YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup12.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 QTRSRRRProof (⇔, 118 ms)
↳8 QTRS
↳9 DependencyPairsProof (⇔, 188 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 QDP
↳13 UsableRulesProof (⇔, 36 ms)
↳14 QDP
↳15 QDPOrderProof (⇔, 167 ms)
↳16 QDP
↳17 DependencyGraphProof (⇔, 0 ms)
↳18 QDP
↳19 QDPOrderProof (⇔, 99 ms)
↳20 QDP
↳21 QDPOrderProof (⇔, 99 ms)
↳22 QDP
↳23 DependencyGraphProof (⇔, 0 ms)
↳24 QDP
↳25 QDPOrderProof (⇔, 61 ms)
↳26 QDP
↳27 DependencyGraphProof (⇔, 0 ms)
↳28 QDP
↳29 QDPOrderProof (⇔, 3990 ms)
↳30 QDP
↳31 PisEmptyProof (⇔, 0 ms)
↳32 YES

(0) Obligation:

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

b(b(d(d(b(b(x)))))) → c(c(d(d(b(b(x))))))
b(b(a(a(c(c(x)))))) → b(b(c(c(x))))
a(a(d(d(x)))) → d(d(c(c(x))))
b(b(b(b(b(b(x)))))) → a(a(b(b(c(c(x))))))
d(d(c(c(x)))) → b(b(d(d(x))))
d(d(c(c(x)))) → d(d(b(b(d(d(x))))))
d(d(a(a(c(c(x)))))) → b(b(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:

b(b(d(d(b(b(x)))))) → b(b(d(d(c(c(x))))))
c(c(a(a(b(b(x)))))) → c(c(b(b(x))))
d(d(a(a(x)))) → c(c(d(d(x))))
b(b(b(b(b(b(x)))))) → c(c(b(b(a(a(x))))))
c(c(d(d(x)))) → d(d(b(b(x))))
c(c(d(d(x)))) → d(d(b(b(d(d(x))))))
c(c(a(a(d(d(x)))))) → b(b(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:

b(b(d(d(b(b(x)))))) → b(b(d(d(c(c(x))))))
c(c(a(a(b(b(x)))))) → c(c(b(b(x))))
b(d(d(a(a(x))))) → b(c(c(d(d(x)))))
d(d(d(a(a(x))))) → d(c(c(d(d(x)))))
c(d(d(a(a(x))))) → c(c(c(d(d(x)))))
a(d(d(a(a(x))))) → a(c(c(d(d(x)))))
b(b(b(b(b(b(b(x))))))) → b(c(c(b(b(a(a(x)))))))
d(b(b(b(b(b(b(x))))))) → d(c(c(b(b(a(a(x)))))))
c(b(b(b(b(b(b(x))))))) → c(c(c(b(b(a(a(x)))))))
a(b(b(b(b(b(b(x))))))) → a(c(c(b(b(a(a(x)))))))
b(c(c(d(d(x))))) → b(d(d(b(b(x)))))
d(c(c(d(d(x))))) → d(d(d(b(b(x)))))
c(c(c(d(d(x))))) → c(d(d(b(b(x)))))
a(c(c(d(d(x))))) → a(d(d(b(b(x)))))
b(c(c(d(d(x))))) → b(d(d(b(b(d(d(x)))))))
d(c(c(d(d(x))))) → d(d(d(b(b(d(d(x)))))))
c(c(c(d(d(x))))) → c(d(d(b(b(d(d(x)))))))
a(c(c(d(d(x))))) → a(d(d(b(b(d(d(x)))))))
b(c(c(a(a(d(d(x))))))) → b(b(b(b(b(x)))))
d(c(c(a(a(d(d(x))))))) → d(b(b(b(b(x)))))
c(c(c(a(a(d(d(x))))))) → c(b(b(b(b(x)))))
a(c(c(a(a(d(d(x))))))) → a(b(b(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:

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{d_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{d_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(a_{c_1}(x1)) = 1 + x1   
POL(a_{d_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = 1 + x1   
POL(b_{c_1}(x1)) = x1   
POL(b_{d_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{b_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1 + x1   
POL(c_{d_1}(x1)) = 1 + x1   
POL(d_{a_1}(x1)) = 2 + x1   
POL(d_{b_1}(x1)) = 1 + x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x))))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{d_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{d_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))) → c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{d_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{c_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x)))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → d_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{a_1}(x))))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))


(8) Obligation:

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

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

Q is empty.

(9) DependencyPairsProof (EQUIVALENT transformation)

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

(10) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{D_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x)))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{D_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{D_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x)))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{D_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{D_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x)))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → D_{D_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → D_{C_1}(c_{c_1}(c_{a_1}(x)))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → C_{C_1}(c_{a_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → D_{D_1}(d_{a_1}(x))
D_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
D_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → D_{D_1}(d_{a_1}(x))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → D_{D_1}(d_{a_1}(x))
A_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
A_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
A_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
A_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → D_{D_1}(d_{a_1}(x))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → C_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{d_1}(x))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → A_{D_1}(x)
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(x))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → A_{C_1}(x)
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))
C_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{d_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → D_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{d_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{c_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{D_1}(d_{d_1}(d_{b_1}(x)))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(d_{d_1}(d_{d_1}(x)))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{D_1}(d_{d_1}(d_{c_1}(x)))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → A_{D_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → D_{D_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
A_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)

The TRS R consists of the following rules:

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 110 less nodes.

