YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

Let C be the following subset of R:

  (empty)

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

  +(y1,s(p(x1_1))) -> +(y1,x1_1)
  +(y1,s(p(x1_1))) -> s(+(y1,p(x1_1)))
  +(y1,p(s(x1_1))) -> +(y1,x1_1)
  +(y1,p(s(x1_1))) -> p(+(y1,s(x1_1)))
  -(p(x0_1),s(y2)) -> p(-(x0_1,s(y2)))
  -(p(x0_1),s(y2)) -> p(-(p(x0_1),y2))
  -(s(x0_1),s(y2)) -> s(-(x0_1,s(y2)))
  -(s(x0_1),s(y2)) -> p(-(s(x0_1),y2))
  -(y1,s(p(x1_1))) -> -(y1,x1_1)
  -(y1,s(p(x1_1))) -> p(-(y1,p(x1_1)))
  -(p(x0_1),p(y2)) -> p(-(x0_1,p(y2)))
  -(p(x0_1),p(y2)) -> s(-(p(x0_1),y2))
  -(s(x0_1),p(y2)) -> s(-(x0_1,p(y2)))
  -(s(x0_1),p(y2)) -> s(-(s(x0_1),y2))
  -(y1,p(s(x1_1))) -> -(y1,x1_1)
  -(y1,p(s(x1_1))) -> s(-(y1,s(x1_1)))
  -(p(y1),s(x0_2)) -> p(-(p(y1),x0_2))
  -(p(y1),s(x0_2)) -> p(-(y1,s(x0_2)))
  -(p(y1),p(x0_2)) -> s(-(p(y1),x0_2))
  -(p(y1),p(x0_2)) -> p(-(y1,p(x0_2)))
  -(p(s(x1_1)),y2) -> -(x1_1,y2)
  -(p(s(x1_1)),y2) -> p(-(s(x1_1),y2))
  -(s(y1),s(x0_2)) -> p(-(s(y1),x0_2))
  -(s(y1),s(x0_2)) -> s(-(y1,s(x0_2)))
  -(s(y1),p(x0_2)) -> s(-(s(y1),x0_2))
  -(s(y1),p(x0_2)) -> s(-(y1,p(x0_2)))
  -(s(p(x1_1)),y2) -> -(x1_1,y2)
  -(s(p(x1_1)),y2) -> s(-(p(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.