YES

# Compositional critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  a(b(x)) -> C(x)
  b(c(x)) -> A(x)
  c(a(x)) -> B(x)
  A(C(x)) -> b(x)
  C(B(x)) -> a(x)
  B(A(x)) -> c(x)
  a(a(a(a(x)))) -> A(A(A(x)))
  A(A(A(A(x)))) -> a(a(a(x)))
  b(b(b(b(x)))) -> B(B(B(x)))
  B(B(B(B(x)))) -> b(b(b(x)))
  c(c(c(c(x)))) -> C(C(C(x)))
  C(C(C(C(x)))) -> c(c(c(x)))
  B(a(a(a(x)))) -> c(A(A(A(x))))
  A(A(A(b(x)))) -> a(a(a(C(x))))
  C(b(b(b(x)))) -> a(B(B(B(x))))
  B(B(B(c(x)))) -> b(b(b(A(x))))
  A(c(c(c(x)))) -> b(C(C(C(x))))
  C(C(C(a(x)))) -> c(c(c(B(x))))
  a(A(x)) -> x
  A(a(x)) -> x
  b(B(x)) -> x
  B(b(x)) -> x
  c(C(x)) -> x
  C(c(x)) -> x

Let C be the following subset of R:

  (empty)

The critical pair system CPS(R,C) is:

  c(a(b(x0))) -> c(C(x0))
  c(a(b(x0))) -> B(b(x0))
  a(a(a(a(b(x0))))) -> a(a(a(C(x0))))
  a(a(a(a(b(x0))))) -> A(A(A(b(x0))))
  B(a(a(a(b(x0))))) -> B(a(a(C(x0))))
  B(a(a(a(b(x0))))) -> c(A(A(A(b(x0)))))
  C(C(C(a(b(x0))))) -> C(C(C(C(x0))))
  C(C(C(a(b(x0))))) -> c(c(c(B(b(x0)))))
  A(a(b(x0))) -> A(C(x0))
  A(a(b(x0))) -> b(x0)
  a(b(c(x0))) -> a(A(x0))
  a(b(c(x0))) -> C(c(x0))
  b(b(b(b(c(x0))))) -> b(b(b(A(x0))))
  b(b(b(b(c(x0))))) -> B(B(B(c(x0))))
  A(A(A(b(c(x0))))) -> A(A(A(A(x0))))
  A(A(A(b(c(x0))))) -> a(a(a(C(c(x0)))))
  C(b(b(b(c(x0))))) -> C(b(b(A(x0))))
  C(b(b(b(c(x0))))) -> a(B(B(B(c(x0)))))
  B(b(c(x0))) -> B(A(x0))
  B(b(c(x0))) -> c(x0)
  b(c(a(x0))) -> b(B(x0))
  b(c(a(x0))) -> A(a(x0))
  c(c(c(c(a(x0))))) -> c(c(c(B(x0))))
  c(c(c(c(a(x0))))) -> C(C(C(a(x0))))
  B(B(B(c(a(x0))))) -> B(B(B(B(x0))))
  B(B(B(c(a(x0))))) -> b(b(b(A(a(x0)))))
  A(c(c(c(a(x0))))) -> A(c(c(B(x0))))
  A(c(c(c(a(x0))))) -> b(C(C(C(a(x0)))))
  C(c(a(x0))) -> C(B(x0))
  C(c(a(x0))) -> a(x0)
  B(A(C(x0))) -> B(b(x0))
  B(A(C(x0))) -> c(C(x0))
  A(A(A(A(C(x0))))) -> A(A(A(b(x0))))
  A(A(A(A(C(x0))))) -> a(a(a(C(x0))))
  a(A(C(x0))) -> a(b(x0))
  a(A(C(x0))) -> C(x0)
  A(C(B(x0))) -> A(a(x0))
  A(C(B(x0))) -> b(B(x0))
  C(C(C(C(B(x0))))) -> C(C(C(a(x0))))
  C(C(C(C(B(x0))))) -> c(c(c(B(x0))))
  c(C(B(x0))) -> c(a(x0))
  c(C(B(x0))) -> B(x0)
  C(B(A(x0))) -> C(c(x0))
  C(B(A(x0))) -> a(A(x0))
  B(B(B(B(A(x0))))) -> B(B(B(c(x0))))
  B(B(B(B(A(x0))))) -> b(b(b(A(x0))))
  b(B(A(x0))) -> b(c(x0))
  b(B(A(x0))) -> A(x0)
  c(a(a(a(a(x0))))) -> c(A(A(A(x0))))
