,x)
  *(+(x,y),z) -> +(*(x,z),*(y,z))
  *(+(y,x),z) -> +(*(x,z),*(y,z))

Let C be the following subset of R:

  *(+(x,y),z) -> +(*(x,z),*(y,z))
  +(x,y) -> +(y,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.

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  *(+(x,y),z) -> +(*(x,z),*(y,z))
  +(x,y) -> +(y,x)

Let C be the following subset of R:

  (empty)

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(*(+(x,y),z) -> +(*(x,z),*(y,z))) = 2
  phi(+(x,y) -> +(y,x)) = 1

  psi(*(+(x,y),z) -> +(*(x,z),*(y,z))) = 4
  psi(+(x,y) -> +(y,x)) = 3


Therefore, the confluence of R follows from that of C.

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  (empty)

Let C be the following subset of R:

  (empty)

The parallel critical pair system PCPS(R,C) is:

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# emptiness

The empty TRS is confluent.