YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  d(0()) -> 0()
  d(s(x)) -> s(s(d(x)))
  f(0()) -> 0()
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  f(g(0())) -> +(+(g(0()),g(0())),g(0()))
  g(x) -> s(d(x))

Let C be the following subset of R:

  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  d(0()) -> 0()
  d(s(x)) -> s(s(d(x)))
  f(0()) -> 0()
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  f(g(0())) -> +(+(g(0()),g(0())),g(0()))
  g(x) -> s(d(x))

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.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  +(x,0()) -> x
  +(x,s(y)) -> s(+(x,y))
  d(0()) -> 0()
  d(s(x)) -> s(s(d(x)))
  f(0()) -> 0()
  f(s(x)) -> +(+(s(x),s(x)),s(x))
  f(g(0())) -> +(+(g(0()),g(0())),g(0()))
  g(x) -> s(d(x))

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

  phi(+(x,0()) -> x) = 1
  phi(+(x,s(y)) -> s(+(x,y))) = 1
  phi(d(0()) -> 0()) = 1
  phi(d(s(x)) -> s(s(d(x)))) = 1
  phi(f(0()) -> 0()) = 1
  phi(f(s(x)) -> +(+(s(x),s(x)),s(x))) = 1
  phi(f(g(0())) -> +(+(g(0()),g(0())),g(0()))) = 2
  phi(g(x) -> s(d(x))) = 2

  psi(+(x,0()) -> x) = 1
  psi(+(x,s(y)) -> s(+(x,y))) = 1
  psi(d(0()) -> 0()) = 1
  psi(d(s(x)) -> s(s(d(x)))) = 1
  psi(f(0()) -> 0()) = 1
  psi(f(s(x)) -> +(+(s(x),s(x)),s(x))) = 1
  psi(f(g(0())) -> +(+(g(0()),g(0())),g(0()))) = 2
  psi(g(x) -> s(d(x))) = 2