Termination w.r.t. Q proof of /tmp/tmpdpvWIf/dup01.srs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A13(a13(x1)) → A12(a23(a23(a12(a12(x1)))))
A12(a12(a34(a34(x1)))) → A34(a12(a12(x1)))
A12(a12(a56(a56(x1)))) → A56(a12(a12(x1)))
A23(a23(a56(a56(x1)))) → A56(a23(a23(x1)))
A26(a26(x1)) → A23(a23(x1))
A36(a36(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
A14(a14(x1)) → A23(a23(a12(a12(x1))))
A25(a25(x1)) → A23(a34(a34(a45(a45(a34(a34(a23(a23(x1)))))))))
A26(a26(x1)) → A45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))
A13(a13(x1)) → A23(a12(a12(x1)))
A35(a35(x1)) → A34(a45(a45(a34(a34(x1)))))
A25(a25(x1)) → A34(a45(a45(a34(a34(a23(a23(x1)))))))
A12(a12(a45(a45(x1)))) → A45(a12(a12(x1)))
A13(a13(x1)) → A12(x1)
A26(a26(x1)) → A56(a56(a45(a45(a34(a34(a23(a23(x1))))))))
A26(a26(x1)) → A45(a34(a34(a23(a23(x1)))))
A15(a15(x1)) → A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))
A36(a36(x1)) → A56(a56(a45(a45(a34(a34(x1))))))
A15(a15(x1)) → A23(a23(a12(a12(x1))))
A14(a14(x1)) → A23(a12(a12(x1)))
A26(a26(x1)) → A56(a45(a45(a34(a34(a23(a23(x1)))))))
A46(a46(x1)) → A56(a56(a45(a45(x1))))
A16(a16(x1)) → A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))
A36(a36(x1)) → A56(a45(a45(a34(a34(x1)))))
A46(a46(x1)) → A45(a56(a56(a45(a45(x1)))))
A12(a12(a56(a56(x1)))) → A12(a12(x1))
A16(a16(x1)) → A56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))
A13(a13(x1)) → A23(a23(a12(a12(x1))))
A23(a23(a56(a56(x1)))) → A56(a56(a23(a23(x1))))
A26(a26(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))
A12(a12(a56(a56(x1)))) → A56(a56(a12(a12(x1))))
A15(a15(x1)) → A23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))
A46(a46(x1)) → A56(a45(a45(x1)))
A35(a35(x1)) → A45(a34(a34(x1)))
A16(a16(x1)) → A23(a12(a12(x1)))
A35(a35(x1)) → A34(a34(x1))
A36(a36(x1)) → A34(x1)
A23(a23(a45(a45(x1)))) → A45(a45(a23(a23(x1))))
A46(a46(x1)) → A45(a45(x1))
A14(a14(x1)) → A23(a23(a34(a34(a23(a23(a12(a12(x1))))))))
A25(a25(x1)) → A23(a23(x1))
A16(a16(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A16(a16(x1)) → A45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))
A12(a12(a34(a34(x1)))) → A12(a12(x1))
A26(a26(x1)) → A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))))
A26(a26(x1)) → A34(a34(a23(a23(x1))))
A26(a26(x1)) → A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
A16(a16(x1)) → A23(a23(a12(a12(x1))))
A36(a36(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(x1)))))))))
A15(a15(x1)) → A45(a34(a34(a23(a23(a12(a12(x1)))))))
A15(a15(x1)) → A34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))
A24(a24(x1)) → A34(a34(a23(a23(x1))))
A24(a24(x1)) → A23(a23(a34(a34(a23(a23(x1))))))
A24(a24(x1)) → A23(x1)
A25(a25(x1)) → A34(a34(a45(a45(a34(a34(a23(a23(x1))))))))
A16(a16(x1)) → A12(x1)
A34(a34(a56(a56(x1)))) → A34(x1)
A26(a26(x1)) → A23(x1)
A16(a16(x1)) → A45(a45(a34(a34(a23(a23(a12(a12(x1))))))))
A15(a15(x1)) → A45(a45(a34(a34(a23(a23(a12(a12(x1))))))))
