Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/nonterminSLASHRubio-innSLASHbn111.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:

f₂(f₂(X, Y), Z) -> f₂(X, f₂(Y, Z))
f₂(X, f₂(Y, Z)) -> f₂(Y, Y)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₂(f₂(X, Y), Z) -> f₂(X, f₂(Y, Z))
f₂(X, f₂(Y, Z)) -> f₂(Y, Y)

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:

F₂(X, f₂(Y, Z)) -> F₂(Y, Y)
F₂(f₂(X, Y), Z) -> F₂(X, f₂(Y, Z))
F₂(f₂(X, Y), Z) -> F₂(Y, Z)

The TRS R consists of the following rules:

f₂(f₂(X, Y), Z) -> f₂(X, f₂(Y, Z))
f₂(X, f₂(Y, Z)) -> f₂(Y, Y)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

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

F₂(X, f₂(Y, Z)) -> F₂(Y, Y)
F₂(f₂(X, Y), Z) -> F₂(X, f₂(Y, Z))
F₂(f₂(X, Y), Z) -> F₂(Y, Z)

The TRS R consists of the following rules:

f₂(f₂(X, Y), Z) -> f₂(X, f₂(Y, Z))
f₂(X, f₂(Y, Z)) -> f₂(Y, Y)

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