YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  (empty)

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

  phi(+(x,0()) -> x) = 12
  phi(+(x,s(y)) -> s(+(x,y))) = 10
  phi(+(0(),y) -> y) = 11
  phi(+(s(x),y) -> s(+(x,y))) = 2
  phi(dbl(0()) -> 0()) = 1
  phi(dbl(s(x)) -> s(s(dbl(x)))) = 3
  phi(+(+(x,y),z) -> +(x,+(y,z))) = 9
  phi(+(x,y) -> +(y,x)) = 8
  phi(dbl(+(x,y)) -> +(dbl(x),dbl(y))) = 6

  psi(+(x,0()) -> x) = 7
  psi(+(x,s(y)) -> s(+(x,y))) = 4
  psi(+(0(),y) -> y) = 6
  psi(+(s(x),y) -> s(+(x,y))) = 5
  psi(dbl(0()) -> 0()) = 1
  psi(dbl(s(x)) -> s(s(dbl(x)))) = 1
  psi(+(+(x,y),z) -> +(x,+(y,z))) = 10
  psi(+(x,y) -> +(y,x)) = 10
  psi(dbl(+(x,y)) -> +(dbl(x),dbl(y))) = 8


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:

  (empty)

Let C be the following subset of R:

  (empty)

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

  (empty)

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

# Emptiness.

The empty TRS is confluent.