YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup14.srs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 FlatCCProof (⇔, 0 ms)
↳4 QTRS
↳5 RootLabelingProof (⇔, 0 ms)
↳6 QTRS
↳7 DependencyPairsProof (⇔, 176 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 2 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 236 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 QDP
↳15 UsableRulesProof (⇔, 2 ms)
↳16 QDP
↳17 QDPSizeChangeProof (⇔, 0 ms)
↳18 YES

(0) Obligation:

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

a(a(b(b(b(b(a(a(x)))))))) → a(a(c(c(a(a(b(b(x))))))))
a(a(c(c(x)))) → c(c(c(c(a(a(x))))))
c(c(c(c(c(c(x)))))) → b(b(c(c(b(b(x))))))

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

a(a(b(b(b(b(a(a(x)))))))) → b(b(a(a(c(c(a(a(x))))))))
c(c(a(a(x)))) → a(a(c(c(c(c(x))))))
c(c(c(c(c(c(x)))))) → b(b(c(c(b(b(x))))))

Q is empty.

(3) FlatCCProof (EQUIVALENT transformation)

We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete.

(4) Obligation:

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

a(a(a(b(b(b(b(a(a(x))))))))) → a(b(b(a(a(c(c(a(a(x)))))))))
b(a(a(b(b(b(b(a(a(x))))))))) → b(b(b(a(a(c(c(a(a(x)))))))))
c(a(a(b(b(b(b(a(a(x))))))))) → c(b(b(a(a(c(c(a(a(x)))))))))
a(c(c(a(a(x))))) → a(a(a(c(c(c(c(x)))))))
b(c(c(a(a(x))))) → b(a(a(c(c(c(c(x)))))))
c(c(c(a(a(x))))) → c(a(a(c(c(c(c(x)))))))
a(c(c(c(c(c(c(x))))))) → a(b(b(c(c(b(b(x)))))))
b(c(c(c(c(c(c(x))))))) → b(b(b(c(c(b(b(x)))))))
c(c(c(c(c(c(c(x))))))) → c(b(b(c(c(b(b(x)))))))

Q is empty.

(5) RootLabelingProof (EQUIVALENT transformation)

We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(6) Obligation:

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

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(8) Obligation:

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

A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{a_1}(x))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{b_1}(x)))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{b_1}(x))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{a_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{b_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{a_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{c_1}(c_{b_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → C_{C_1}(c_{b_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 48 less nodes.

(10) Obligation:

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

B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → C_{A_1}(a_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(c_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(c_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{C_1}(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = x1   
POL(A_{C_1}(x1)) = x1   
POL(B_{A_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(C_{A_1}(x1)) = x1   
POL(C_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(a_{c_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = 0   
POL(c_{a_1}(x1)) = x1   
POL(c_{b_1}(x1)) = 0   
POL(c_{c_1}(x1)) = x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))

(12) Obligation:

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

A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → B_{A_1}(x)
B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.

(14) Obligation:

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

B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))))) → c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x)))))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x)))))))

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

(15) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

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

B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)

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

(17) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x))))))) → B_{C_1}(x)
    The graph contains the following edges 1 > 1

(18) YES