YES

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

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

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.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

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

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

  phi(+(x,y) -> +(y,x)) = 3
  phi(*(+(x,y),z) -> +(*(x,z),*(y,z))) = 6
  phi(*(+(y,x),z) -> +(*(x,z),*(y,z))) = 4

  psi(+(x,y) -> +(y,x)) = 5
  psi(*(+(x,y),z) -> +(*(x,z),*(y,z))) = 2
  psi(*(+(y,x),z) -> +(*(x,z),*(y,z))) = 1