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:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(x1)))))
0(x1) → x1

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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:

THRICE(s(x1)) → P(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))
SIXTIMES(s(x1)) → P(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))))
HALF(s(s(x1))) → P(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
HALF(s(x1)) → P(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))
HALF(0(x1)) → 01(p(s(s(s(s(x1))))))
SIXTIMES(s(x1)) → P(p(s(s(s(x1)))))
THRICE(s(x1)) → HALF(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))
THRICE(s(x1)) → P(s(s(x1)))
THRICE(s(x1)) → P(s(p(p(s(s(x1))))))
THRICE(s(x1)) → P(p(s(s(x1))))
THRICE(0(x1)) → P(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
HALF(s(x1)) → P(s(x1))
HALF(s(s(x1))) → P(p(s(s(half(p(p(s(s(p(s(x1)))))))))))
SIXTIMES(0(x1)) → P(s(p(s(x1))))
HALF(s(s(x1))) → P(p(s(s(p(s(x1))))))
SIXTIMES(0(x1)) → 01(s(s(s(s(s(p(s(p(s(x1))))))))))
SIXTIMES(s(x1)) → P(s(p(p(p(s(s(s(x1))))))))
SIXTIMES(s(x1)) → P(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))
HALF(0(x1)) → P(s(0(p(s(s(s(s(x1))))))))
HALF(s(x1)) → P(p(s(s(half(p(p(s(s(p(s(x1)))))))))))
P(p(s(x1))) → P(x1)
THRICE(s(x1)) → P(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))
THRICE(0(x1)) → P(s(p(s(x1))))
THRICE(s(x1)) → SIXTIMES(p(s(p(p(s(s(x1)))))))
THRICE(s(x1)) → P(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))
SIXTIMES(0(x1)) → P(s(x1))
HALF(s(x1)) → P(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))
HALF(0(x1)) → P(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
THRICE(0(x1)) → P(p(s(s(s(0(p(s(p(s(x1))))))))))
HALF(0(x1)) → P(s(s(p(s(0(p(s(s(s(s(x1)))))))))))
THRICE(s(x1)) → P(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
HALF(s(s(x1))) → P(s(s(p(s(x1)))))
THRICE(0(x1)) → P(p(p(s(s(s(0(p(s(p(s(x1)))))))))))
HALF(s(x1)) → P(p(s(s(p(s(x1))))))
SIXTIMES(s(x1)) → P(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))))))))))))))
HALF(s(x1)) → P(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
HALF(s(s(x1))) → HALF(p(p(s(s(p(s(x1)))))))
SIXTIMES(s(x1)) → P(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))
THRICE(0(x1)) → P(s(s(s(0(p(s(p(s(x1)))))))))
SIXTIMES(0(x1)) → P(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
P(0(x1)) → 01(s(s(s(s(x1)))))
SIXTIMES(0(x1)) → P(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))
SIXTIMES(s(x1)) → P(s(s(s(x1))))
SIXTIMES(s(x1)) → P(p(p(s(s(s(x1))))))
HALF(s(x1)) → HALF(p(p(s(s(p(s(x1)))))))
HALF(s(x1)) → P(s(s(half(p(p(s(s(p(s(x1))))))))))
HALF(s(x1)) → P(s(s(p(s(x1)))))
HALF(0(x1)) → P(s(s(s(s(x1)))))
THRICE(0(x1)) → 01(p(s(p(s(x1)))))
HALF(s(s(x1))) → P(s(x1))
SIXTIMES(s(x1)) → SIXTIMES(p(s(p(p(p(s(s(s(x1)))))))))
HALF(s(s(x1))) → P(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))
THRICE(s(x1)) → P(s(sixtimes(p(s(p(p(s(s(x1)))))))))
THRICE(0(x1)) → P(s(x1))
SIXTIMES(s(x1)) → P(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))
HALF(s(s(x1))) → P(s(s(half(p(p(s(s(p(s(x1))))))))))

The TRS R consists of the following rules:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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:

