Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/nonterminSLASHHMSLASHn001.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:

h₁(f₁(f₁(x))) -> h₁(f₁(g₁(f₁(x))))
f₁(g₁(f₁(x))) -> f₁(f₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

h₁(f₁(f₁(x))) -> h₁(f₁(g₁(f₁(x))))
f₁(g₁(f₁(x))) -> f₁(f₁(x))

Q is empty.

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

H₁(f₁(f₁(x))) -> H₁(f₁(g₁(f₁(x))))
H₁(f₁(f₁(x))) -> F₁(g₁(f₁(x)))
F₁(g₁(f₁(x))) -> F₁(f₁(x))

The TRS R consists of the following rules:

h₁(f₁(f₁(x))) -> h₁(f₁(g₁(f₁(x))))
f₁(g₁(f₁(x))) -> f₁(f₁(x))

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:

H₁(f₁(f₁(x))) -> H₁(f₁(g₁(f₁(x))))
H₁(f₁(f₁(x))) -> F₁(g₁(f₁(x)))
F₁(g₁(f₁(x))) -> F₁(f₁(x))

The TRS R consists of the following rules:

h₁(f₁(f₁(x))) -> h₁(f₁(g₁(f₁(x))))
f₁(g₁(f₁(x))) -> f₁(f₁(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

H₁(f₁(f₁(x))) -> H₁(f₁(g₁(f₁(x))))

The TRS R consists of the following rules:

h₁(f₁(f₁(x))) -> h₁(f₁(g₁(f₁(x))))
f₁(g₁(f₁(x))) -> f₁(f₁(x))

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