Termination w.r.t. Q proof of TRS/nontermin/caribooex3.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(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:

G₁(x) -> F₁(g₁(x))
F₁(g₁(a)) -> G₁(b)
G₁(x) -> G₁(x)
F₁(g₁(a)) -> F₁(s₁(g₁(b)))

The TRS R consists of the following rules:

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

G₁(x) -> F₁(g₁(x))
F₁(g₁(a)) -> G₁(b)
G₁(x) -> G₁(x)
F₁(g₁(a)) -> F₁(s₁(g₁(b)))

The TRS R consists of the following rules:

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

G₁(x) -> F₁(g₁(x))
F₁(g₁(a)) -> G₁(b)
G₁(x) -> G₁(x)

The TRS R consists of the following rules:

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(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(g₁(a)) -> G₁(b)

The remaining pairs can at least be oriented weakly.

G₁(x) -> F₁(g₁(x))
G₁(x) -> G₁(x)

Used ordering: Polynomial interpretation [21]:

POL(F₁(x₁)) = x₁
POL(G₁(x₁)) = x₁
POL(a) = 1
POL(b) = 0
POL(f₁(x₁)) = 0
POL(g₁(x₁)) = x₁
POL(s₁(x₁)) = 0

The following usable rules [14] were oriented:

f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))
f₁(g₁(a)) -> f₁(s₁(g₁(b)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

G₁(x) -> F₁(g₁(x))
G₁(x) -> G₁(x)

The TRS R consists of the following rules:

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

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

G₁(x) -> G₁(x)

The TRS R consists of the following rules:

f₁(g₁(a)) -> f₁(s₁(g₁(b)))
f₁(f₁(x)) -> b
g₁(x) -> f₁(g₁(x))

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