Termination w.r.t. Q proof of TRS/nontermin/AG01#4.3.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:

f₃(g₂(x, y), x, z) -> f₃(z, z, z)
g₂(x, y) -> x
g₂(x, y) -> y

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₃(g₂(x, y), x, z) -> f₃(z, z, z)
g₂(x, y) -> x
g₂(x, 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:

F₃(g₂(x, y), x, z) -> F₃(z, z, z)

The TRS R consists of the following rules:

f₃(g₂(x, y), x, z) -> f₃(z, z, z)
g₂(x, y) -> x
g₂(x, 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:

F₃(g₂(x, y), x, z) -> F₃(z, z, z)

The TRS R consists of the following rules:

f₃(g₂(x, y), x, z) -> f₃(z, z, z)
g₂(x, y) -> x
g₂(x, y) -> y

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