Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/SK90SLASH4.40.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(f₂(a, x), y), z) -> f₂(f₂(x, z), f₂(y, z))
f₂(f₂(b, x), y) -> x
f₂(c, y) -> y

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

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

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

f₂(f₂(f₂(a, x0), x1), x2)
f₂(f₂(b, x0), x1)
f₂(c, x0)

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f₂(f₂(f₂(a, x0), x1), x2)
f₂(f₂(b, x0), x1)
f₂(c, x0)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f₂(f₂(f₂(a, x0), x1), x2)
f₂(f₂(b, x0), x1)
f₂(c, x0)

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