YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

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

  phi(+(+(x,y),z) -> +(x,+(y,z))) = 5
  phi(+(x,+(y,z)) -> +(+(x,y),z)) = 6
  phi(+(x,y) -> +(y,x)) = 4

  psi(+(+(x,y),z) -> +(x,+(y,z))) = 1
  psi(+(x,+(y,z)) -> +(+(x,y),z)) = 2
  psi(+(x,y) -> +(y,x)) = 3