  c(a(a(a(a(x0))))) -> B(a(a(a(x0))))
  a(a(a(a(a(x0))))) -> a(A(A(A(x0))))
  a(a(a(a(a(x0))))) -> A(A(A(a(x0))))
  a(a(a(a(a(a(x0)))))) -> a(a(A(A(A(x0)))))
  a(a(a(a(a(a(x0)))))) -> A(A(A(a(a(x0)))))
  a(a(a(a(a(a(a(x0))))))) -> a(a(a(A(A(A(x0))))))
  a(a(a(a(a(a(a(x0))))))) -> A(A(A(a(a(a(x0))))))
  B(a(a(a(a(x0))))) -> B(A(A(A(x0))))
  B(a(a(a(a(x0))))) -> c(A(A(A(a(x0)))))
  B(a(a(a(a(a(x0)))))) -> B(a(A(A(A(x0)))))
  B(a(a(a(a(a(x0)))))) -> c(A(A(A(a(a(x0))))))
  B(a(a(a(a(a(a(x0))))))) -> B(a(a(A(A(A(x0))))))
  B(a(a(a(a(a(a(x0))))))) -> c(A(A(A(a(a(a(x0)))))))
  C(C(C(a(a(a(a(x0))))))) -> C(C(C(A(A(A(x0))))))
  C(C(C(a(a(a(a(x0))))))) -> c(c(c(B(a(a(a(x0)))))))
  A(a(a(a(a(x0))))) -> A(A(A(A(x0))))
  A(a(a(a(a(x0))))) -> a(a(a(x0)))
  B(A(A(A(A(x0))))) -> B(a(a(a(x0))))
  B(A(A(A(A(x0))))) -> c(A(A(A(x0))))
  A(A(A(A(A(x0))))) -> A(a(a(a(x0))))
  A(A(A(A(A(x0))))) -> a(a(a(A(x0))))
  A(A(A(A(A(A(x0)))))) -> A(A(a(a(a(x0)))))
  A(A(A(A(A(A(x0)))))) -> a(a(a(A(A(x0)))))
  A(A(A(A(A(A(A(x0))))))) -> A(A(A(a(a(a(x0))))))
  A(A(A(A(A(A(A(x0))))))) -> a(a(a(A(A(A(x0))))))
  a(A(A(A(A(x0))))) -> a(a(a(a(x0))))
  a(A(A(A(A(x0))))) -> A(A(A(x0)))
  a(b(b(b(b(x0))))) -> a(B(B(B(x0))))
  a(b(b(b(b(x0))))) -> C(b(b(b(x0))))
  b(b(b(b(b(x0))))) -> b(B(B(B(x0))))
  b(b(b(b(b(x0))))) -> B(B(B(b(x0))))
  b(b(b(b(b(b(x0)))))) -> b(b(B(B(B(x0)))))
  b(b(b(b(b(b(x0)))))) -> B(B(B(b(b(x0)))))
  b(b(b(b(b(b(b(x0))))))) -> b(b(b(B(B(B(x0))))))
  b(b(b(b(b(b(b(x0))))))) -> B(B(B(b(b(b(x0))))))
  A(A(A(b(b(b(b(x0))))))) -> A(A(A(B(B(B(x0))))))
  A(A(A(b(b(b(b(x0))))))) -> a(a(a(C(b(b(b(x0)))))))
  C(b(b(b(b(x0))))) -> C(B(B(B(x0))))
  C(b(b(b(b(x0))))) -> a(B(B(B(b(x0)))))
  C(b(b(b(b(b(x0)))))) -> C(b(B(B(B(x0)))))
  C(b(b(b(b(b(x0)))))) -> a(B(B(B(b(b(x0))))))
  C(b(b(b(b(b(b(x0))))))) -> C(b(b(B(B(B(x0))))))
  C(b(b(b(b(b(b(x0))))))) -> a(B(B(B(b(b(b(x0)))))))
  B(b(b(b(b(x0))))) -> B(B(B(B(x0))))
  B(b(b(b(b(x0))))) -> b(b(b(x0)))
  C(B(B(B(B(x0))))) -> C(b(b(b(x0))))
  C(B(B(B(B(x0))))) -> a(B(B(B(x0))))
  B(B(B(B(B(x0))))) -> B(b(b(b(x0))))
  B(B(B(B(B(x0))))) -> b(b(b(B(x0))))
  B(B(B(B(B(B(x0)))))) -> B(B(b(b(b(x0)))))
  B(B(B(B(B(B(x0)))))) -> b(b(b(B(B(x0)))))
  B(B(B(B(B(B(B(x0))))))) -> B(B(B(b(b(b(x0))))))
  B(B(B(B(B(B(B(x0))))))) -> b(b(b(B(B(B(x0))))))
  b(B(B(B(B(x0))))) -> b(b(b(b(x0))))
