Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/AG01SLASHHASH3.31.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(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:

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

The set Q consists of the following terms:

f₁(g₁(x0))

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

F₁(g₁(x)) -> F₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))
F₁(g₁(x)) -> F₁(x)

The TRS R consists of the following rules:

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

The set Q consists of the following terms:

f₁(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:

F₁(g₁(x)) -> F₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))
F₁(g₁(x)) -> F₁(x)

The TRS R consists of the following rules:

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

The set Q consists of the following terms:

f₁(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

              ↳ QDPAfsSolverProof

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

F₁(g₁(x)) -> F₁(x)

The TRS R consists of the following rules:

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

The set Q consists of the following terms:

f₁(g₁(x0))

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

F₁(g₁(x)) -> F₁(x)

Used argument filtering: F₁(x1) = x1
g₁(x1) = g₁(x1)
Used ordering: Quasi Precedence: trivial

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ QDPAfsSolverProof

                ↳ QDP

                  ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f₁(g₁(x)) -> f₁(a₂(g₁(g₁(f₁(x))), g₁(f₁(x))))

The set Q consists of the following terms:

f₁(g₁(x0))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.