Termination w.r.t. Q proof of TRS/Endrullislinear1.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:

a₁(a₁(f₂(b, a₁(x)))) -> f₂(a₁(a₁(a₁(x))), b)
a₁(a₁(x)) -> f₂(b, a₁(f₂(a₁(x), b)))
f₂(a₁(x), b) -> f₂(b, a₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a₁(a₁(f₂(b, a₁(x)))) -> f₂(a₁(a₁(a₁(x))), b)
a₁(a₁(x)) -> f₂(b, a₁(f₂(a₁(x), b)))
f₂(a₁(x), b) -> f₂(b, a₁(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:

A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(x))
A₁(a₁(x)) -> F₂(a₁(x), b)
A₁(a₁(x)) -> A₁(f₂(a₁(x), b))
A₁(a₁(x)) -> F₂(b, a₁(f₂(a₁(x), b)))
A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(a₁(x)))
A₁(a₁(f₂(b, a₁(x)))) -> F₂(a₁(a₁(a₁(x))), b)
F₂(a₁(x), b) -> F₂(b, a₁(x))

The TRS R consists of the following rules:

a₁(a₁(f₂(b, a₁(x)))) -> f₂(a₁(a₁(a₁(x))), b)
a₁(a₁(x)) -> f₂(b, a₁(f₂(a₁(x), b)))
f₂(a₁(x), b) -> f₂(b, a₁(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:

A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(x))
A₁(a₁(x)) -> F₂(a₁(x), b)
A₁(a₁(x)) -> A₁(f₂(a₁(x), b))
A₁(a₁(x)) -> F₂(b, a₁(f₂(a₁(x), b)))
A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(a₁(x)))
A₁(a₁(f₂(b, a₁(x)))) -> F₂(a₁(a₁(a₁(x))), b)
F₂(a₁(x), b) -> F₂(b, a₁(x))

The TRS R consists of the following rules:

a₁(a₁(f₂(b, a₁(x)))) -> f₂(a₁(a₁(a₁(x))), b)
a₁(a₁(x)) -> f₂(b, a₁(f₂(a₁(x), b)))
f₂(a₁(x), b) -> f₂(b, a₁(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(x))
A₁(a₁(f₂(b, a₁(x)))) -> A₁(a₁(a₁(x)))

The TRS R consists of the following rules:

a₁(a₁(f₂(b, a₁(x)))) -> f₂(a₁(a₁(a₁(x))), b)
a₁(a₁(x)) -> f₂(b, a₁(f₂(a₁(x), b)))
f₂(a₁(x), b) -> f₂(b, a₁(x))

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