  b(B(B(B(B(x0))))) -> B(B(B(x0)))
  b(c(c(c(c(x0))))) -> b(C(C(C(x0))))
  b(c(c(c(c(x0))))) -> A(c(c(c(x0))))
  c(c(c(c(c(x0))))) -> c(C(C(C(x0))))
  c(c(c(c(c(x0))))) -> C(C(C(c(x0))))
  c(c(c(c(c(c(x0)))))) -> c(c(C(C(C(x0)))))
  c(c(c(c(c(c(x0)))))) -> C(C(C(c(c(x0)))))
  c(c(c(c(c(c(c(x0))))))) -> c(c(c(C(C(C(x0))))))
  c(c(c(c(c(c(c(x0))))))) -> C(C(C(c(c(c(x0))))))
  B(B(B(c(c(c(c(x0))))))) -> B(B(B(C(C(C(x0))))))
  B(B(B(c(c(c(c(x0))))))) -> b(b(b(A(c(c(c(x0)))))))
  A(c(c(c(c(x0))))) -> A(C(C(C(x0))))
  A(c(c(c(c(x0))))) -> b(C(C(C(c(x0)))))
  A(c(c(c(c(c(x0)))))) -> A(c(C(C(C(x0)))))
  A(c(c(c(c(c(x0)))))) -> b(C(C(C(c(c(x0))))))
  A(c(c(c(c(c(c(x0))))))) -> A(c(c(C(C(C(x0))))))
  A(c(c(c(c(c(c(x0))))))) -> b(C(C(C(c(c(c(x0)))))))
  C(c(c(c(c(x0))))) -> C(C(C(C(x0))))
  C(c(c(c(c(x0))))) -> c(c(c(x0)))
  A(C(C(C(C(x0))))) -> A(c(c(c(x0))))
  A(C(C(C(C(x0))))) -> b(C(C(C(x0))))
  C(C(C(C(C(x0))))) -> C(c(c(c(x0))))
  C(C(C(C(C(x0))))) -> c(c(c(C(x0))))
  C(C(C(C(C(C(x0)))))) -> C(C(c(c(c(x0)))))
  C(C(C(C(C(C(x0)))))) -> c(c(c(C(C(x0)))))
  C(C(C(C(C(C(C(x0))))))) -> C(C(C(c(c(c(x0))))))
  C(C(C(C(C(C(C(x0))))))) -> c(c(c(C(C(C(x0))))))
  c(C(C(C(C(x0))))) -> c(c(c(c(x0))))
  c(C(C(C(C(x0))))) -> C(C(C(x0)))
  C(B(a(a(a(x0))))) -> C(c(A(A(A(x0)))))
  C(B(a(a(a(x0))))) -> a(a(a(a(x0))))
  B(B(B(B(a(a(a(x0))))))) -> B(B(B(c(A(A(A(x0)))))))
  B(B(B(B(a(a(a(x0))))))) -> b(b(b(a(a(a(x0))))))
  b(B(a(a(a(x0))))) -> b(c(A(A(A(x0)))))
  b(B(a(a(a(x0))))) -> a(a(a(x0)))
  B(A(A(A(b(x0))))) -> B(a(a(a(C(x0)))))
  B(A(A(A(b(x0))))) -> c(A(A(b(x0))))
  A(A(A(A(b(x0))))) -> A(a(a(a(C(x0)))))
  A(A(A(A(b(x0))))) -> a(a(a(b(x0))))
  A(A(A(A(A(b(x0)))))) -> A(A(a(a(a(C(x0))))))
  A(A(A(A(A(b(x0)))))) -> a(a(a(A(b(x0)))))
  A(A(A(A(A(A(b(x0))))))) -> A(A(A(a(a(a(C(x0)))))))
  A(A(A(A(A(A(b(x0))))))) -> a(a(a(A(A(b(x0))))))
  a(A(A(A(b(x0))))) -> a(a(a(a(C(x0)))))
  a(A(A(A(b(x0))))) -> A(A(b(x0)))
  A(C(b(b(b(x0))))) -> A(a(B(B(B(x0)))))
  A(C(b(b(b(x0))))) -> b(b(b(b(x0))))
  C(C(C(C(b(b(b(x0))))))) -> C(C(C(a(B(B(B(x0)))))))
  C(C(C(C(b(b(b(x0))))))) -> c(c(c(b(b(b(x0))))))
  c(C(b(b(b(x0))))) -> c(a(B(B(B(x0)))))
  c(C(b(b(b(x0))))) -> b(b(b(x0)))
  C(B(B(B(c(x0))))) -> C(b(b(b(A(x0)))))
  C(B(B(B(c(x0))))) -> a(B(B(c(x0))))
  B(B(B(B(c(x0))))) -> B(b(b(b(A(x0)))))
  B(B(B(B(c(x0))))) -> b(b(b(c(x0))))