(12) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → D_{C_1}(c_{c_1}(c_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
a_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{d_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x)))))))
c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))) → c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
a_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → a_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

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

(13) 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.

(14) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → D_{C_1}(c_{c_1}(c_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(15) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{b_1}(b_{d_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{b_1}(b_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → B_{C_1}(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{B_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(B_{D_1}(x1)) = 0   
POL(C_{C_1}(x1)) = x1   
POL(C_{D_1}(x1)) = 0   
POL(D_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{d_1}(x1)) = x1   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(b_{d_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{c_1}(x1)) = x1   
POL(c_{d_1}(x1)) = x1   
POL(d_{a_1}(x1)) = 0   
POL(d_{b_1}(x1)) = x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = x1   

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

c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

(16) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → D_{C_1}(c_{c_1}(c_{a_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(19) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{B_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(B_{D_1}(x1)) = x1   
POL(C_{C_1}(x1)) = x1   
POL(C_{D_1}(x1)) = 0   
POL(D_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{d_1}(x1)) = x1   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(b_{d_1}(x1)) = x1   
POL(c_{a_1}(x1)) = x1   
POL(c_{c_1}(x1)) = x1   
POL(c_{d_1}(x1)) = x1   
POL(d_{a_1}(x1)) = 0   
POL(d_{b_1}(x1)) = x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = x1   

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

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

(20) Obligation:

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

D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(21) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{C_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{D_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → C_{D_1}(d_{d_1}(d_{a_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{B_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(B_{D_1}(x1)) = x1   
POL(C_{C_1}(x1)) = x1   
POL(C_{D_1}(x1)) = x1   
POL(D_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{d_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(b_{d_1}(x1)) = x1   
POL(c_{a_1}(x1)) = x1   
POL(c_{c_1}(x1)) = x1   
POL(c_{d_1}(x1)) = x1   
POL(d_{a_1}(x1)) = x1   
POL(d_{b_1}(x1)) = x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = x1   

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

b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

(22) Obligation:

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

D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{D_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{D_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{D_1}(d_{d_1}(d_{a_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(24) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(25) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → C_{C_1}(c_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(c_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → C_{C_1}(x)
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{b_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
B_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
C_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{c_1}(x))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{C_1}(x)
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{d_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(x))))
D_{C_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{a_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{B_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(C_{C_1}(x1)) = x1   
POL(D_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{d_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = 1 + x1   
POL(b_{c_1}(x1)) = x1   
POL(b_{d_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 1 + x1   
POL(c_{c_1}(x1)) = 1 + x1   
POL(c_{d_1}(x1)) = 1 + x1   
POL(d_{a_1}(x1)) = x1   
POL(d_{b_1}(x1)) = 1 + x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = x1   

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

c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

(26) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → D_{C_1}(c_{c_1}(c_{d_1}(x)))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → D_{C_1}(c_{c_1}(c_{c_1}(x)))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(28) Obligation:

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

B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))

The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(29) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
B_{B_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → B_{B_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(B_{B_1}(x1)) = 0A +
[0A,0A,0A]
·x1

POL(b_{d_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I0A\
|0A-I-I|
\-I-I-I/
·x1

POL(d_{d_1}(x1)) =
/0A\
|0A|
\0A/
 +
/-I-I0A\
|0A-I-I|
\-I0A-I/
·x1

POL(d_{b_1}(x1)) =
/0A\
|0A|
\-I/
 +
/0A0A0A\
|0A1A0A|
\-I-I0A/
·x1

POL(b_{b_1}(x1)) =
/0A\
|-I|
\-I/
 +
/-I-I0A\
|-I-I0A|
\-I-I-I/
·x1

POL(b_{c_1}(x1)) =
/1A\
|0A|
\0A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(d_{c_1}(x1)) =
/1A\
|0A|
\0A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(c_{c_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I0A\
|-I0A0A|
\0A-I0A/
·x1

POL(c_{d_1}(x1)) =
/0A\
|0A|
\0A/
 +
/-I-I-I\
|0A0A0A|
\0A0A0A/
·x1

POL(d_{a_1}(x1)) =
/-I\
|1A|
\1A/
 +
/0A0A0A\
|0A0A0A|
\0A0A-I/
·x1

POL(c_{a_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|-I-I-I|
\0A0A0A/
·x1

POL(a_{a_1}(x1)) =
/-I\
|-I|
\-I/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

POL(a_{d_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

POL(b_{a_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

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

d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))

(30) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

b_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
c_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → c_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
b_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x))))))
b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{b_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{b_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{d_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{d_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{c_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{c_1}(x)))))))
d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x))))) → d_{d_1}(d_{d_1}(d_{b_1}(b_{b_1}(b_{d_1}(d_{d_1}(d_{a_1}(x)))))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{b_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{d_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x)))))
d_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(d_{d_1}(d_{c_1}(x))))))) → d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x)))))
d_{d_1}(d_{d_1}(d_{a_1}(a_{a_1}(a_{a_1}(x))))) → d_{c_1}(c_{c_1}(c_{d_1}(d_{d_1}(d_{a_1}(x)))))

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

(31) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(32) YES