YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_06_SRS/secr5.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 (⇔, 69 ms)
↳8 QTRS
↳9 DependencyPairsProof (⇔, 67 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 520 ms)
↳14 QDP
↳15 DependencyGraphProof (⇔, 0 ms)
↳16 AND
    ↳17 QDP
        ↳18 UsableRulesProof (⇔, 0 ms)
        ↳19 QDP
        ↳20 QDPOrderProof (⇔, 0 ms)
        ↳21 QDP
        ↳22 PisEmptyProof (⇔, 0 ms)
        ↳23 YES
    ↳24 QDP
        ↳25 QDPOrderProof (⇔, 70 ms)
        ↳26 QDP
        ↳27 QDPOrderProof (⇔, 257 ms)
        ↳28 QDP
        ↳29 QDPOrderProof (⇔, 3878 ms)
        ↳30 QDP
        ↳31 QDPOrderProof (⇔, 35 ms)
        ↳32 QDP
        ↳33 QDPOrderProof (⇔, 32 ms)
        ↳34 QDP
        ↳35 PisEmptyProof (⇔, 0 ms)
        ↳36 YES

(0) Obligation:

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

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

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

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

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


(8) Obligation:

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

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{B_1}(b_{c_1}(c_{c_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → B_{C_1}(c_{c_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → B_{C_1}(c_{b_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → B_{C_1}(c_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{A_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{B_1}(b_{c_1}(c_{d_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → B_{C_1}(c_{d_1}(x))
A_{C_1}(c_{a_1}(a_{c_1}(x))) → A_{A_1}(a_{c_1}(x))
A_{C_1}(c_{a_1}(a_{b_1}(x))) → A_{A_1}(a_{b_1}(x))
A_{C_1}(c_{a_1}(a_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{C_1}(c_{a_1}(a_{d_1}(x))) → A_{A_1}(a_{d_1}(x))
D_{C_1}(c_{a_1}(a_{c_1}(x))) → D_{A_1}(a_{c_1}(x))
D_{C_1}(c_{a_1}(a_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{C_1}(c_{a_1}(a_{a_1}(x))) → D_{A_1}(a_{a_1}(x))
D_{C_1}(c_{a_1}(a_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
C_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
C_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
D_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
D_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{C_1}(c_{c_1}(x)) → C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
C_{C_1}(c_{c_1}(x)) → A_{A_1}(a_{b_1}(b_{c_1}(x)))
C_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
C_{C_1}(c_{b_1}(x)) → C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
C_{C_1}(c_{b_1}(x)) → A_{A_1}(a_{b_1}(b_{b_1}(x)))
C_{C_1}(c_{a_1}(x)) → C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{C_1}(c_{a_1}(x)) → A_{A_1}(a_{b_1}(b_{a_1}(x)))
A_{C_1}(c_{c_1}(x)) → A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
A_{C_1}(c_{c_1}(x)) → A_{A_1}(a_{b_1}(b_{c_1}(x)))
A_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
A_{C_1}(c_{b_1}(x)) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
A_{C_1}(c_{b_1}(x)) → A_{A_1}(a_{b_1}(b_{b_1}(x)))
A_{C_1}(c_{a_1}(x)) → A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
A_{C_1}(c_{a_1}(x)) → A_{A_1}(a_{b_1}(b_{a_1}(x)))
D_{C_1}(c_{c_1}(x)) → D_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
D_{C_1}(c_{c_1}(x)) → A_{A_1}(a_{b_1}(b_{c_1}(x)))
D_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
D_{C_1}(c_{b_1}(x)) → D_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
D_{C_1}(c_{b_1}(x)) → A_{A_1}(a_{b_1}(b_{b_1}(x)))
D_{C_1}(c_{a_1}(x)) → D_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
D_{C_1}(c_{a_1}(x)) → A_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{C_1}(c_{c_1}(x)) → A_{A_1}(a_{b_1}(b_{c_1}(x)))
B_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{b_1}(x)) → A_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{C_1}(c_{a_1}(x)) → A_{A_1}(a_{b_1}(b_{a_1}(x)))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{C_1}(c_{d_1}(d_{d_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
B_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
B_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
B_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))

