Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar4/nonterminSLASHAG01SLASHHASH4.20.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:

f₁(f₁(x)) -> f₁(x)
g₁(0) -> g₁(f₁(0))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(f₁(x)) -> f₁(x)
g₁(0) -> g₁(f₁(0))

Q is empty.

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

G₁(0) -> G₁(f₁(0))
G₁(0) -> F₁(0)

The TRS R consists of the following rules:

f₁(f₁(x)) -> f₁(x)
g₁(0) -> g₁(f₁(0))

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:

G₁(0) -> G₁(f₁(0))
G₁(0) -> F₁(0)

The TRS R consists of the following rules:

f₁(f₁(x)) -> f₁(x)
g₁(0) -> g₁(f₁(0))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 0 SCCs with 2 less nodes.