Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/variousSLASH25.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:

g₁(a) -> g₁(b)
b -> f₂(a, a)
f₂(a, a) -> g₁(d)

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

g₁(a) -> g₁(b)
b -> f₂(a, a)
f₂(a, a) -> g₁(d)

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:

g₁(a) -> g₁(b)
b -> f₂(a, a)
f₂(a, a) -> g₁(d)

The set Q consists of the following terms:

g₁(a)
b
f₂(a, a)

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F₂(a, a) -> G₁(d)
B -> F₂(a, a)
G₁(a) -> B
G₁(a) -> G₁(b)

The TRS R consists of the following rules:

g₁(a) -> g₁(b)
b -> f₂(a, a)
f₂(a, a) -> g₁(d)

The set Q consists of the following terms:

g₁(a)
b
f₂(a, a)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₂(a, a) -> G₁(d)
B -> F₂(a, a)
G₁(a) -> B
G₁(a) -> G₁(b)

The TRS R consists of the following rules:

g₁(a) -> g₁(b)
b -> f₂(a, a)
f₂(a, a) -> g₁(d)

The set Q consists of the following terms:

g₁(a)
b
f₂(a, a)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 4 less nodes.