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:

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

twoto(0(x1)) → p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
twoto(s(x1)) → p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
twice(0(x1)) → p(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
twice(s(x1)) → s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(p(s(x1)))))))
0(x1) → 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)) → TWICE(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))
TWOTO(0(x1)) → P(p(p(s(s(s(0(p(p(s(s(x1)))))))))))
TWOTO(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))
TWOTO(0(x1)) → P(s(s(x1)))
TWOTO(0(x1)) → P(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1)))))))))))))))
TWICE(s(x1)) → P(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))
TWICE(s(x1)) → TWICE(p(s(p(s(p(s(p(s(x1)))))))))
TWOTO(0(x1)) → P(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
TWICE(0(x1)) → 01(s(p(s(s(s(s(p(s(x1)))))))))
TWOTO(0(x1)) → P(s(s(s(0(p(p(s(s(x1)))))))))
TWOTO(s(x1)) → TWOTO(p(s(p(s(x1)))))
TWOTO(0(x1)) → P(p(s(s(x1))))
P(0(x1)) → 01(s(s(s(s(p(s(x1)))))))
TWICE(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(s(s(twoto(p(s(p(s(x1)))))))))
TWOTO(s(x1)) → P(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))
TWICE(0(x1)) → P(s(s(s(s(p(s(x1)))))))
TWOTO(s(x1)) → P(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))
TWOTO(s(x1)) → P(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(p(s(p(s(x1))))))))
TWOTO(0(x1)) → P(p(s(s(s(0(p(p(s(s(x1))))))))))
P(p(s(x1))) → P(x1)
TWICE(0(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(p(s(x1))))))
TWOTO(s(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))
TWICE(s(x1)) → P(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))
TWOTO(s(x1)) → P(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(x1))))
TWOTO(0(x1)) → 01(p(p(s(s(x1)))))
TWOTO(s(x1)) → P(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
TWICE(s(x1)) → P(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))
TWOTO(s(x1)) → P(p(s(s(s(twoto(p(s(p(s(x1))))))))))
P(0(x1)) → P(s(x1))
TWICE(0(x1)) → P(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
TWICE(0(x1)) → P(s(0(s(p(s(s(s(s(p(s(x1)))))))))))

The TRS R consists of the following rules:

twoto(0(x1)) → p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
twoto(s(x1)) → p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
twice(0(x1)) → p(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
twice(s(x1)) → s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(p(s(x1)))))))
0(x1) → 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)) → TWICE(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))
TWOTO(0(x1)) → P(p(p(s(s(s(0(p(p(s(s(x1)))))))))))
TWOTO(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))
TWOTO(0(x1)) → P(s(s(x1)))
TWOTO(0(x1)) → P(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1)))))))))))))))
TWICE(s(x1)) → P(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))
TWICE(s(x1)) → TWICE(p(s(p(s(p(s(p(s(x1)))))))))
TWOTO(0(x1)) → P(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
TWICE(0(x1)) → 01(s(p(s(s(s(s(p(s(x1)))))))))
TWOTO(0(x1)) → P(s(s(s(0(p(p(s(s(x1)))))))))
TWOTO(s(x1)) → TWOTO(p(s(p(s(x1)))))
TWOTO(0(x1)) → P(p(s(s(x1))))
P(0(x1)) → 01(s(s(s(s(p(s(x1)))))))
TWICE(s(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(s(s(twoto(p(s(p(s(x1)))))))))
TWOTO(s(x1)) → P(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))
TWICE(0(x1)) → P(s(s(s(s(p(s(x1)))))))
TWOTO(s(x1)) → P(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))
TWOTO(s(x1)) → P(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(p(s(p(s(x1))))))))
TWOTO(0(x1)) → P(p(s(s(s(0(p(p(s(s(x1))))))))))
P(p(s(x1))) → P(x1)
TWICE(0(x1)) → P(s(x1))
TWOTO(s(x1)) → P(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(p(s(x1))))))
TWOTO(s(x1)) → P(s(p(s(x1))))
TWOTO(s(x1)) → P(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))
TWICE(s(x1)) → P(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))
TWICE(s(x1)) → P(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))
TWOTO(s(x1)) → P(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))
TWICE(s(x1)) → P(s(p(s(x1))))
TWOTO(0(x1)) → 01(p(p(s(s(x1)))))
TWOTO(s(x1)) → P(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
TWICE(s(x1)) → P(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))
TWOTO(s(x1)) → P(p(s(s(s(twoto(p(s(p(s(x1))))))))))
P(0(x1)) → P(s(x1))
TWICE(0(x1)) → P(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
TWICE(0(x1)) → P(s(0(s(p(s(s(s(s(p(s(x1)))))))))))

The TRS R consists of the following rules:

twoto(0(x1)) → p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
twoto(s(x1)) → p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
twice(0(x1)) → p(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
twice(s(x1)) → s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(p(s(x1)))))))
0(x1) → 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 36 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ 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:

twoto(0(x1)) → p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1))))))))))))))))
twoto(s(x1)) → p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))
twice(0(x1)) → p(s(p(s(0(s(p(s(s(s(s(p(s(x1)))))))))))))
twice(s(x1)) → s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))))
p(p(s(x1))) → p(x1)
p(s(x1)) → x1
p(0(x1)) → 0(s(s(s(s(p(s(x1)))))))
0(x1) → 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)) = 1 + x_1   
POL(s(x1)) = x_1   
The value of delta used in the strict ordering is 2.
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:

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

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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:

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

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