YES

# Compositional 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 critical pair system CPS(R,C) is:

  p(s(p(x0))) -> p(x0)
  +(y0,s(p(x0))) -> +(y0,x0)
  +(y0,s(p(x0))) -> s(+(y0,p(x0)))
  +(s(p(x0)),y1) -> +(x0,y1)
  +(s(p(x0)),y1) -> s(+(p(x0),y1))
  s(p(s(x0))) -> s(x0)
  +(y0,p(s(x0))) -> +(y0,x0)
  +(y0,p(s(x0))) -> p(+(y0,s(x0)))
  +(p(s(x0)),y1) -> +(x0,y1)
  +(p(s(x0)),y1) -> p(+(s(x0),y1))
  +(0(),0()) -> 0()
  +(p(y0),0()) -> p(y0)
  +(p(y0),0()) -> p(+(y0,0()))
  +(s(y0),0()) -> s(y0)
  +(s(y0),0()) -> s(+(y0,0()))
  +(0(),s(x1)) -> s(+(0(),x1))
  +(0(),s(x1)) -> s(x1)
  +(p(y0),s(x1)) -> s(+(p(y0),x1))
  +(p(y0),s(x1)) -> p(+(y0,s(x1)))
  +(s(y0),s(x1)) -> s(+(s(y0),x1))
  +(s(y0),s(x1)) -> s(+(y0,s(x1)))
  +(0(),p(x1)) -> p(+(0(),x1))
  +(0(),p(x1)) -> p(x1)
  +(p(y0),p(x1)) -> p(+(p(y0),x1))
  +(p(y0),p(x1)) -> p(+(y0,p(x1)))
  +(s(y0),p(x1)) -> p(+(s(y0),x1))
  +(s(y0),p(x1)) -> s(+(y0,p(x1)))
  +(0(),s(y1)) -> s(y1)
  +(0(),s(y1)) -> s(+(0(),y1))
  +(0(),p(y1)) -> p(y1)
  +(0(),p(y1)) -> p(+(0(),y1))
  +(p(x0),0()) -> p(+(x0,0()))
  +(p(x0),0()) -> p(x0)
  +(p(x0),s(y1)) -> p(+(x0,s(y1)))
  +(p(x0),s(y1)) -> s(+(p(x0),y1))
  +(p(x0),p(y1)) -> p(+(x0,p(y1)))
  +(p(x0),p(y1)) -> p(+(p(x0),y1))
  +(s(x0),0()) -> s(+(x0,0()))
  +(s(x0),0()) -> s(x0)
  +(s(x0),s(y1)) -> s(+(x0,s(y1)))
  +(s(x0),s(y1)) -> s(+(s(x0),y1))
  +(s(x0),p(y1)) -> s(+(x0,p(y1)))
  +(s(x0),p(y1)) -> p(+(s(x0),y1))

The TRS R is locally confluent and CPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Emptiness.

The empty TRS is confluent.