Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/ZantemaSLASHz20.trs

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:

f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F₂(x, f₂(y, z)) -> F₂(x, y)
F₂(f₂(a, b), x) -> F₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
F₂(f₂(a, b), x) -> F₂(a, x)
F₂(f₂(a, b), x) -> F₂(a, f₂(c, f₂(b, f₂(a, x))))
F₂(x, f₂(y, z)) -> F₂(f₂(x, y), z)
F₂(f₂(a, b), x) -> F₂(b, f₂(a, x))
F₂(f₂(a, b), x) -> F₂(c, f₂(b, f₂(a, x)))

The TRS R consists of the following rules:

f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

F₂(x, f₂(y, z)) -> F₂(x, y)
F₂(f₂(a, b), x) -> F₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
F₂(f₂(a, b), x) -> F₂(a, x)
F₂(f₂(a, b), x) -> F₂(a, f₂(c, f₂(b, f₂(a, x))))
F₂(x, f₂(y, z)) -> F₂(f₂(x, y), z)
F₂(f₂(a, b), x) -> F₂(b, f₂(a, x))
F₂(f₂(a, b), x) -> F₂(c, f₂(b, f₂(a, x)))

The TRS R consists of the following rules:

f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

F₂(f₂(a, b), x) -> F₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
F₂(f₂(a, b), x) -> F₂(b, f₂(a, x))
F₂(f₂(a, b), x) -> F₂(c, f₂(b, f₂(a, x)))

The remaining pairs can at least by weakly be oriented.

F₂(x, f₂(y, z)) -> F₂(x, y)
F₂(f₂(a, b), x) -> F₂(a, x)
F₂(f₂(a, b), x) -> F₂(a, f₂(c, f₂(b, f₂(a, x))))
F₂(x, f₂(y, z)) -> F₂(f₂(x, y), z)

Used ordering: Combined order from the following AFS and order.
F₂(x1, x2) = x1
f₂(x1, x2) = x1
a = a
b = b
c = c

Lexicographic Path Order [19].
Precedence:

a > b > c

The following usable rules [14] were oriented:

f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)
f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

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

F₂(x, f₂(y, z)) -> F₂(x, y)
F₂(f₂(a, b), x) -> F₂(a, f₂(c, f₂(b, f₂(a, x))))
F₂(f₂(a, b), x) -> F₂(a, x)
F₂(x, f₂(y, z)) -> F₂(f₂(x, y), z)

The TRS R consists of the following rules:

f₂(f₂(a, b), x) -> f₂(b, f₂(a, f₂(c, f₂(b, f₂(a, x)))))
f₂(x, f₂(y, z)) -> f₂(f₂(x, y), z)

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