YES

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

Consider the left-linear TRS R:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  g(a()) -> g(b())
  b() -> a()

Let C be the following subset of R:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  g(a()) -> g(b())
  b() -> a()

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:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  g(a()) -> g(b())
  b() -> a()

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

  phi(f(x1,g(x2)) -> f(x1,g(x1))) = 1
  phi(f(g(y1),y2) -> f(g(y1),g(y1))) = 1
  phi(g(a()) -> g(b())) = 1
  phi(b() -> a()) = 1

  psi(f(x1,g(x2)) -> f(x1,g(x1))) = 1
  psi(f(g(y1),y2) -> f(g(y1),g(y1))) = 1
  psi(g(a()) -> g(b())) = 1
  psi(b() -> a()) = 1