Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(C(C(x1)))))))))
B1(b(c(c(x1)))) → A2(x1)
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(a(x1))))))))
C2(c(a(a(x1)))) → B2(B(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(a(C(C(x1))))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(C(C(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(C(C(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(A(A(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(x1)))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(x1)))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B2(B(A(A(x1)))) → C2(x1)
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → C1(C(x1))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(A(A(x1)))))))))
B2(B(A(A(x1)))) → C2(c(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(x1))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(c(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(A(A(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(C(C(x1))))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(A(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(x1)))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(C(C(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(B(B(x1)))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(c(B(B(x1))))))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(A(A(x1))))))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(b(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(C(C(x1)))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(x1)))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(B(B(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(x1)))))))
C1(C(B(B(x1)))) → A1(x1)
A2(A(A(A(A(A(b(b(x1)))))))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(x1)))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(B(B(x1))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(A(A(A(A(A(A(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(x1))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
A1(a(b(b(x1)))) → C1(C(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(x1))
A1(a(b(b(x1)))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(C(C(x1))))
A2(A(C(C(x1)))) → B1(b(x1))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(a(B(B(B(B(B(B(x1))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(B(B(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(b(A(A(x1))))))))))
C2(c(a(a(x1)))) → B2(x1)
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(A(A(x1)))
B1(b(c(c(x1)))) → A2(A(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(B(B(B(B(B(B(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(B(B(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(C(C(C(C(C(C(x1)))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(c(A(A(A(A(A(A(x1))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
A2(A(C(C(x1)))) → B1(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(b(C(C(C(C(C(C(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(A(A(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(B(x1))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
C1(C(B(B(x1)))) → A1(a(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(C(C(x1)))))))))
B1(b(c(c(x1)))) → A2(x1)
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(a(x1))))))))
C2(c(a(a(x1)))) → B2(B(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(a(C(C(x1))))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(C(C(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(C(C(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(A(A(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(x1)))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(x1)))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B2(B(A(A(x1)))) → C2(x1)
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → C1(C(x1))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(A(A(x1)))))))))
B2(B(A(A(x1)))) → C2(c(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(x1))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(c(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(A(A(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(C(C(x1))))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(A(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(x1)))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(C(C(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(B(B(x1)))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(c(B(B(x1))))))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(A(A(x1))))))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(b(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(C(C(x1)))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(x1)))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(B(B(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(x1)))))))
C1(C(B(B(x1)))) → A1(x1)
A2(A(A(A(A(A(b(b(x1)))))))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(x1)))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(B(B(x1))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(A(A(A(A(A(A(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(x1))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
A1(a(b(b(x1)))) → C1(C(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(x1))
A1(a(b(b(x1)))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(C(C(x1))))
A2(A(C(C(x1)))) → B1(b(x1))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(a(B(B(B(B(B(B(x1))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(B(B(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(b(A(A(x1))))))))))
C2(c(a(a(x1)))) → B2(x1)
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(A(A(x1)))
B1(b(c(c(x1)))) → A2(A(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(B(B(B(B(B(B(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(B(B(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(C(C(C(C(C(C(x1)))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(c(A(A(A(A(A(A(x1))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
A2(A(C(C(x1)))) → B1(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(b(C(C(C(C(C(C(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(A(A(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(B(x1))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
C1(C(B(B(x1)))) → A1(a(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(C(C(x1)))))))))
B1(b(c(c(x1)))) → A2(x1)
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(a(x1))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(C(C(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(C(C(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(A(A(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(x1)))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(x1)
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(x1)))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B2(B(A(A(x1)))) → C2(x1)
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(B(B(x1)))))
A2(A(A(A(A(A(b(b(x1)))))))) → C1(C(x1))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(x1))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(A(A(x1)))))))))
B2(B(A(A(x1)))) → C2(c(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(x1))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(a(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(x1)))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(B(B(x1)))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(c(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(A(A(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(C(C(x1))))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(A(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(x1)))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(C(C(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(B(B(x1)))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(c(c(B(B(x1))))))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(A(A(x1))))))))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(b(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(C(C(x1)))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(x1)))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(x1))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(B(B(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(c(x1)))))))
C1(C(B(B(x1)))) → A1(x1)
A2(A(A(A(A(A(b(b(x1)))))))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(b(x1)))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(x1)
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(B(B(x1))))))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(x1)))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(A(A(A(A(A(A(x1)))))))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(x1))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(x1)))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(c(c(c(c(x1))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(x1))))
A1(a(b(b(x1)))) → C1(C(x1))
A2(A(A(A(A(A(A(A(x1)))))))) → A1(a(a(a(a(a(x1))))))
C1(C(C(C(C(C(C(C(x1)))))))) → C2(c(x1))
A1(a(b(b(x1)))) → C1(x1)
B2(B(B(B(B(B(B(B(x1)))))))) → B1(b(b(b(b(b(x1))))))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(C(C(x1))))
A2(A(C(C(x1)))) → B1(b(x1))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(a(B(B(B(B(B(B(x1))))))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(B(B(x1))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(b(b(b(A(A(x1))))))))))
C2(c(a(a(x1)))) → B2(x1)
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(x1)))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(x1)))))
B2(B(B(B(B(B(c(c(x1)))))))) → A2(x1)
B2(B(B(B(B(B(c(c(x1)))))))) → B1(A(A(x1)))
B1(b(c(c(x1)))) → A2(A(x1))
B2(B(B(B(B(B(B(B(x1)))))))) → B1(x1)
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(x1))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → A1(B(B(B(B(B(B(x1)))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(x1)))))
C1(C(C(C(C(C(a(a(x1)))))))) → C2(c(c(c(c(c(B(B(x1))))))))
C1(C(b(b(b(b(b(b(b(b(x1)))))))))) → B2(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(C(C(C(C(C(C(x1)))))))
B2(B(a(a(a(a(a(a(a(a(x1)))))))))) → C2(c(A(A(A(A(A(A(x1))))))))
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(x1))
A2(A(C(C(x1)))) → B1(x1)
A2(A(c(c(c(c(c(c(c(c(x1)))))))))) → B1(b(C(C(C(C(C(C(x1))))))))
B2(B(B(B(B(B(c(c(x1)))))))) → B1(b(b(b(b(A(A(x1)))))))
C1(C(C(C(C(C(a(a(x1)))))))) → B2(B(x1))
A1(a(a(a(a(a(a(a(a(a(x1)))))))))) → A2(A(A(A(A(A(x1))))))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(x1)))
C1(C(B(B(x1)))) → A1(a(x1))
The remaining pairs can at least be oriented weakly.

