(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(b(b(b(a(x))))) → b(b(b(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(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(b(b(a(x))))) → b(b(b(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(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(a(a(x))) → a(b(x))
b(a(a(x))) → b(b(x))
a(a(b(a(x)))) → a(b(b(b(x))))
b(a(b(a(x)))) → b(b(b(b(x))))
a(a(b(a(a(x))))) → a(b(a(b(b(a(x))))))
b(a(b(a(a(x))))) → b(b(a(b(b(a(x))))))
a(a(b(b(a(x))))) → a(b(b(b(b(b(x))))))
b(a(b(b(a(x))))) → b(b(b(b(b(b(x))))))
a(a(b(a(a(a(x)))))) → a(b(a(a(b(b(a(a(x))))))))
b(a(b(a(a(a(x)))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(b(b(a(a(x)))))) → a(b(b(a(b(b(b(a(x))))))))
b(a(b(b(a(a(x)))))) → b(b(b(a(b(b(b(a(x))))))))
a(a(b(a(b(a(x)))))) → a(b(a(b(b(b(a(b(x))))))))
b(a(b(a(b(a(x)))))) → b(b(a(b(b(b(a(b(x))))))))
a(a(b(b(b(a(x)))))) → a(b(b(b(b(b(b(b(x))))))))
b(a(b(b(b(a(x)))))) → b(b(b(b(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_{a_1}(x)))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → 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_{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_{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}(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}(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_{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_{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_{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}(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_{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_{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_{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}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{a_1}(x))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{a_1}(x))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{a_1}(x)))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → 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_{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_{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}(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}(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_{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}(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_{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_{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}(x)))))))
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{a_1}(x))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{a_1}(x))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{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_{a_1}(a_{a_1}(a_{a_1}(x)))))) → B_{A_1}(a_{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_{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_{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_{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_{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)))
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_{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_{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_{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_{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_{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_{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_{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_{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_{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 17 less nodes.
(12) TRUE