The TRS R consists of the following rules:

c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x))))
c_{c_1}(c_{a_1}(a_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
c_{c_1}(c_{a_1}(a_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{c_1}(c_{a_1}(a_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{c_1}(c_{a_1}(a_{d_1}(x))) → c_{a_1}(a_{d_1}(x))
a_{c_1}(c_{a_1}(a_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
a_{c_1}(c_{a_1}(a_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{c_1}(c_{a_1}(a_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{c_1}(c_{a_1}(a_{d_1}(x))) → a_{a_1}(a_{d_1}(x))
d_{c_1}(c_{a_1}(a_{c_1}(x))) → d_{a_1}(a_{c_1}(x))
d_{c_1}(c_{a_1}(a_{b_1}(x))) → d_{a_1}(a_{b_1}(x))
d_{c_1}(c_{a_1}(a_{a_1}(x))) → d_{a_1}(a_{a_1}(x))
d_{c_1}(c_{a_1}(a_{d_1}(x))) → d_{a_1}(a_{d_1}(x))
b_{c_1}(c_{a_1}(a_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
b_{c_1}(c_{a_1}(a_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{c_1}(c_{a_1}(a_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{c_1}(c_{a_1}(a_{d_1}(x))) → b_{a_1}(a_{d_1}(x))
c_{a_1}(a_{d_1}(d_{c_1}(x))) → c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))))
c_{a_1}(a_{d_1}(d_{a_1}(x))) → c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))))
c_{a_1}(a_{d_1}(d_{d_1}(x))) → c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))))
a_{a_1}(a_{d_1}(d_{c_1}(x))) → a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))))
a_{a_1}(a_{d_1}(d_{a_1}(x))) → a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))))
a_{a_1}(a_{d_1}(d_{d_1}(x))) → a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))))
d_{a_1}(a_{d_1}(d_{c_1}(x))) → d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))))
d_{a_1}(a_{d_1}(d_{a_1}(x))) → d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))))
d_{a_1}(a_{d_1}(d_{d_1}(x))) → d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))))
c_{c_1}(c_{c_1}(x)) → c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
c_{c_1}(c_{b_1}(x)) → c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
c_{c_1}(c_{a_1}(x)) → c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
a_{c_1}(c_{c_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
a_{c_1}(c_{b_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
a_{c_1}(c_{a_1}(x)) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
d_{c_1}(c_{c_1}(x)) → d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
d_{c_1}(c_{b_1}(x)) → d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
d_{c_1}(c_{a_1}(x)) → d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{c_1}(c_{c_1}(x)) → b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x))))
b_{c_1}(c_{b_1}(x)) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{c_1}(c_{a_1}(x)) → b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
c_{c_1}(c_{d_1}(d_{c_1}(x))) → c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
c_{c_1}(c_{d_1}(d_{a_1}(x))) → c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
c_{c_1}(c_{d_1}(d_{d_1}(x))) → c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
a_{c_1}(c_{d_1}(d_{c_1}(x))) → a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
a_{c_1}(c_{d_1}(d_{a_1}(x))) → a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
a_{c_1}(c_{d_1}(d_{d_1}(x))) → a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
d_{c_1}(c_{d_1}(d_{c_1}(x))) → d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
d_{c_1}(c_{d_1}(d_{a_1}(x))) → d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
d_{c_1}(c_{d_1}(d_{d_1}(x))) → d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
b_{c_1}(c_{d_1}(d_{c_1}(x))) → b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
b_{c_1}(c_{d_1}(d_{a_1}(x))) → b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
b_{c_1}(c_{d_1}(d_{d_1}(x))) → b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_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 23 less nodes.

