YES

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

Consider the left-linear TRS R:

  F(c(x)) -> G(x)
  G(x) -> F(x)
  c(x) -> x

Let C be the following subset of R:

  (empty)

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

  phi(F(c(x)) -> G(x)) = 2
  phi(G(x) -> F(x)) = 1
  phi(c(x) -> x) = 2

  psi(F(c(x)) -> G(x)) = 2
  psi(G(x) -> F(x)) = 1
  psi(c(x) -> x) = 2


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.