YES

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

Consider the left-linear TRS R:

  -(0(),0()) -> 0()
  -(s(x),0()) -> s(x)
  -(x,s(y)) -> -(d(x),y)
  d(s(x)) -> x
  -(s(x),s(y)) -> -(x,y)
  -(d(x),y) -> -(x,s(y))

Let C be the following subset of R:

  (empty)

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

  phi(-(0(),0()) -> 0()) = 1
  phi(-(s(x),0()) -> s(x)) = 1
  phi(-(x,s(y)) -> -(d(x),y)) = 1
  phi(d(s(x)) -> x) = 2
  phi(-(s(x),s(y)) -> -(x,y)) = 3
  phi(-(d(x),y) -> -(x,s(y))) = 6

  psi(-(0(),0()) -> 0()) = 1
  psi(-(s(x),0()) -> s(x)) = 1
  psi(-(x,s(y)) -> -(d(x),y)) = 4
  psi(d(s(x)) -> x) = 5
  psi(-(s(x),s(y)) -> -(x,y)) = 5
  psi(-(d(x),y) -> -(x,s(y))) = 6


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.