YES

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

Consider the left-linear TRS R:

  F(x,y) -> c(y)
  G(x) -> x
  f(x) -> g(x)
  g(x) -> c(x)

Let C be the following subset of R:

  (empty)

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

  phi(F(x,y) -> c(y)) = 1
  phi(G(x) -> x) = 1
  phi(f(x) -> g(x)) = 1
  phi(g(x) -> c(x)) = 1

  psi(F(x,y) -> c(y)) = 1
  psi(G(x) -> x) = 1
  psi(f(x) -> g(x)) = 1
  psi(g(x) -> c(x)) = 1


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)

The TRS R is locally confluent and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Emptiness.

The empty TRS is confluent.