(12) Obligation:

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

C_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{a_1}(a_{c_1}(x))) → D_{A_1}(a_{c_1}(x))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{C_1}(c_{a_1}(a_{a_1}(x))) → D_{A_1}(a_{a_1}(x))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{C_1}(c_{a_1}(a_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{B_1}(b_{c_1}(c_{c_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → B_{C_1}(c_{c_1}(x))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{a_1}(a_{c_1}(x))) → A_{A_1}(a_{c_1}(x))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{a_1}(a_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(x)
C_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{a_1}(a_{d_1}(x))) → A_{A_1}(a_{d_1}(x))
A_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
B_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → B_{C_1}(c_{b_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → B_{C_1}(c_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{A_1}(x)
C_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x))))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{B_1}(b_{c_1}(c_{d_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → B_{C_1}(c_{d_1}(x))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
D_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{C_1}(c_{d_1}(d_{d_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
B_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → B_{C_1}(c_{a_1}(a_{c_1}(x)))
B_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
B_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
B_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{C_1}(c_{a_1}(a_{c_1}(x)))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x))))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x))))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → B_{C_1}(c_{a_1}(a_{a_1}(x)))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
D_{A_1}(a_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
D_{A_1}(a_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x))))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → B_{C_1}(c_{a_1}(a_{d_1}(x)))
D_{A_1}(a_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{c_1}(x))
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{C_1}(x)
A_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{C_1}(c_{a_1}(a_{a_1}(x)))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(x)
A_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{C_1}(c_{a_1}(a_{d_1}(x)))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = x1   
POL(A_{C_1}(x1)) = x1   
POL(B_{C_1}(x1)) = x1   
POL(C_{A_1}(x1)) = x1   
POL(C_{B_1}(x1)) = x1   
POL(C_{C_1}(x1)) = x1   
POL(D_{A_1}(x1)) = x1   
POL(D_{C_1}(x1)) = x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = x1   
POL(a_{d_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = 0   
POL(b_{b_1}(x1)) = 0   
POL(b_{c_1}(x1)) = x1   
POL(c_{a_1}(x1)) = x1   
POL(c_{b_1}(x1)) = 0   
POL(c_{c_1}(x1)) = x1   
POL(c_{d_1}(x1)) = 1 + x1   
POL(d_{a_1}(x1)) = x1   
POL(d_{c_1}(x1)) = x1   
POL(d_{d_1}(x1)) = 1 + x1   

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

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

(14) Obligation:

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

C_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
B_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
D_{C_1}(c_{a_1}(a_{c_1}(x))) → D_{A_1}(a_{c_1}(x))
D_{C_1}(c_{a_1}(a_{a_1}(x))) → D_{A_1}(a_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x))))
C_{C_1}(c_{d_1}(d_{c_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
D_{C_1}(c_{a_1}(a_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{B_1}(b_{c_1}(c_{c_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → B_{C_1}(c_{c_1}(x))
A_{C_1}(c_{a_1}(a_{c_1}(x))) → A_{A_1}(a_{c_1}(x))
D_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
D_{C_1}(c_{d_1}(d_{c_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
A_{C_1}(c_{a_1}(a_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{C_1}(x)
A_{C_1}(c_{a_1}(a_{d_1}(x))) → A_{A_1}(a_{d_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x))))
D_{C_1}(c_{d_1}(d_{a_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{c_1}(x)) → B_{C_1}(x)
A_{C_1}(c_{d_1}(d_{c_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x)))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → B_{C_1}(c_{b_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x))))
D_{C_1}(c_{d_1}(d_{d_1}(x))) → D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → B_{C_1}(c_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{A_1}(x)
C_{C_1}(c_{d_1}(d_{a_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{B_1}(b_{c_1}(c_{d_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → B_{C_1}(c_{d_1}(x))
C_{C_1}(c_{d_1}(d_{d_1}(x))) → C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))
A_{C_1}(c_{d_1}(d_{a_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x)))))
A_{C_1}(c_{d_1}(d_{d_1}(x))) → A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x)))))

The TRS R consists of the following rules:

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

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes.

(16) Complex Obligation (AND)

(17) Obligation:

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

B_{C_1}(c_{c_1}(x)) → B_{C_1}(x)

The TRS R consists of the following rules:

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

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

(18) 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.

