YES

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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

  phi(+(x,0()) -> x) = 1
  phi(+(x,s(y)) -> s(+(x,y))) = 0
  phi(+(x,p(y)) -> p(+(x,y))) = 0
  phi(+(0(),y) -> y) = 0
  phi(+(s(x),y) -> s(+(x,y))) = 0
  phi(+(p(x),y) -> p(+(x,y))) = 0
  phi(s(p(x)) -> x) = 0
  phi(p(s(x)) -> x) = 0
  phi(+(+(x,y),z) -> +(x,+(y,z))) = 0
  phi(+(x,+(y,z)) -> +(+(x,y),z)) = 1

  psi(+(x,0()) -> x) = 1
  psi(+(x,s(y)) -> s(+(x,y))) = 0
  psi(+(x,p(y)) -> p(+(x,y))) = 0
  psi(+(0(),y) -> y) = 0
  psi(+(s(x),y) -> s(+(x,y))) = 0
  psi(+(p(x),y) -> p(+(x,y))) = 0
  psi(s(p(x)) -> x) = 0
  psi(p(s(x)) -> x) = 0
  psi(+(+(x,y),z) -> +(x,+(y,z))) = 0
  psi(+(x,+(y,z)) -> +(+(x,y),z)) = 1


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:

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

Let C be the following subset of R:

  (empty)

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

  +(0(),s(y2)) -> s(y2)
  +(0(),s(y2)) -> s(+(0(),y2))
  +(s(x0_1),s(y2)) -> s(+(x0_1,s(y2)))
  +(s(x0_1),s(y2)) -> s(+(s(x0_1),y2))
  +(p(x0_1),s(y2)) -> p(+(x0_1,s(y2)))
  +(p(x0_1),s(y2)) -> s(+(p(x0_1),y2))
  +(+(x0_1,x0_2),s(y2)) -> +(x0_1,+(x0_2,s(y2)))
  +(+(x0_1,x0_2),s(y2)) -> s(+(+(x0_1,x0_2),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))
  +(s(x0_1),p(y2)) -> s(+(x0_1,p(y2)))
  +(s(x0_1),p(y2)) -> p(+(s(x0_1),y2))
  +(p(x0_1),p(y2)) -> p(+(x0_1,p(y2)))
  +(p(x0_1),p(y2)) -> p(+(p(x0_1),y2))
  +(+(x0_1,x0_2),p(y2)) -> +(x0_1,+(x0_2,p(y2)))
  +(+(x0_1,x0_2),p(y2)) -> p(+(+(x0_1,x0_2),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)
  +(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))
  +(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))
  +(+(y1,y2),s(x0_2)) -> s(+(+(y1,y2),x0_2))
  +(+(y1,y2),s(x0_2)) -> +(y1,+(y2,s(x0_2)))
  +(+(y1,y2),p(x0_2)) -> p(+(+(y1,y2),x0_2))
  +(+(y1,y2),p(x0_2)) -> +(y1,+(y2,p(x0_2)))
  +(+(y1,s(x1_2)),y3) -> +(s(+(y1,x1_2)),y3)
  +(+(y1,s(x1_2)),y3) -> +(y1,+(s(x1_2),y3))
  +(+(y1,p(x1_2)),y3) -> +(p(+(y1,x1_2)),y3)
  +(+(y1,p(x1_2)),y3) -> +(y1,+(p(x1_2),y3))
  +(+(0(),y2),y3) -> +(y2,y3)
  +(+(0(),y2),y3) -> +(0(),+(y2,y3))
  +(+(s(x1_1),y2),y3) -> +(s(+(x1_1,y2)),y3)
  +(+(s(x1_1),y2),y3) -> +(s(x1_1),+(y2,y3))
  +(+(p(x1_1),y2),y3) -> +(p(+(x1_1,y2)),y3)
  +(+(p(x1_1),y2),y3) -> +(p(x1_1),+(y2,y3))
  +(+(+(x1_1,x1_2),y2),y3) -> +(+(x1_1,+(x1_2,y2)),y3)
  +(+(+(x1_1,x1_2),y2),y3) -> +(+(x1_1,x1_2),+(y2,y3))

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.