Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema

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 (⇔, 62 ms)

↳8 QTRS

↳9 DependencyPairsProof (⇔, 14 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 TRUE

(0) Obligation:

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

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

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

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

Q is empty.

(3) FlatCCProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

Q is empty.

(5) RootLabelingProof (EQUIVALENT transformation)

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

(6) Obligation:

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

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{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_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x₁)) = 1 + x₁
POL(a_{b_1}(x₁)) = x₁
POL(b_{a_1}(x₁)) = x₁
POL(b_{b_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}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x)))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{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_{a_1}(a_{b_1}(x)))))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))))))))

(8) Obligation:

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

a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_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:

A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → A_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → B_{A_1}(a_{a_1}(a_{a_1}(x)))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{a_1}(a_{b_1}(x)))
B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x)))))) → B_{A_1}(a_{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}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))))))
a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x)))))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_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 0 SCCs with 26 less nodes.

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) FlatCCProof (EQUIVALENT transformation)

(4) Obligation:

(5) RootLabelingProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyPairsProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) TRUE