(19) Obligation:

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

B_{C_1}(c_{c_1}(x)) → B_{C_1}(x)

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

(20) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(B_{C_1}(x1)) = x1   
POL(c_{c_1}(x1)) = 1 + x1   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(21) Obligation:

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

(22) PisEmptyProof (EQUIVALENT transformation)

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

(23) YES

(24) Obligation:

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → C_{B_1}(b_{c_1}(c_{c_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{B_1}(b_{c_1}(c_{d_1}(x)))

The TRS R consists of the following rules:

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


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

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

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

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

(26) Obligation:

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x)))) → C_{B_1}(b_{c_1}(c_{d_1}(x)))

The TRS R consists of the following rules:

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

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

(27) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(C_{B_1}(x1)) = 2·x1   
POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = 2·x1   
POL(a_{c_1}(x1)) = 3 + x1   
POL(a_{d_1}(x1)) = 4   
POL(b_{a_1}(x1)) = 0   
POL(b_{b_1}(x1)) = 0   
POL(b_{c_1}(x1)) = 2·x1   
POL(c_{a_1}(x1)) = 2·x1   
POL(c_{b_1}(x1)) = 0   
POL(c_{c_1}(x1)) = 4·x1   
POL(c_{d_1}(x1)) = 5   
POL(d_{a_1}(x1)) = 0   
POL(d_{c_1}(x1)) = 0   
POL(d_{d_1}(x1)) = 0   

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

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

(28) Obligation:

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))

The TRS R consists of the following rules:

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


C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → C_{B_1}(b_{c_1}(c_{a_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

POL(b_{c_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I-I\
|-I0A-I|
\0A0A0A/
·x1

POL(c_{a_1}(x1)) =
/0A\
|1A|
\1A/
 +
/0A0A0A\
|-I0A0A|
\1A0A0A/
·x1

POL(a_{b_1}(x1)) =
/0A\
|-I|
\1A/
 +
/0A-I-I\
|1A0A0A|
\0A0A0A/
·x1

POL(c_{b_1}(x1)) =
/0A\
|1A|
\1A/
 +
/0A-I-I\
|0A0A0A|
\1A0A0A/
·x1

POL(a_{a_1}(x1)) =
/1A\
|-I|
\0A/
 +
/1A0A0A\
|-I0A0A|
\-I0A0A/
·x1

POL(a_{c_1}(x1)) =
/0A\
|1A|
\0A/
 +
/0A-I1A\
|0A1A1A|
\1A0A0A/
·x1

POL(c_{c_1}(x1)) =
/0A\
|1A|
\1A/
 +
/0A0A-I\
|1A0A0A|
\1A1A0A/
·x1

POL(a_{d_1}(x1)) =
/0A\
|0A|
\1A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(c_{d_1}(x1)) =
/0A\
|1A|
\1A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

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

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

POL(d_{c_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

POL(d_{a_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A1A\
|0A1A0A|
\0A1A-I/
·x1

POL(d_{d_1}(x1)) =
/0A\
|0A|
\-I/
 +
/-I-I0A\
|-I0A0A|
\-I-I-I/
·x1

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

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

(30) Obligation:

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(x)

The TRS R consists of the following rules:

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

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

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

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

(32) Obligation:

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

C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → C_{B_1}(b_{c_1}(c_{b_1}(x)))

The TRS R consists of the following rules:

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

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

(33) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(C_{B_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 0   
POL(a_{b_1}(x1)) = 1   
POL(a_{c_1}(x1)) = 0   
POL(a_{d_1}(x1)) = 0   
POL(b_{a_1}(x1)) = 0   
POL(b_{b_1}(x1)) = x1   
POL(b_{c_1}(x1)) = x1   
POL(c_{a_1}(x1)) = x1   
POL(c_{b_1}(x1)) = 0   
POL(c_{c_1}(x1)) = x1   
POL(c_{d_1}(x1)) = 0   
POL(d_{a_1}(x1)) = 0   
POL(d_{c_1}(x1)) = 0   
POL(d_{d_1}(x1)) = 0   

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

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

(34) Obligation:

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

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

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) YES