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.

G₁(g₁(x)) -> F₁(x)

The remaining pairs can at least be oriented weakly.

F₁(f₁(x)) -> G₁(f₁(x))

Used ordering: Polynomial interpretation [21]:

POL(F₁(x₁)) = x₁
POL(G₁(x₁)) = x₁
POL(f₁(x₁)) = 1 + x₁
POL(g₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₁(f₁(x)) -> G₁(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.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.