C2(c(a(a(x1)))) → B2(B(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(a(C(C(x1))))))))))
C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))
Used ordering: Polynomial interpretation [25,35]:

POL(C(x1)) = 3 + x_1   
POL(B2(x1)) = 4 + x_1   
POL(c(x1)) = 2 + x_1   
POL(A1(x1)) = 2 + x_1   
POL(B(x1)) = 3 + x_1   
POL(a(x1)) = 2 + x_1   
POL(C2(x1)) = 1 + x_1   
POL(A(x1)) = 3 + x_1   
POL(B1(x1)) = 1 + x_1   
POL(b(x1)) = 2 + x_1   
POL(A2(x1)) = 3 + x_1   
POL(C1(x1)) = 4 + x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

b(b(B(B(x1)))) → x1
A(A(a(a(x1)))) → x1
c(c(C(C(x1)))) → x1
B(B(b(b(x1)))) → x1
C(C(c(c(x1)))) → x1
B(B(B(B(B(B(B(B(x1)))))))) → b(b(b(b(b(b(b(b(x1))))))))
C(C(B(B(x1)))) → a(a(x1))
B(B(A(A(x1)))) → c(c(x1))
c(c(c(c(c(c(c(c(c(c(x1)))))))))) → 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(a(a(a(a(x1)))))))))) → A(A(A(A(A(A(x1))))))
c(c(a(a(x1)))) → B(B(x1))
A(A(A(A(A(A(A(A(x1)))))))) → a(a(a(a(a(a(a(a(x1))))))))
b(b(c(c(x1)))) → 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))))))))))
C(C(C(C(C(C(C(C(x1)))))))) → c(c(c(c(c(c(c(c(x1))))))))
a(a(b(b(x1)))) → C(C(x1))
b(b(b(b(b(b(b(b(b(b(x1)))))))))) → B(B(B(B(B(B(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))))))))
A(A(C(C(x1)))) → 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(A(A(x1)))) → x1



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

C2(c(c(c(c(c(c(c(c(c(x1)))))))))) → C1(C(C(C(C(C(x1))))))
C2(c(a(a(x1)))) → B2(B(x1))
A2(A(A(A(A(A(b(b(x1)))))))) → A1(a(a(a(a(a(a(a(C(C(x1))))))))))
B1(b(b(b(b(b(b(b(b(b(x1)))))))))) → B2(B(B(B(B(B(x1))))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.