YES

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

Consider the left-linear TRS R:

  +(0(),y) -> y
  +(s(0()),y) -> s(y)
  +(s(s(x)),y) -> s(s(+(y,x)))

Let C be the following subset of R:

  +(0(),y) -> y
  +(s(0()),y) -> s(y)
  +(s(s(x)),y) -> s(s(+(y,x)))

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

  (empty)

The TRS R is locally confluent 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:

  +(0(),y) -> y
  +(s(0()),y) -> s(y)
  +(s(s(x)),y) -> s(s(+(y,x)))

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

  phi(+(0(),y) -> y) = 1
  phi(+(s(0()),y) -> s(y)) = 1
  phi(+(s(s(x)),y) -> s(s(+(y,x)))) = 1

  psi(+(0(),y) -> y) = 1
  psi(+(s(0()),y) -> s(y)) = 1
  psi(+(s(s(x)),y) -> s(s(+(y,x)))) = 1