Termination w.r.t. Q proof of /tmp/tmpfuGDPp/s6.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(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(a34(x1))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

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

A46(x1) → A56(a45(x1))
A23(a45(x1)) → A23(x1)
A12(a45(x1)) → A12(x1)
A26(x1) → A34(a45(a56(a45(a34(a23(x1))))))
A26(x1) → A23(a34(a45(a56(a45(a34(a23(x1)))))))
A36(x1) → A45(a34(x1))
A16(x1) → A34(a23(a12(x1)))
A36(x1) → A34(x1)
A25(x1) → A34(a23(x1))
A26(x1) → A23(x1)
A12(a34(x1)) → A12(x1)
A16(x1) → A23(a12(x1))
A16(x1) → A23(a34(a45(a56(a45(a34(a23(a12(x1))))))))
A16(x1) → A56(a45(a34(a23(a12(x1)))))
A24(x1) → A23(a34(a23(x1)))
A15(x1) → A12(x1)
A26(x1) → A56(a45(a34(a23(x1))))
A35(x1) → A34(x1)
A25(x1) → A23(a34(a45(a34(a23(x1)))))
A26(x1) → A45(a56(a45(a34(a23(x1)))))
A23(a56(x1)) → A56(a23(x1))
A36(x1) → A34(a45(a56(a45(a34(x1)))))
A15(x1) → A12(a23(a34(a45(a34(a23(a12(x1)))))))
A15(x1) → A23(a34(a45(a34(a23(a12(x1))))))
A13(x1) → A23(a12(x1))
A24(x1) → A23(x1)
A34(a56(x1)) → A34(x1)
A16(x1) → A45(a34(a23(a12(x1))))
A25(x1) → A34(a45(a34(a23(x1))))
A23(a56(x1)) → A23(x1)
A14(x1) → A23(a34(a23(a12(x1))))
A15(x1) → A34(a23(a12(x1)))
A46(x1) → A45(a56(a45(x1)))
A14(x1) → A12(x1)
A15(x1) → A23(a12(x1))
A35(x1) → A34(a45(a34(x1)))
A15(x1) → A45(a34(a23(a12(x1))))
A26(x1) → A34(a23(x1))
A12(a56(x1)) → A12(x1)
A16(x1) → A34(a45(a56(a45(a34(a23(a12(x1)))))))
A12(a34(x1)) → A34(a12(x1))
A13(x1) → A12(x1)
A25(x1) → A23(x1)
A12(a45(x1)) → A45(a12(x1))
A23(a45(x1)) → A45(a23(x1))
A16(x1) → A12(x1)
A14(x1) → A34(a23(a12(x1)))
A35(x1) → A45(a34(x1))
A14(x1) → A12(a23(a34(a23(a12(x1)))))
A26(x1) → A45(a34(a23(x1)))
A16(x1) → A45(a56(a45(a34(a23(a12(x1))))))
A12(a56(x1)) → A56(a12(x1))
A14(x1) → A23(a12(x1))
A16(x1) → A12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
A46(x1) → A45(x1)
A15(x1) → A34(a45(a34(a23(a12(x1)))))
A25(x1) → A45(a34(a23(x1)))
A34(a56(x1)) → A56(a34(x1))
A13(x1) → A12(a23(a12(x1)))
A24(x1) → A34(a23(x1))
A36(x1) → A45(a56(a45(a34(x1))))
A36(x1) → A56(a45(a34(x1)))

The TRS R consists of the following rules:

