Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/ZantemaSLASHz28.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(0, x), 1) -> f₂(g₁(f₂(x, x)), x)
f₂(g₁(x), y) -> g₁(f₂(x, y))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₂(f₂(0, x), 1) -> f₂(g₁(f₂(x, x)), x)
f₂(g₁(x), y) -> g₁(f₂(x, 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(0, x), 1) -> f₂(g₁(f₂(x, x)), x)
f₂(g₁(x), y) -> g₁(f₂(x, y))

The set Q consists of the following terms:

f₂(f₂(0, x0), 1)
f₂(g₁(x0), x1)

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(g₁(x), y) -> F₂(x, y)
F₂(f₂(0, x), 1) -> F₂(x, x)
F₂(f₂(0, x), 1) -> F₂(g₁(f₂(x, x)), x)

The TRS R consists of the following rules:

f₂(f₂(0, x), 1) -> f₂(g₁(f₂(x, x)), x)
f₂(g₁(x), y) -> g₁(f₂(x, y))

The set Q consists of the following terms:

f₂(f₂(0, x0), 1)
f₂(g₁(x0), x1)

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(g₁(x), y) -> F₂(x, y)
F₂(f₂(0, x), 1) -> F₂(x, x)
F₂(f₂(0, x), 1) -> F₂(g₁(f₂(x, x)), x)

The TRS R consists of the following rules:

f₂(f₂(0, x), 1) -> f₂(g₁(f₂(x, x)), x)
f₂(g₁(x), y) -> g₁(f₂(x, y))

The set Q consists of the following terms:

f₂(f₂(0, x0), 1)
f₂(g₁(x0), x1)

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