  B(B(B(B(B(c(x0)))))) -> B(B(b(b(b(A(x0))))))
  B(B(B(B(B(c(x0)))))) -> b(b(b(B(c(x0)))))
  B(B(B(B(B(B(c(x0))))))) -> B(B(B(b(b(b(A(x0)))))))
  B(B(B(B(B(B(c(x0))))))) -> b(b(b(B(B(c(x0))))))
  b(B(B(B(c(x0))))) -> b(b(b(b(A(x0)))))
  b(B(B(B(c(x0))))) -> B(B(c(x0)))
  B(A(c(c(c(x0))))) -> B(b(C(C(C(x0)))))
  B(A(c(c(c(x0))))) -> c(c(c(c(x0))))
  A(A(A(A(c(c(c(x0))))))) -> A(A(A(b(C(C(C(x0)))))))
  A(A(A(A(c(c(c(x0))))))) -> a(a(a(c(c(c(x0))))))
  a(A(c(c(c(x0))))) -> a(b(C(C(C(x0)))))
  a(A(c(c(c(x0))))) -> c(c(c(x0)))
  A(C(C(C(a(x0))))) -> A(c(c(c(B(x0)))))
  A(C(C(C(a(x0))))) -> b(C(C(a(x0))))
  C(C(C(C(a(x0))))) -> C(c(c(c(B(x0)))))
  C(C(C(C(a(x0))))) -> c(c(c(a(x0))))
  C(C(C(C(C(a(x0)))))) -> C(C(c(c(c(B(x0))))))
  C(C(C(C(C(a(x0)))))) -> c(c(c(C(a(x0)))))
  C(C(C(C(C(C(a(x0))))))) -> C(C(C(c(c(c(B(x0)))))))
  C(C(C(C(C(C(a(x0))))))) -> c(c(c(C(C(a(x0))))))
  c(C(C(C(a(x0))))) -> c(c(c(c(B(x0)))))
  c(C(C(C(a(x0))))) -> C(C(a(x0)))
  c(a(A(x0))) -> c(x0)
  c(a(A(x0))) -> B(A(x0))
  a(a(a(a(A(x0))))) -> a(a(a(x0)))
  a(a(a(a(A(x0))))) -> A(A(A(A(x0))))
  B(a(a(a(A(x0))))) -> B(a(a(x0)))
  B(a(a(a(A(x0))))) -> c(A(A(A(A(x0)))))
  C(C(C(a(A(x0))))) -> C(C(C(x0)))
  C(C(C(a(A(x0))))) -> c(c(c(B(A(x0)))))
  A(a(A(x0))) -> A(x0)
  B(A(a(x0))) -> B(x0)
  B(A(a(x0))) -> c(a(x0))
  A(A(A(A(a(x0))))) -> A(A(A(x0)))
  A(A(A(A(a(x0))))) -> a(a(a(a(x0))))
  a(A(a(x0))) -> a(x0)
  a(b(B(x0))) -> a(x0)
  a(b(B(x0))) -> C(B(x0))
  b(b(b(b(B(x0))))) -> b(b(b(x0)))
  b(b(b(b(B(x0))))) -> B(B(B(B(x0))))
  A(A(A(b(B(x0))))) -> A(A(A(x0)))
  A(A(A(b(B(x0))))) -> a(a(a(C(B(x0)))))
  C(b(b(b(B(x0))))) -> C(b(b(x0)))
  C(b(b(b(B(x0))))) -> a(B(B(B(B(x0)))))
  B(b(B(x0))) -> B(x0)
  C(B(b(x0))) -> C(x0)
  C(B(b(x0))) -> a(b(x0))
  B(B(B(B(b(x0))))) -> B(B(B(x0)))
  B(B(B(B(b(x0))))) -> b(b(b(b(x0))))
  b(B(b(x0))) -> b(x0)
  b(c(C(x0))) -> b(x0)
  b(c(C(x0))) -> A(C(x0))
  c(c(c(c(C(x0))))) -> c(c(c(x0)))
  c(c(c(c(C(x0))))) -> C(C(C(C(x0))))
  B(B(B(c(C(x0))))) -> B(B(B(x0)))
  B(B(B(c(C(x0))))) -> b(b(b(A(C(x0)))))
  A(c(c(c(C(x0))))) -> A(c(c(x0)))
  A(c(c(c(C(x0))))) -> b(C(C(C(C(x0)))))
  C(c(C(x0))) -> C(x0)
  A(C(c(x0))) -> A(x0)
  A(C(c(x0))) -> b(c(x0))
  C(C(C(C(c(x0))))) -> C(C(C(x0)))
  C(C(C(C(c(x0))))) -> c(c(c(c(x0))))
  c(C(c(x0))) -> c(x0)

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.