Termination w.r.t. Q proof of /tmp/tmp67nyCc/aprove06.srs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(x1)))))))))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(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:

TWOTO(s(x1)) → P(s(x1))
TOWER(s(x1)) → P(s(tower(p(s(p(s(x1)))))))
TWICE(s(x1)) → P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
TOWER(s(x1)) → P(s(p(s(x1))))
TOWER(0(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))
TWOTO(s(x1)) → P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))
TWOTO(s(x1)) → P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))
TOWER(s(x1)) → P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
TWOTO(s(x1)) → TWOTO(p(s(p(s(x1)))))
TWOTO(s(x1)) → P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))
TWOTO(s(x1)) → P(s(s(s(twoto(p(s(p(s(x1)))))))))
TWOTO(s(x1)) → P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
TOWER(s(x1)) → TOWER(p(s(p(s(x1)))))
TWOTO(s(x1)) → TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))
TWOTO(s(x1)) → P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))
P(p(s(x1))) → P(x1)
TOWER(s(x1)) → P(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))
TOWER(0(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))
TOWER(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))
TWOTO(s(x1)) → P(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))
TWICE(s(x1)) → P(s(s(s(x1))))
TWOTO(s(x1)) → P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))
TOWER(s(x1)) → TWOTO(p(s(p(s(tower(p(s(p(s(x1))))))))))
TWOTO(s(x1)) → P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))
TWOTO(s(x1)) → P(p(s(s(s(twoto(p(s(p(s(x1))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(x1))))))
TWICE(s(x1)) → P(p(s(s(s(x1)))))
TWICE(s(x1)) → TWICE(p(p(p(s(s(s(x1)))))))
TOWER(s(x1)) → P(s(p(s(tower(p(s(p(s(x1)))))))))

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(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:

TWOTO(s(x1)) → P(s(x1))
TOWER(s(x1)) → P(s(tower(p(s(p(s(x1)))))))
TWICE(s(x1)) → P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
TOWER(s(x1)) → P(s(p(s(x1))))
TOWER(0(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))
TWOTO(s(x1)) → P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))
TWOTO(s(x1)) → P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))
TOWER(s(x1)) → P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
TWOTO(s(x1)) → TWOTO(p(s(p(s(x1)))))
TWOTO(s(x1)) → P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))
TWOTO(s(x1)) → P(s(s(s(twoto(p(s(p(s(x1)))))))))
TWOTO(s(x1)) → P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
TOWER(s(x1)) → TOWER(p(s(p(s(x1)))))
TWOTO(s(x1)) → TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))
TWOTO(s(x1)) → P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))
P(p(s(x1))) → P(x1)
TOWER(s(x1)) → P(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))
TOWER(0(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))
TWICE(s(x1)) → P(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))
TOWER(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))
TWOTO(s(x1)) → P(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))
TWICE(s(x1)) → P(s(s(s(x1))))
TWOTO(s(x1)) → P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))
TOWER(s(x1)) → TWOTO(p(s(p(s(tower(p(s(p(s(x1))))))))))
TWOTO(s(x1)) → P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))
TWOTO(s(x1)) → P(p(s(s(s(twoto(p(s(p(s(x1))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(x1))))))
TWICE(s(x1)) → P(p(s(s(s(x1)))))
TWICE(s(x1)) → TWICE(p(p(p(s(s(s(x1)))))))
TOWER(s(x1)) → P(s(p(s(tower(p(s(p(s(x1)))))))))

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(x1)))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 31 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

P(p(s(x1))) → P(x1)

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(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.

P(p(s(x1))) → P(x1)

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

POL(P(x₁)) = (2)x_1
POL(p(x₁)) = 2 + (2)x_1
POL(s(x₁)) = 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

          ↳ QDP

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

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(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

          ↳ QDP

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

TWICE(s(x1)) → TWICE(p(p(p(s(s(s(x1)))))))

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(x1)))))))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TWOTO(s(x1)) → TWOTO(p(s(p(s(x1)))))

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(x1)))))))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TOWER(s(x1)) → TOWER(p(s(p(s(x1)))))

The TRS R consists of the following rules:

tower(0(x1)) → s(0(p(s(p(s(x1))))))
tower(s(x1)) → p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1))))))))))))))
twoto(0(x1)) → s(0(x1))
twoto(s(x1)) → p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))))
twice(0(x1)) → 0(x1)
twice(s(x1)) → p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(s(s(s(s(x1)))))))))

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