Termination w.r.t. Q proof of TRS/HMt006.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:

f₂(g₂(x, y), f₂(y, y)) -> f₂(g₂(y, x), y)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₂(g₂(x, y), f₂(y, y)) -> f₂(g₂(y, x), y)

Q is empty.

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(g₂(x, y), f₂(y, y)) -> F₂(g₂(y, x), y)

The TRS R consists of the following rules:

f₂(g₂(x, y), f₂(y, y)) -> f₂(g₂(y, x), y)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

F₂(g₂(x, y), f₂(y, y)) -> F₂(g₂(y, x), y)

The TRS R consists of the following rules:

f₂(g₂(x, y), f₂(y, y)) -> f₂(g₂(y, x), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

F₂(g₂(x, y), f₂(y, y)) -> F₂(g₂(y, x), y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(F₂(x₁, x₂)) = 2·x₁ + 3·x₁·x₂ + 2·x₂
POL(f₂(x₁, x₂)) = 2 + 3·x₁ + 2·x₁·x₂ + 3·x₂
POL(g₂(x₁, x₂)) = 2·x₁ + 3·x₁·x₂ + x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ PisEmptyProof

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

f₂(g₂(x, y), f₂(y, y)) -> f₂(g₂(y, x), y)

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.