Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar1/SK90SLASH4.54.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:

g₁(f₂(x, y)) -> f₂(f₂(g₁(g₁(x)), g₁(g₁(y))), f₂(g₁(g₁(x)), g₁(g₁(y))))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

g₁(f₂(x, y)) -> f₂(f₂(g₁(g₁(x)), g₁(g₁(y))), f₂(g₁(g₁(x)), g₁(g₁(y))))

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(f₂(x, y)) -> f₂(f₂(g₁(g₁(x)), g₁(g₁(y))), f₂(g₁(g₁(x)), g₁(g₁(y))))

The set Q consists of the following terms:

g₁(f₂(x0, x1))

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

G₁(f₂(x, y)) -> G₁(x)
G₁(f₂(x, y)) -> G₁(g₁(x))
G₁(f₂(x, y)) -> G₁(y)
G₁(f₂(x, y)) -> G₁(g₁(y))

The TRS R consists of the following rules:

g₁(f₂(x, y)) -> f₂(f₂(g₁(g₁(x)), g₁(g₁(y))), f₂(g₁(g₁(x)), g₁(g₁(y))))

The set Q consists of the following terms:

g₁(f₂(x0, x1))

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:

G₁(f₂(x, y)) -> G₁(x)
G₁(f₂(x, y)) -> G₁(g₁(x))
G₁(f₂(x, y)) -> G₁(y)
G₁(f₂(x, y)) -> G₁(g₁(y))

The TRS R consists of the following rules:

g₁(f₂(x, y)) -> f₂(f₂(g₁(g₁(x)), g₁(g₁(y))), f₂(g₁(g₁(x)), g₁(g₁(y))))

The set Q consists of the following terms:

g₁(f₂(x0, x1))

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