Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/AG01SLASHHASH3.29.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:

f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₃(s₁(x), y, y) -> f₃(y, 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:

f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

The set Q consists of the following terms:

f₃(s₁(x0), x1, 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:

F₃(s₁(x), y, y) -> F₃(y, x, s₁(x))

The TRS R consists of the following rules:

f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

The set Q consists of the following terms:

f₃(s₁(x0), x1, x1)

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:

F₃(s₁(x), y, y) -> F₃(y, x, s₁(x))

The TRS R consists of the following rules:

f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

The set Q consists of the following terms:

f₃(s₁(x0), x1, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.