Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/AG01SLASHHASH3.26.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(x) -> s₁(x)
f₁(s₁(s₁(x))) -> s₁(f₁(f₁(x)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(x) -> s₁(x)
f₁(s₁(s₁(x))) -> s₁(f₁(f₁(x)))

Q is empty.

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

F₁(s₁(s₁(x))) -> F₁(x)
F₁(s₁(s₁(x))) -> F₁(f₁(x))

The TRS R consists of the following rules:

f₁(x) -> s₁(x)
f₁(s₁(s₁(x))) -> s₁(f₁(f₁(x)))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPAfsSolverProof

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

F₁(s₁(s₁(x))) -> F₁(x)
F₁(s₁(s₁(x))) -> F₁(f₁(x))

The TRS R consists of the following rules:

f₁(x) -> s₁(x)
f₁(s₁(s₁(x))) -> s₁(f₁(f₁(x)))

Q is empty.
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(s₁(s₁(x))) -> F₁(x)
F₁(s₁(s₁(x))) -> F₁(f₁(x))

Used argument filtering: F₁(x1) = x1
s₁(x1) = s₁(x1)
f₁(x1) = f₁(x1)
Used ordering: Quasi Precedence: [s_1, f_1]

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPAfsSolverProof

        ↳ QDP

          ↳ PisEmptyProof

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

f₁(x) -> s₁(x)
f₁(s₁(s₁(x))) -> s₁(f₁(f₁(x)))

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