YES(?,O(n^1))

Problem:
 a(a(b(b(x1)))) -> C(C(x1))
 b(b(c(c(x1)))) -> A(A(x1))
 c(c(a(a(x1)))) -> B(B(x1))
 A(A(C(C(x1)))) -> b(b(x1))
 C(C(B(B(x1)))) -> a(a(x1))
 B(B(A(A(x1)))) -> c(c(x1))
 a(a(a(a(a(a(a(a(a(a(x1)))))))))) -> A(A(A(A(A(A(x1))))))
 A(A(A(A(A(A(A(A(x1)))))))) -> a(a(a(a(a(a(a(a(x1))))))))
 b(b(b(b(b(b(b(b(b(b(x1)))))))))) -> B(B(B(B(B(B(x1))))))
 B(B(B(B(B(B(B(B(x1)))))))) -> b(b(b(b(b(b(b(b(x1))))))))
 c(c(c(c(c(c(c(c(c(c(x1)))))))))) -> C(C(C(C(C(C(x1))))))
 C(C(C(C(C(C(C(C(x1)))))))) -> c(c(c(c(c(c(c(c(x1))))))))
 B(B(a(a(a(a(a(a(a(a(x1)))))))))) -> c(c(A(A(A(A(A(A(x1))))))))
 A(A(A(A(A(A(b(b(x1)))))))) -> a(a(a(a(a(a(a(a(C(C(x1))))))))))
 C(C(b(b(b(b(b(b(b(b(x1)))))))))) -> a(a(B(B(B(B(B(B(x1))))))))
 B(B(B(B(B(B(c(c(x1)))))))) -> b(b(b(b(b(b(b(b(A(A(x1))))))))))
 A(A(c(c(c(c(c(c(c(c(x1)))))))))) -> b(b(C(C(C(C(C(C(x1))))))))
 C(C(C(C(C(C(a(a(x1)))))))) -> c(c(c(c(c(c(c(c(B(B(x1))))))))))
 a(a(A(A(x1)))) -> x1
 A(A(a(a(x1)))) -> x1
 b(b(B(B(x1)))) -> x1
 B(B(b(b(x1)))) -> x1
 c(c(C(C(x1)))) -> x1
 C(C(c(c(x1)))) -> x1

Proof:
 Bounds Processor:
  bound: 0
  enrichment: match
  automaton:
   final states: {7,6,5,4,3,2}
   transitions:
    a0(1) -> 2*
    b0(1) -> 3*
    C0(1) -> 6*
    c0(1) -> 4*
    A0(1) -> 5*
    B0(1) -> 7*
    f120() -> 1*
  problem:
   
  Qed