YES

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  f(x) -> g(x)
  f(x) -> h(f(x))
  h(f(x)) -> h(g(x))
  g(x) -> h(g(x))

Let C be the following subset of R:

  f(x) -> h(f(x))
  f(x) -> g(x)
  g(x) -> h(g(x))

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

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

Consider the left-linear TRS R:

  f(x) -> h(f(x))
  f(x) -> g(x)
  g(x) -> h(g(x))

Let C be the following subset of R:

  f(x) -> h(f(x))
  f(x) -> g(x)
  g(x) -> h(g(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:

  f(x) -> h(f(x))
  f(x) -> g(x)
  g(x) -> h(g(x))

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

  phi(f(x) -> h(f(x))) = 1
  phi(f(x) -> g(x)) = 3
  phi(g(x) -> h(g(x))) = 2

  psi(f(x) -> h(f(x))) = 1
  psi(f(x) -> g(x)) = 3
  psi(g(x) -> h(g(x))) = 2