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

A46(x1) → A56(a45(x1))
A23(a45(x1)) → A23(x1)
A12(a45(x1)) → A12(x1)
A26(x1) → A34(a45(a56(a45(a34(a23(x1))))))
A26(x1) → A23(a34(a45(a56(a45(a34(a23(x1)))))))
A36(x1) → A45(a34(x1))
A16(x1) → A34(a23(a12(x1)))
A36(x1) → A34(x1)
A25(x1) → A34(a23(x1))
A26(x1) → A23(x1)
A12(a34(x1)) → A12(x1)
A16(x1) → A23(a12(x1))
A16(x1) → A23(a34(a45(a56(a45(a34(a23(a12(x1))))))))
A16(x1) → A56(a45(a34(a23(a12(x1)))))
A24(x1) → A23(a34(a23(x1)))
A15(x1) → A12(x1)
A26(x1) → A56(a45(a34(a23(x1))))
A35(x1) → A34(x1)
A25(x1) → A23(a34(a45(a34(a23(x1)))))
A26(x1) → A45(a56(a45(a34(a23(x1)))))
A23(a56(x1)) → A56(a23(x1))
A36(x1) → A34(a45(a56(a45(a34(x1)))))
A15(x1) → A12(a23(a34(a45(a34(a23(a12(x1)))))))
A15(x1) → A23(a34(a45(a34(a23(a12(x1))))))
A13(x1) → A23(a12(x1))
A24(x1) → A23(x1)
A34(a56(x1)) → A34(x1)
A16(x1) → A45(a34(a23(a12(x1))))
A25(x1) → A34(a45(a34(a23(x1))))
A23(a56(x1)) → A23(x1)
A14(x1) → A23(a34(a23(a12(x1))))
A15(x1) → A34(a23(a12(x1)))
A46(x1) → A45(a56(a45(x1)))
A14(x1) → A12(x1)
A15(x1) → A23(a12(x1))
A35(x1) → A34(a45(a34(x1)))
A15(x1) → A45(a34(a23(a12(x1))))
A26(x1) → A34(a23(x1))
A12(a56(x1)) → A12(x1)
A16(x1) → A34(a45(a56(a45(a34(a23(a12(x1)))))))
A12(a34(x1)) → A34(a12(x1))
A13(x1) → A12(x1)
A25(x1) → A23(x1)
A12(a45(x1)) → A45(a12(x1))
A23(a45(x1)) → A45(a23(x1))
A16(x1) → A12(x1)
A14(x1) → A34(a23(a12(x1)))
A35(x1) → A45(a34(x1))
A14(x1) → A12(a23(a34(a23(a12(x1)))))
A26(x1) → A45(a34(a23(x1)))
A16(x1) → A45(a56(a45(a34(a23(a12(x1))))))
A12(a56(x1)) → A56(a12(x1))
A14(x1) → A23(a12(x1))
A16(x1) → A12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
A46(x1) → A45(x1)
A15(x1) → A34(a45(a34(a23(a12(x1)))))
A25(x1) → A45(a34(a23(x1)))
A34(a56(x1)) → A56(a34(x1))
A13(x1) → A12(a23(a12(x1)))
A24(x1) → A34(a23(x1))
A36(x1) → A45(a56(a45(a34(x1))))
A36(x1) → A56(a45(a34(x1)))

The TRS R consists of the following rules:

a12(a12(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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 56 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

A34(a56(x1)) → A34(x1)

The TRS R consists of the following rules:

a12(a12(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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(a56(x1)) → A34(x1)

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

POL(A34(x₁)) = (4)x_1
POL(a56(x₁)) = 1 + (4)x_1

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented: none

↳ 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(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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(a56(x1)) → A23(x1)
A23(a45(x1)) → A23(x1)

The TRS R consists of the following rules:

a12(a12(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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(a56(x1)) → A23(x1)
A23(a45(x1)) → A23(x1)

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

POL(a45(x₁)) = 4 + (4)x_1
POL(A23(x₁)) = (4)x_1
POL(a56(x₁)) = 4 + x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

a12(a12(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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(a34(x1)) → A12(x1)
A12(a56(x1)) → A12(x1)
A12(a45(x1)) → A12(x1)

The TRS R consists of the following rules:

a12(a12(x1)) → x1
a13(a13(x1)) → x1
a14(a14(x1)) → x1
a15(a15(x1)) → x1
a16(a16(x1)) → x1
a23(a23(x1)) → x1
a24(a24(x1)) → x1
a25(a25(x1)) → x1
a26(a26(x1)) → x1
a34(a34(x1)) → x1
a35(a35(x1)) → x1
a36(a36(x1)) → x1
a45(a45(x1)) → x1
a46(a46(x1)) → x1
a56(a56(x1)) → x1
a13(x1) → a12(a23(a12(x1)))
a14(x1) → a12(a23(a34(a23(a12(x1)))))
a15(x1) → a12(a23(a34(a45(a34(a23(a12(x1)))))))
a16(x1) → a12(a23(a34(a45(a56(a45(a34(a23(a12(x1)))))))))
a24(x1) → a23(a34(a23(x1)))
a25(x1) → a23(a34(a45(a34(a23(x1)))))
a26(x1) → a23(a34(a45(a56(a45(a34(a23(x1)))))))
a35(x1) → a34(a45(a34(x1)))
a36(x1) → a34(a45(a56(a45(a34(x1)))))
a46(x1) → a45(a56(a45(x1)))
a12(a23(a12(a23(a12(a23(x1)))))) → x1
a23(a34(a23(a34(a23(a34(x1)))))) → x1
a34(a45(a34(a45(a34(a45(x1)))))) → x1
a45(a56(a45(a56(a45(a56(x1)))))) → x1
a12(a34(x1)) → a34(a12(x1))
a12(a45(x1)) → a45(a12(x1))
a12(a56(x1)) → a56(a12(x1))
a23(a45(x1)) → a45(a23(x1))
a23(a56(x1)) → a56(a23(x1))
a34(a56(x1)) → a56(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(a34(x1)) → A12(x1)
A12(a56(x1)) → A12(x1)
A12(a45(x1)) → A12(x1)

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

POL(a45(x₁)) = 4 + (4)x_1
POL(A12(x₁)) = (4)x_1
POL(a34(x₁)) = 4 + (4)x_1
POL(a56(x₁)) = 4 + (3)x_1

The value of delta used in the strict ordering is 16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

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

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