Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar4/nonterminSLASHAG01SLASHHASH4.25.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(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₂(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

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(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

The set Q consists of the following terms:

f₂(x0, x0)
g₁(x0)

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

F₂(x, x) -> F₂(g₁(x), x)
F₂(x, x) -> G₁(x)

The TRS R consists of the following rules:

f₂(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

The set Q consists of the following terms:

f₂(x0, x0)
g₁(x0)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₂(x, x) -> F₂(g₁(x), x)
F₂(x, x) -> G₁(x)

The TRS R consists of the following rules:

f₂(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

The set Q consists of the following terms:

f₂(x0, x0)
g₁(x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 1 less node.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

F₂(x, x) -> F₂(g₁(x), x)

The TRS R consists of the following rules:

f₂(x, x) -> f₂(g₁(x), x)
g₁(x) -> s₁(x)

The set Q consists of the following terms:

f₂(x0, x0)
g₁(x0)

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