YES

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

Consider the left-linear TRS R:

  F(H(x),y) -> G(H(x))
  H(I(x)) -> I(x)
  F(I(x),y) -> G(I(x))

Let C be the following subset of R:

  F(H(x),y) -> G(H(x))
  H(I(x)) -> I(x)
  F(I(x),y) -> G(I(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(H(x),y) -> G(H(x))
  H(I(x)) -> I(x)
  F(I(x),y) -> G(I(x))

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

  phi(F(H(x),y) -> G(H(x))) = 2
  phi(H(I(x)) -> I(x)) = 3
  phi(F(I(x),y) -> G(I(x))) = 1

  psi(F(H(x),y) -> G(H(x))) = 1
  psi(H(I(x)) -> I(x)) = 2
  psi(F(I(x),y) -> G(I(x))) = 2