A12(a12(a45(a45(x1)))) → A12(x1)
A25(a25(x1)) → A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
A23(a23(a45(a45(x1)))) → A23(x1)
A35(a35(x1)) → A34(x1)
A26(a26(x1)) → A34(a23(a23(x1)))
A46(a46(x1)) → A45(a45(a56(a56(a45(a45(x1))))))
A12(a12(a45(a45(x1)))) → A12(a12(x1))
A16(a16(x1)) → A12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))))
A16(a16(x1)) → A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))
A34(a34(a56(a56(x1)))) → A34(a34(x1))
A26(a26(x1)) → A45(a45(a34(a34(a23(a23(x1))))))
A24(a24(x1)) → A23(a34(a34(a23(a23(x1)))))
A25(a25(x1)) → A45(a34(a34(a23(a23(x1)))))
A35(a35(x1)) → A45(a45(a34(a34(x1))))
A25(a25(x1)) → A23(x1)
A12(a12(a34(a34(x1)))) → A12(x1)
A15(a15(x1)) → A12(x1)
A25(a25(x1)) → A34(a23(a23(x1)))
A35(a35(x1)) → A34(a34(a45(a45(a34(a34(x1))))))
A23(a23(a45(a45(x1)))) → A45(a23(a23(x1)))
A13(a13(x1)) → A12(a12(x1))
A15(a15(x1)) → A12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))
A25(a25(x1)) → A45(a45(a34(a34(a23(a23(x1))))))
A36(a36(x1)) → A45(a34(a34(x1)))
A36(a36(x1)) → A45(a45(a34(a34(x1))))
A12(a12(a45(a45(x1)))) → A45(a45(a12(a12(x1))))
A24(a24(x1)) → A34(a23(a23(x1)))
A14(a14(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A23(a23(a56(a56(x1)))) → A23(x1)
A12(a12(a34(a34(x1)))) → A34(a34(a12(a12(x1))))
A14(a14(x1)) → A34(a23(a23(a12(a12(x1)))))
A34(a34(a56(a56(x1)))) → A56(a34(a34(x1)))
A36(a36(x1)) → A34(a34(x1))
A34(a34(a56(a56(x1)))) → A56(a56(a34(a34(x1))))
A12(a12(a56(a56(x1)))) → A12(x1)
A16(a16(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))
A14(a14(x1)) → A12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
A16(a16(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
A15(a15(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A26(a26(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))
A13(a13(x1)) → A12(a12(a23(a23(a12(a12(x1))))))
A23(a23(a45(a45(x1)))) → A23(a23(x1))
A15(a15(x1)) → A34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))
A16(a16(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))
A15(a15(x1)) → A12(a12(x1))
A15(a15(x1)) → A23(a12(a12(x1)))
A36(a36(x1)) → A45(a56(a56(a45(a45(a34(a34(x1)))))))
A46(a46(x1)) → A45(x1)
A16(a16(x1)) → A12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
A16(a16(x1)) → A56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))
A16(a16(x1)) → A34(a23(a23(a12(a12(x1)))))
A15(a15(x1)) → A34(a23(a23(a12(a12(x1)))))
A26(a26(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))
A36(a36(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(x1))))))))
A14(a14(x1)) → A12(a12(x1))
A14(a14(x1)) → A23(a34(a34(a23(a23(a12(a12(x1)))))))
A23(a23(a56(a56(x1)))) → A23(a23(x1))
A14(a14(x1)) → A12(a23(a23(a34(a34(a23(a23(a12(a12(x1)))))))))
A16(a16(x1)) → A45(a34(a34(a23(a23(a12(a12(x1)))))))
A14(a14(x1)) → A12(x1)
A15(a15(x1)) → A12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
A24(a24(x1)) → A23(a23(x1))
A25(a25(x1)) → A34(a34(a23(a23(x1))))
A16(a16(x1)) → A12(a12(x1))

