YES

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

Consider the left-linear TRS R:

  a() -> a'()
  h(x,a'(),y) -> h(x,y,y)
  h(x,y,a'()) -> h(x,y,y)
  g() -> f()
  h(f(),a(),a()) -> h(g(),a(),a())
  h(g(),a(),a()) -> h(f(),a(),a())

Let C be the following subset of R:

  g() -> f()
  h(x,a'(),y) -> h(x,y,y)
  a() -> a'()
  h(x,y,a'()) -> h(x,y,y)
  h(f(),a(),a()) -> h(g(),a(),a())

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 rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  g() -> f()
  h(x,a'(),y) -> h(x,y,y)
  a() -> a'()
  h(x,y,a'()) -> h(x,y,y)
  h(f(),a(),a()) -> h(g(),a(),a())

Let C be the following subset of R:

  (empty)

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

  phi(g() -> f()) = 1
  phi(h(x,a'(),y) -> h(x,y,y)) = 1
  phi(a() -> a'()) = 6
  phi(h(x,y,a'()) -> h(x,y,y)) = 1
  phi(h(f(),a(),a()) -> h(g(),a(),a())) = 2

  psi(g() -> f()) = 1
  psi(h(x,a'(),y) -> h(x,y,y)) = 4
  psi(a() -> a'()) = 3
  psi(h(x,y,a'()) -> h(x,y,y)) = 1
  psi(h(f(),a(),a()) -> h(g(),a(),a())) = 5


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.