YES

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

Consider the left-linear TRS R:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  hd(:(x,y)) -> x
  tl(:(x,y)) -> y
  inc(tl(nats())) -> tl(inc(nats()))

Let C be the following subset of R:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  hd(:(x,y)) -> x
  tl(:(x,y)) -> y
  inc(tl(nats())) -> tl(inc(nats()))

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:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  hd(:(x,y)) -> x
  tl(:(x,y)) -> y
  inc(tl(nats())) -> tl(inc(nats()))

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

  phi(nats() -> :(0(),inc(nats()))) = 2
  phi(inc(:(x,y)) -> :(s(x),inc(y))) = 1
  phi(hd(:(x,y)) -> x) = 1
  phi(tl(:(x,y)) -> y) = 1
  phi(inc(tl(nats())) -> tl(inc(nats()))) = 4

  psi(nats() -> :(0(),inc(nats()))) = 3
  psi(inc(:(x,y)) -> :(s(x),inc(y))) = 1
  psi(hd(:(x,y)) -> x) = 1
  psi(tl(:(x,y)) -> y) = 1
  psi(inc(tl(nats())) -> tl(inc(nats()))) = 3