YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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 critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

  phi(+(x,y) -> +(y,x)) = 5
  phi(+(x,0()) -> x) = 3
  phi(+(x,s(y)) -> s(+(x,y))) = 2
  phi(+(0(),y) -> y) = 2
  phi(+(s(x),y) -> s(+(x,y))) = 2
  phi(+(+(x,y),z) -> +(x,+(y,z))) = 7

  psi(+(x,y) -> +(y,x)) = 4
  psi(+(x,0()) -> x) = 6
  psi(+(x,s(y)) -> s(+(x,y))) = 3
  psi(+(0(),y) -> y) = 6
  psi(+(s(x),y) -> s(+(x,y))) = 1
  psi(+(+(x,y),z) -> +(x,+(y,z))) = 6