YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  -(x,s(y)) -> -(d(x),y)
  d(s(x)) -> x
  -(s(x),s(y)) -> -(x,y)
  -(d(x),y) -> -(x,s(y))

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,s(y)) -> -(d(x),y)
  d(s(x)) -> x
  -(s(x),s(y)) -> -(x,y)
  -(d(x),y) -> -(x,s(y))

Let C be the following subset of R:

  -(x,s(y)) -> -(d(x),y)
  d(s(x)) -> x
  -(s(x),s(y)) -> -(x,y)
  -(d(x),y) -> -(x,s(y))

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,s(y)) -> -(d(x),y)
  d(s(x)) -> x
  -(s(x),s(y)) -> -(x,y)
  -(d(x),y) -> -(x,s(y))

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

  phi(-(x,s(y)) -> -(d(x),y)) = 1
  phi(d(s(x)) -> x) = 2
  phi(-(s(x),s(y)) -> -(x,y)) = 3
  phi(-(d(x),y) -> -(x,s(y))) = 7

  psi(-(x,s(y)) -> -(d(x),y)) = 4
  psi(d(s(x)) -> x) = 5
  psi(-(s(x),s(y)) -> -(x,y)) = 6
  psi(-(d(x),y) -> -(x,s(y))) = 7