THRICE(s(x1)) → P(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))
SIXTIMES(s(x1)) → P(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))))
HALF(s(s(x1))) → P(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
HALF(s(x1)) → P(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))
HALF(0(x1)) → 01(p(s(s(s(s(x1))))))
SIXTIMES(s(x1)) → P(p(s(s(s(x1)))))
THRICE(s(x1)) → HALF(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))
THRICE(s(x1)) → P(s(s(x1)))
THRICE(s(x1)) → P(s(p(p(s(s(x1))))))
THRICE(s(x1)) → P(p(s(s(x1))))
THRICE(0(x1)) → P(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
HALF(s(x1)) → P(s(x1))
HALF(s(s(x1))) → P(p(s(s(half(p(p(s(s(p(s(x1)))))))))))
SIXTIMES(0(x1)) → P(s(p(s(x1))))
HALF(s(s(x1))) → P(p(s(s(p(s(x1))))))
SIXTIMES(0(x1)) → 01(s(s(s(s(s(p(s(p(s(x1))))))))))
SIXTIMES(s(x1)) → P(s(p(p(p(s(s(s(x1))))))))
SIXTIMES(s(x1)) → P(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))
HALF(0(x1)) → P(s(0(p(s(s(s(s(x1))))))))
HALF(s(x1)) → P(p(s(s(half(p(p(s(s(p(s(x1)))))))))))
P(p(s(x1))) → P(x1)
THRICE(s(x1)) → P(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))
THRICE(0(x1)) → P(s(p(s(x1))))
THRICE(s(x1)) → SIXTIMES(p(s(p(p(s(s(x1)))))))
THRICE(s(x1)) → P(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))
SIXTIMES(0(x1)) → P(s(x1))
HALF(s(x1)) → P(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))
HALF(0(x1)) → P(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
THRICE(0(x1)) → P(p(s(s(s(0(p(s(p(s(x1))))))))))
HALF(0(x1)) → P(s(s(p(s(0(p(s(s(s(s(x1)))))))))))
THRICE(s(x1)) → P(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
HALF(s(s(x1))) → P(s(s(p(s(x1)))))
THRICE(0(x1)) → P(p(p(s(s(s(0(p(s(p(s(x1)))))))))))
HALF(s(x1)) → P(p(s(s(p(s(x1))))))
SIXTIMES(s(x1)) → P(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))))))))))))))
HALF(s(x1)) → P(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
HALF(s(s(x1))) → HALF(p(p(s(s(p(s(x1)))))))
SIXTIMES(s(x1)) → P(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1)))))))))))))
THRICE(0(x1)) → P(s(s(s(0(p(s(p(s(x1)))))))))
SIXTIMES(0(x1)) → P(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
P(0(x1)) → 01(s(s(s(s(x1)))))
SIXTIMES(0(x1)) → P(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))
SIXTIMES(s(x1)) → P(s(s(s(x1))))
SIXTIMES(s(x1)) → P(p(p(s(s(s(x1))))))
HALF(s(x1)) → HALF(p(p(s(s(p(s(x1)))))))
HALF(s(x1)) → P(s(s(half(p(p(s(s(p(s(x1))))))))))
HALF(s(x1)) → P(s(s(p(s(x1)))))
HALF(0(x1)) → P(s(s(s(s(x1)))))
THRICE(0(x1)) → 01(p(s(p(s(x1)))))
HALF(s(s(x1))) → P(s(x1))
SIXTIMES(s(x1)) → SIXTIMES(p(s(p(p(p(s(s(s(x1)))))))))
HALF(s(s(x1))) → P(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))
THRICE(s(x1)) → P(s(sixtimes(p(s(p(p(s(s(x1)))))))))
THRICE(0(x1)) → P(s(x1))
SIXTIMES(s(x1)) → P(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))
HALF(s(s(x1))) → P(s(s(half(p(p(s(s(p(s(x1))))))))))

The TRS R consists of the following rules:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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 52 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:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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)) = (1/2)x_1   
POL(p(x1)) = (1/4)x_1   
POL(s(x1)) = 1/4 + (4)x_1   
The value of delta used in the strict ordering is 1/32.
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:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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:

SIXTIMES(s(x1)) → SIXTIMES(p(s(p(p(p(s(s(s(x1)))))))))

The TRS R consists of the following rules:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(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:

HALF(s(s(x1))) → HALF(p(p(s(s(p(s(x1)))))))
HALF(s(x1)) → HALF(p(p(s(s(p(s(x1)))))))

The TRS R consists of the following rules:

thrice(0(x1)) → p(s(p(p(p(s(s(s(0(p(s(p(s(x1)))))))))))))
thrice(s(x1)) → p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1))))))))))))))))))
half(0(x1)) → p(p(s(s(p(s(0(p(s(s(s(s(x1))))))))))))
half(s(x1)) → p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))))
half(s(s(x1))) → p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))
sixtimes(0(x1)) → p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1))))))))))))))
sixtimes(s(x1)) → p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(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(x1)))))
0(x1) → x1

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