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

A13(a13(x1)) → A12(a23(a23(a12(a12(x1)))))
A12(a12(a34(a34(x1)))) → A34(a12(a12(x1)))
A12(a12(a56(a56(x1)))) → A56(a12(a12(x1)))
A23(a23(a56(a56(x1)))) → A56(a23(a23(x1)))
A26(a26(x1)) → A23(a23(x1))
A36(a36(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
A14(a14(x1)) → A23(a23(a12(a12(x1))))
A25(a25(x1)) → A23(a34(a34(a45(a45(a34(a34(a23(a23(x1)))))))))
A26(a26(x1)) → A45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))
A13(a13(x1)) → A23(a12(a12(x1)))
A35(a35(x1)) → A34(a45(a45(a34(a34(x1)))))
A25(a25(x1)) → A34(a45(a45(a34(a34(a23(a23(x1)))))))
A12(a12(a45(a45(x1)))) → A45(a12(a12(x1)))
A13(a13(x1)) → A12(x1)
A26(a26(x1)) → A56(a56(a45(a45(a34(a34(a23(a23(x1))))))))
A26(a26(x1)) → A45(a34(a34(a23(a23(x1)))))
A15(a15(x1)) → A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))
A36(a36(x1)) → A56(a56(a45(a45(a34(a34(x1))))))
A15(a15(x1)) → A23(a23(a12(a12(x1))))
A14(a14(x1)) → A23(a12(a12(x1)))
A26(a26(x1)) → A56(a45(a45(a34(a34(a23(a23(x1)))))))
A46(a46(x1)) → A56(a56(a45(a45(x1))))
A16(a16(x1)) → A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))
A36(a36(x1)) → A56(a45(a45(a34(a34(x1)))))
A46(a46(x1)) → A45(a56(a56(a45(a45(x1)))))
A12(a12(a56(a56(x1)))) → A12(a12(x1))
A16(a16(x1)) → A56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))
A13(a13(x1)) → A23(a23(a12(a12(x1))))
A23(a23(a56(a56(x1)))) → A56(a56(a23(a23(x1))))
A26(a26(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))
A12(a12(a56(a56(x1)))) → A56(a56(a12(a12(x1))))
A15(a15(x1)) → A23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))
A46(a46(x1)) → A56(a45(a45(x1)))
A35(a35(x1)) → A45(a34(a34(x1)))
A16(a16(x1)) → A23(a12(a12(x1)))
A35(a35(x1)) → A34(a34(x1))
A36(a36(x1)) → A34(x1)
A23(a23(a45(a45(x1)))) → A45(a45(a23(a23(x1))))
A46(a46(x1)) → A45(a45(x1))
A14(a14(x1)) → A23(a23(a34(a34(a23(a23(a12(a12(x1))))))))
A25(a25(x1)) → A23(a23(x1))
A16(a16(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A16(a16(x1)) → A45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))
A12(a12(a34(a34(x1)))) → A12(a12(x1))
A26(a26(x1)) → A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))))
A26(a26(x1)) → A34(a34(a23(a23(x1))))
A26(a26(x1)) → A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
A16(a16(x1)) → A23(a23(a12(a12(x1))))
A36(a36(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(x1)))))))))
A15(a15(x1)) → A45(a34(a34(a23(a23(a12(a12(x1)))))))
A15(a15(x1)) → A34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))
A24(a24(x1)) → A34(a34(a23(a23(x1))))
A24(a24(x1)) → A23(a23(a34(a34(a23(a23(x1))))))
A24(a24(x1)) → A23(x1)
A25(a25(x1)) → A34(a34(a45(a45(a34(a34(a23(a23(x1))))))))
A16(a16(x1)) → A12(x1)
A34(a34(a56(a56(x1)))) → A34(x1)
A26(a26(x1)) → A23(x1)
A16(a16(x1)) → A45(a45(a34(a34(a23(a23(a12(a12(x1))))))))
A15(a15(x1)) → A45(a45(a34(a34(a23(a23(a12(a12(x1))))))))
A12(a12(a45(a45(x1)))) → A12(x1)
