Termination w.r.t. Q proof of TRS/SK904.54.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

  ↳ 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))))

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:

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))))

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:

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))))

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