Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/nonterminSLASHAG01SLASHHASH4.31.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(d₁(x)) -> d₁(c₁(b₁(a₁(x))))
b₁(c₁(x)) -> c₁(d₁(a₁(b₁(x))))
a₁(c₁(x)) -> x
b₁(d₁(x)) -> x

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

a₁(d₁(x)) -> d₁(c₁(b₁(a₁(x))))
b₁(c₁(x)) -> c₁(d₁(a₁(b₁(x))))
a₁(c₁(x)) -> x
b₁(d₁(x)) -> x

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

a₁(d₁(x)) -> d₁(c₁(b₁(a₁(x))))
b₁(c₁(x)) -> c₁(d₁(a₁(b₁(x))))
a₁(c₁(x)) -> x
b₁(d₁(x)) -> x

The set Q consists of the following terms:

a₁(d₁(x0))
b₁(c₁(x0))
a₁(c₁(x0))
b₁(d₁(x0))

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

A₁(d₁(x)) -> B₁(a₁(x))
B₁(c₁(x)) -> A₁(b₁(x))
B₁(c₁(x)) -> B₁(x)
A₁(d₁(x)) -> A₁(x)

The TRS R consists of the following rules:

a₁(d₁(x)) -> d₁(c₁(b₁(a₁(x))))
b₁(c₁(x)) -> c₁(d₁(a₁(b₁(x))))
a₁(c₁(x)) -> x
b₁(d₁(x)) -> x

The set Q consists of the following terms:

a₁(d₁(x0))
b₁(c₁(x0))
a₁(c₁(x0))
b₁(d₁(x0))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

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

A₁(d₁(x)) -> B₁(a₁(x))
B₁(c₁(x)) -> A₁(b₁(x))
B₁(c₁(x)) -> B₁(x)
A₁(d₁(x)) -> A₁(x)

The TRS R consists of the following rules:

a₁(d₁(x)) -> d₁(c₁(b₁(a₁(x))))
b₁(c₁(x)) -> c₁(d₁(a₁(b₁(x))))
a₁(c₁(x)) -> x
b₁(d₁(x)) -> x

The set Q consists of the following terms:

a₁(d₁(x0))
b₁(c₁(x0))
a₁(c₁(x0))
b₁(d₁(x0))

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