A25(a25(x1)) → A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
A23(a23(a45(a45(x1)))) → A23(x1)
A35(a35(x1)) → A34(x1)
A26(a26(x1)) → A34(a23(a23(x1)))
A46(a46(x1)) → A45(a45(a56(a56(a45(a45(x1))))))
A12(a12(a45(a45(x1)))) → A12(a12(x1))
A16(a16(x1)) → A12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))))))
A16(a16(x1)) → A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))
A34(a34(a56(a56(x1)))) → A34(a34(x1))
A26(a26(x1)) → A45(a45(a34(a34(a23(a23(x1))))))
A24(a24(x1)) → A23(a34(a34(a23(a23(x1)))))
A25(a25(x1)) → A45(a34(a34(a23(a23(x1)))))
A35(a35(x1)) → A45(a45(a34(a34(x1))))
A25(a25(x1)) → A23(x1)
A12(a12(a34(a34(x1)))) → A12(x1)
A15(a15(x1)) → A12(x1)
A25(a25(x1)) → A34(a23(a23(x1)))
A35(a35(x1)) → A34(a34(a45(a45(a34(a34(x1))))))
A23(a23(a45(a45(x1)))) → A45(a23(a23(x1)))
A13(a13(x1)) → A12(a12(x1))
A15(a15(x1)) → A12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))
A25(a25(x1)) → A45(a45(a34(a34(a23(a23(x1))))))
A36(a36(x1)) → A45(a34(a34(x1)))
A36(a36(x1)) → A45(a45(a34(a34(x1))))
A12(a12(a45(a45(x1)))) → A45(a45(a12(a12(x1))))
A24(a24(x1)) → A34(a23(a23(x1)))
A14(a14(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A23(a23(a56(a56(x1)))) → A23(x1)
A12(a12(a34(a34(x1)))) → A34(a34(a12(a12(x1))))
A14(a14(x1)) → A34(a23(a23(a12(a12(x1)))))
A34(a34(a56(a56(x1)))) → A56(a34(a34(x1)))
A36(a36(x1)) → A34(a34(x1))
A34(a34(a56(a56(x1)))) → A56(a56(a34(a34(x1))))
A12(a12(a56(a56(x1)))) → A12(x1)
A16(a16(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))
A14(a14(x1)) → A12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
A16(a16(x1)) → A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
A15(a15(x1)) → A34(a34(a23(a23(a12(a12(x1))))))
A26(a26(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))
A13(a13(x1)) → A12(a12(a23(a23(a12(a12(x1))))))
A23(a23(a45(a45(x1)))) → A23(a23(x1))
A15(a15(x1)) → A34(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))
A16(a16(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1)))))))))))))
A15(a15(x1)) → A12(a12(x1))
A15(a15(x1)) → A23(a12(a12(x1)))
A36(a36(x1)) → A45(a56(a56(a45(a45(a34(a34(x1)))))))
A46(a46(x1)) → A45(x1)
A16(a16(x1)) → A12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
A16(a16(x1)) → A56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))
A16(a16(x1)) → A34(a23(a23(a12(a12(x1)))))
A15(a15(x1)) → A34(a23(a23(a12(a12(x1)))))
A26(a26(x1)) → A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1)))))))))))
A36(a36(x1)) → A45(a45(a56(a56(a45(a45(a34(a34(x1))))))))
A14(a14(x1)) → A12(a12(x1))
A14(a14(x1)) → A23(a34(a34(a23(a23(a12(a12(x1)))))))
A23(a23(a56(a56(x1)))) → A23(a23(x1))
A14(a14(x1)) → A12(a23(a23(a34(a34(a23(a23(a12(a12(x1)))))))))
A16(a16(x1)) → A45(a34(a34(a23(a23(a12(a12(x1)))))))
A14(a14(x1)) → A12(x1)
A15(a15(x1)) → A12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
A24(a24(x1)) → A23(a23(x1))
A25(a25(x1)) → A34(a34(a23(a23(x1))))
A16(a16(x1)) → A12(a12(x1))

