YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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.

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

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:

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

Let C be the following subset of R:

  (empty)

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

  phi(h(x) -> g(x)) = 8
  phi(g(x) -> h(x)) = 2
  phi(f(x,y) -> f(y,x)) = 4
  phi(f(g(x),h(y)) -> f(x,y)) = 8
  phi(f(h(x),h(y)) -> f(y,x)) = 8

  psi(h(x) -> g(x)) = 9
  psi(g(x) -> h(x)) = 3
  psi(f(x,y) -> f(y,x)) = 5
  psi(f(g(x),h(y)) -> f(x,y)) = 10
  psi(f(h(x),h(y)) -> f(y,x)) = 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.