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))
  inc(x) -> s(x)
  +(x,y) -> +(y,x)
  inc(+(x,y)) -> +(inc(x),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) = 4
  phi(+(x,s(y)) -> s(+(x,y))) = 2
  phi(+(0(),y) -> y) = 5
  phi(+(s(x),y) -> s(+(x,y))) = 7
  phi(inc(x) -> s(x)) = 3
  phi(+(x,y) -> +(y,x)) = 3
  phi(inc(+(x,y)) -> +(inc(x),y)) = 8

  psi(+(x,0()) -> x) = 7
  psi(+(x,s(y)) -> s(+(x,y))) = 6
  psi(+(0(),y) -> y) = 7
  psi(+(s(x),y) -> s(+(x,y))) = 6
  psi(inc(x) -> s(x)) = 1
  psi(+(x,y) -> +(y,x)) = 6
  psi(inc(+(x,y)) -> +(inc(x),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)

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.