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 112 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

A34(a34(a56(a56(x1)))) → A34(x1)
A34(a34(a56(a56(x1)))) → A34(a34(x1))

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A34(a34(a56(a56(x1)))) → A34(x1)
A34(a34(a56(a56(x1)))) → A34(a34(x1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(A34(x₁)) = (1/4)x_1
POL(a45(x₁)) = (1/2)x_1
POL(a34(x₁)) = (4)x_1
POL(a56(x₁)) = 4 + x_1

The value of delta used in the strict ordering is 8.
The following usable rules [17] were oriented:

a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))
a34(a34(a34(a34(x1)))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

A23(a23(a45(a45(x1)))) → A23(a23(x1))
A23(a23(a56(a56(x1)))) → A23(a23(x1))
A23(a23(a56(a56(x1)))) → A23(x1)
A23(a23(a45(a45(x1)))) → A23(x1)

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A23(a23(a45(a45(x1)))) → A23(a23(x1))
A23(a23(a45(a45(x1)))) → A23(x1)

The remaining pairs can at least be oriented weakly.

A23(a23(a56(a56(x1)))) → A23(a23(x1))
A23(a23(a56(a56(x1)))) → A23(x1)

Used ordering: Polynomial interpretation [25,35]:

POL(a23(x₁)) = (4)x_1
POL(a45(x₁)) = 1/4 + (2)x_1
POL(A23(x₁)) = (2)x_1
POL(a34(x₁)) = (3/2)x_1
POL(a56(x₁)) = (4)x_1

The value of delta used in the strict ordering is 6.
The following usable rules [17] were oriented:

a45(a45(a45(a45(x1)))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a23(a23(a23(a23(x1)))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

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

A23(a23(a56(a56(x1)))) → A23(a23(x1))
A23(a23(a56(a56(x1)))) → A23(x1)

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A23(a23(a56(a56(x1)))) → A23(a23(x1))
A23(a23(a56(a56(x1)))) → A23(x1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(a23(x₁)) = x_1
POL(a45(x₁)) = (4)x_1
POL(A23(x₁)) = (2)x_1
POL(a34(x₁)) = (4)x_1
POL(a56(x₁)) = 1/4 + (2)x_1

The value of delta used in the strict ordering is 3/2.
The following usable rules [17] were oriented:

a45(a45(a45(a45(x1)))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a23(a23(a23(a23(x1)))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

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

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

A12(a12(a56(a56(x1)))) → A12(x1)
A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a34(a34(x1)))) → A12(x1)
A12(a12(a45(a45(x1)))) → A12(x1)
A12(a12(a34(a34(x1)))) → A12(a12(x1))
A12(a12(a56(a56(x1)))) → A12(a12(x1))

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A12(a12(a34(a34(x1)))) → A12(x1)
A12(a12(a34(a34(x1)))) → A12(a12(x1))

The remaining pairs can at least be oriented weakly.

