YES

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  H(F(x,y)) -> F(H(R(x)),y)
  F(x,K(y,z)) -> G(P(y),Q(z,x))
  H(Q(x,y)) -> Q(x,H(R(y)))
  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(y))

Let C be the following subset of R:

  (empty)

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

  phi(H(F(x,y)) -> F(H(R(x)),y)) = 3
  phi(F(x,K(y,z)) -> G(P(y),Q(z,x))) = 2
  phi(H(Q(x,y)) -> Q(x,H(R(y)))) = 1
  phi(Q(x,H(R(y))) -> H(Q(x,y))) = 1
  phi(H(G(x,y)) -> G(x,H(y))) = 1

  psi(H(F(x,y)) -> F(H(R(x)),y)) = 3
  psi(F(x,K(y,z)) -> G(P(y),Q(z,x))) = 2
  psi(H(Q(x,y)) -> Q(x,H(R(y)))) = 1
  psi(Q(x,H(R(y))) -> H(Q(x,y))) = 1
  psi(H(G(x,y)) -> G(x,H(y))) = 3


Therefore, the confluence of R follows from that of C.

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

Consider the left-linear TRS R:

  (empty)

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.