Termination w.r.t. Q proof of TRS/D3304.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(f₁(x)) -> g₁(f₁(x))
g₁(g₁(x)) -> f₁(x)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(f₁(x)) -> g₁(f₁(x))
g₁(g₁(x)) -> f₁(x)

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:

F₁(f₁(x)) -> G₁(f₁(x))
G₁(g₁(x)) -> F₁(x)

The TRS R consists of the following rules:

f₁(f₁(x)) -> g₁(f₁(x))
g₁(g₁(x)) -> f₁(x)

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:

F₁(f₁(x)) -> G₁(f₁(x))
G₁(g₁(x)) -> F₁(x)

The TRS R consists of the following rules:

f₁(f₁(x)) -> g₁(f₁(x))
g₁(g₁(x)) -> f₁(x)

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.

F₁(f₁(x)) -> G₁(f₁(x))
G₁(g₁(x)) -> F₁(x)

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

POL(F₁(x₁)) = 1 + 3·x₁ + 3·x₁²
POL(G₁(x₁)) = 3·x₁ + 3·x₁²
POL(f₁(x₁)) = 1 + 2·x₁ + 3·x₁²
POL(g₁(x₁)) = 3 + x₁ + x₁²

The following usable rules [14] were oriented:

g₁(g₁(x)) -> f₁(x)
f₁(f₁(x)) -> g₁(f₁(x))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ PisEmptyProof

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

f₁(f₁(x)) -> g₁(f₁(x))
g₁(g₁(x)) -> 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.