A12(a12(a56(a56(x1)))) → A12(x1)
A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a45(a45(x1)))) → A12(x1)
A12(a12(a56(a56(x1)))) → A12(a12(x1))

Used ordering: Polynomial interpretation [25,35]:

POL(a23(x₁)) = (1/2)x_1
POL(a45(x₁)) = (2)x_1
POL(A12(x₁)) = (2)x_1
POL(a12(x₁)) = (2)x_1
POL(a34(x₁)) = 1/4 + (4)x_1
POL(a56(x₁)) = x_1

The value of delta used in the strict ordering is 5.
The following usable rules [17] were oriented:

a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a45(a45(x1)))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))
a12(a12(a12(a12(x1)))) → x1
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a34(a34(a34(a34(x1)))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a56(a56(x1)))) → A12(x1)
A12(a12(a45(a45(x1)))) → A12(x1)
A12(a12(a56(a56(x1)))) → A12(a12(x1))

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A12(a12(a56(a56(x1)))) → A12(x1)
A12(a12(a56(a56(x1)))) → A12(a12(x1))

The remaining pairs can at least be oriented weakly.

A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a45(a45(x1)))) → A12(x1)

Used ordering: Polynomial interpretation [25,35]:

POL(a23(x₁)) = (9/4)x_1
POL(a45(x₁)) = x_1
POL(A12(x₁)) = x_1
POL(a12(x₁)) = x_1
POL(a34(x₁)) = x_1
POL(a56(x₁)) = 7/4 + (4)x_1

The value of delta used in the strict ordering is 35/4.
The following usable rules [17] were oriented:

a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a45(a45(x1)))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))
a12(a12(a12(a12(x1)))) → x1
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a34(a34(a34(a34(x1)))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a45(a45(x1)))) → A12(x1)

The TRS R consists of the following rules:

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A12(a12(a45(a45(x1)))) → A12(a12(x1))
A12(a12(a45(a45(x1)))) → A12(x1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(a23(x₁)) = (3/4)x_1
POL(a45(x₁)) = 1/4 + x_1
POL(A12(x₁)) = (1/4)x_1
POL(a12(x₁)) = (13/4)x_1
POL(a34(x₁)) = x_1
POL(a56(x₁)) = (4)x_1

The value of delta used in the strict ordering is 13/32.
The following usable rules [17] were oriented:

a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a45(a45(x1)))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a56(a56(a56(a56(x1)))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))
a12(a12(a12(a12(x1)))) → x1
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a34(a34(a34(a34(x1)))) → x1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

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

a12(a12(a12(a12(x1)))) → x1
a13(a13(a13(a13(x1)))) → x1
a14(a14(a14(a14(x1)))) → x1
a15(a15(a15(a15(x1)))) → x1
a16(a16(a16(a16(x1)))) → x1
a23(a23(a23(a23(x1)))) → x1
a24(a24(a24(a24(x1)))) → x1
a25(a25(a25(a25(x1)))) → x1
a26(a26(a26(a26(x1)))) → x1
a34(a34(a34(a34(x1)))) → x1
a35(a35(a35(a35(x1)))) → x1
a36(a36(a36(a36(x1)))) → x1
a45(a45(a45(a45(x1)))) → x1
a46(a46(a46(a46(x1)))) → x1
a56(a56(a56(a56(x1)))) → x1
a13(a13(x1)) → a12(a12(a23(a23(a12(a12(x1))))))
a14(a14(x1)) → a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))
a15(a15(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))
a16(a16(x1)) → a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))
a24(a24(x1)) → a23(a23(a34(a34(a23(a23(x1))))))
a25(a25(x1)) → a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))
a26(a26(x1)) → a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))
a35(a35(x1)) → a34(a34(a45(a45(a34(a34(x1))))))
a36(a36(x1)) → a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))
a46(a46(x1)) → a45(a45(a56(a56(a45(a45(x1))))))
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1)))))))))))) → x1
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1)))))))))))) → x1
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1)))))))))))) → x1
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1)))))))))))) → x1
a12(a12(a34(a34(x1)))) → a34(a34(a12(a12(x1))))
a12(a12(a45(a45(x1)))) → a45(a45(a12(a12(x1))))
a12(a12(a56(a56(x1)))) → a56(a56(a12(a12(x1))))
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1))))
a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1))))
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.