YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup13.srs

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

0 QTRS
↳1 RootLabelingProof (⇔, 0 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 4 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 58 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 1349 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 4526 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 TRUE

(0) Obligation:

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

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

Q is empty.

(1) 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.

(2) Obligation:

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

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

Q is empty.

(3) 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)) = 1 + x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_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   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(4) Obligation:

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

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

Q is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

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

A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(x)
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x))))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))) → B_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x))))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))) → B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → B_{B_1}(b_{a_1}(x))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → B_{B_1}(b_{b_1}(x))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → B_{B_1}(x)
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → B_{B_1}(b_{c_1}(x))

The TRS R consists of the following rules:

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

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 23 less nodes.

(8) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(9) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x)))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A_{A_1}(x1) ) = 2x1

POL( a_{b_1}(x1) ) = x1

POL( b_{b_1}(x1) ) = max{0, x1 - 1}

POL( b_{a_1}(x1) ) = x1 + 2

POL( a_{a_1}(x1) ) = x1

POL( b_{c_1}(x1) ) = x1 + 2

POL( c_{c_1}(x1) ) = x1 + 2

POL( c_{a_1}(x1) ) = x1

POL( c_{b_1}(x1) ) = x1


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

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

(10) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(11) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x)))))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

POL(A_{A_1}(x1)) = 0 +
[1,0,0]
·x1

POL(a_{b_1}(x1)) =
/0\
|0|
\1/
 +
/010\
|111|
\010/
·x1

POL(b_{b_1}(x1)) =
/0\
|0|
\0/
 +
/010\
|001|
\100/
·x1

POL(b_{c_1}(x1)) =
/0\
|1|
\0/
 +
/000\
|100|
\010/
·x1

POL(c_{c_1}(x1)) =
/1\
|0|
\0/
 +
/110\
|000|
\000/
·x1

POL(c_{a_1}(x1)) =
/0\
|0|
\1/
 +
/010\
|000|
\111/
·x1

POL(a_{a_1}(x1)) =
/0\
|1|
\0/
 +
/100\
|011|
\000/
·x1

POL(b_{a_1}(x1)) =
/0\
|0|
\0/
 +
/000\
|000|
\010/
·x1

POL(c_{b_1}(x1)) =
/0\
|0|
\1/
 +
/111\
|010|
\111/
·x1

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

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

(12) Obligation:

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

A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))

The TRS R consists of the following rules:

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

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(14) TRUE