YES

Problem:
 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

Proof:
 Church Rosser Transformation Processor (ac):
  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
  AC critical peaks: joinable
  AC-KBO Processor:
   precedence:
    B > A > C > c > b > a
    weight function:
     w0 = 2
     w(B) = w(A) = w(c) = w(C) = w(a) = w(b) = 1
    problem:
     
    Qed