YES

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

Consider the left-linear TRS R:

  g(a(),y) -> y
  f(x,a()) -> f(x,g(x,b()))
  g(h(x),y) -> g(x,h(y))

Let C be the following subset of R:

  g(a(),y) -> y
  f(x,a()) -> f(x,g(x,b()))
  g(h(x),y) -> g(x,h(y))

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:

  g(a(),y) -> y
  f(x,a()) -> f(x,g(x,b()))
  g(h(x),y) -> g(x,h(y))

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

  phi(g(a(),y) -> y) = 1
  phi(f(x,a()) -> f(x,g(x,b()))) = 1
  phi(g(h(x),y) -> g(x,h(y))) = 1

  psi(g(a(),y) -> y) = 1
  psi(f(x,a()) -> f(x,g(x,b()))) = 1
  psi(g(h(x),y) -> g(x,h(y))) = 1