YES

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

Consider the left-linear TRS R:

  c(b(x)) -> c(a(x))
  b(b(x)) -> b(b(x))
  b(b(x)) -> c(a(x))
  a(a(x)) -> c(a(x))
  c(c(x)) -> a(c(x))
  b(a(x)) -> c(b(x))
  c(a(x)) -> a(c(x))
  c(c(x)) -> c(a(x))

Let C be the following subset of R:

  c(b(x)) -> c(a(x))
  b(b(x)) -> c(a(x))
  a(a(x)) -> c(a(x))
  c(c(x)) -> a(c(x))
  b(a(x)) -> c(b(x))
  c(a(x)) -> a(c(x))
  c(c(x)) -> c(a(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:

  c(b(x)) -> c(a(x))
  b(b(x)) -> c(a(x))
  a(a(x)) -> c(a(x))
  c(c(x)) -> a(c(x))
  b(a(x)) -> c(b(x))
  c(a(x)) -> a(c(x))
  c(c(x)) -> c(a(x))

Let C be the following subset of R:

  (empty)

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

  phi(c(b(x)) -> c(a(x))) = 5
  phi(b(b(x)) -> c(a(x))) = 12
  phi(a(a(x)) -> c(a(x))) = 5
  phi(c(c(x)) -> a(c(x))) = 2
  phi(b(a(x)) -> c(b(x))) = 9
  phi(c(a(x)) -> a(c(x))) = 1
  phi(c(c(x)) -> c(a(x))) = 10

  psi(c(b(x)) -> c(a(x))) = 8
  psi(b(b(x)) -> c(a(x))) = 12
  psi(a(a(x)) -> c(a(x))) = 3
  psi(c(c(x)) -> a(c(x))) = 7
  psi(b(a(x)) -> c(b(x))) = 11
  psi(c(a(x)) -> a(c(x))) = 4
  psi(c(c(x)) -> c(a(x))) = 6


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.