YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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 rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

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

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


Therefore, the confluence of R follows from that of C.

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  (empty)

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

  +(s(s(x1_1)),y2) -> +(x1_1,y2)
  +(s(s(x1_1)),y2) -> s(+(s(x1_1),y2))

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# emptiness

The empty TRS is confluent.