YES

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

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(),y) -> 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:

  +(p(x0_1),0()) -> p(+(x0_1,0()))
  +(p(x0_1),0()) -> p(x0_1)
  +(s(x0_1),0()) -> s(+(x0_1,0()))
  +(s(x0_1),0()) -> s(x0_1)
  +(0(),s(y2)) -> s(y2)
  +(0(),s(y2)) -> s(+(0(),y2))
  +(p(x0_1),s(y2)) -> p(+(x0_1,s(y2)))
  +(p(x0_1),s(y2)) -> s(+(p(x0_1),y2))
  +(s(x0_1),s(y2)) -> s(+(x0_1,s(y2)))
  +(s(x0_1),s(y2)) -> s(+(s(x0_1),y2))
  +(y1,s(p(x1_1))) -> +(y1,x1_1)
  +(y1,s(p(x1_1))) -> s(+(y1,p(x1_1)))
  +(0(),p(y2)) -> p(y2)
  +(0(),p(y2)) -> p(+(0(),y2))
  +(p(x0_1),p(y2)) -> p(+(x0_1,p(y2)))
  +(p(x0_1),p(y2)) -> p(+(p(x0_1),y2))
  +(s(x0_1),p(y2)) -> s(+(x0_1,p(y2)))
  +(s(x0_1),p(y2)) -> p(+(s(x0_1),y2))
  +(y1,p(s(x1_1))) -> +(y1,x1_1)
  +(y1,p(s(x1_1))) -> p(+(y1,s(x1_1)))
  +(0(),s(x0_2)) -> s(+(0(),x0_2))
  +(0(),s(x0_2)) -> s(x0_2)
  +(0(),p(x0_2)) -> p(+(0(),x0_2))
  +(0(),p(x0_2)) -> p(x0_2)
  +(p(y1),0()) -> p(y1)
  +(p(y1),0()) -> p(+(y1,0()))
  +(p(y1),s(x0_2)) -> s(+(p(y1),x0_2))
  +(p(y1),s(x0_2)) -> p(+(y1,s(x0_2)))
  +(p(y1),p(x0_2)) -> p(+(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),0()) -> s(y1)
  +(s(y1),0()) -> s(+(y1,0()))
  +(s(y1),s(x0_2)) -> s(+(s(y1),x0_2))
  +(s(y1),s(x0_2)) -> s(+(y1,s(x0_2)))
  +(s(y1),p(x0_2)) -> p(+(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.