Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/SK90SLASH2.46.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:

a₁(a₁(x)) -> b₁(b₁(x))
b₁(b₁(a₁(x))) -> a₁(b₁(b₁(x)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a₁(a₁(x)) -> b₁(b₁(x))
b₁(b₁(a₁(x))) -> a₁(b₁(b₁(x)))

Q is empty.

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

A₁(a₁(x)) -> B₁(x)
B₁(b₁(a₁(x))) -> A₁(b₁(b₁(x)))
B₁(b₁(a₁(x))) -> B₁(x)
B₁(b₁(a₁(x))) -> B₁(b₁(x))
A₁(a₁(x)) -> B₁(b₁(x))

The TRS R consists of the following rules:

a₁(a₁(x)) -> b₁(b₁(x))
b₁(b₁(a₁(x))) -> a₁(b₁(b₁(x)))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPAfsSolverProof

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

A₁(a₁(x)) -> B₁(x)
B₁(b₁(a₁(x))) -> A₁(b₁(b₁(x)))
B₁(b₁(a₁(x))) -> B₁(x)
B₁(b₁(a₁(x))) -> B₁(b₁(x))
A₁(a₁(x)) -> B₁(b₁(x))

The TRS R consists of the following rules:

a₁(a₁(x)) -> b₁(b₁(x))
b₁(b₁(a₁(x))) -> a₁(b₁(b₁(x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

A₁(a₁(x)) -> B₁(x)
B₁(b₁(a₁(x))) -> A₁(b₁(b₁(x)))
B₁(b₁(a₁(x))) -> B₁(x)
B₁(b₁(a₁(x))) -> B₁(b₁(x))
A₁(a₁(x)) -> B₁(b₁(x))

Used argument filtering: A₁(x1) = x1
a₁(x1) = a₁(x1)
B₁(x1) = x1
b₁(x1) = x1
Used ordering: Quasi Precedence: trivial

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPAfsSolverProof

        ↳ QDP

          ↳ PisEmptyProof

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

a₁(a₁(x)) -> b₁(b₁(x))
b₁(b₁(a₁(x))) -> a₁(b₁(b₁(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.