Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/AG01SLASHHASH3.49.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:

f₁(c₂(s₁(x), y)) -> f₁(c₂(x, s₁(y)))
f₁(c₂(s₁(x), s₁(y))) -> g₁(c₂(x, y))
g₁(c₂(x, s₁(y))) -> g₁(c₂(s₁(x), y))
g₁(c₂(s₁(x), s₁(y))) -> f₁(c₂(x, y))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(c₂(s₁(x), y)) -> f₁(c₂(x, s₁(y)))
f₁(c₂(s₁(x), s₁(y))) -> g₁(c₂(x, y))
g₁(c₂(x, s₁(y))) -> g₁(c₂(s₁(x), y))
g₁(c₂(s₁(x), s₁(y))) -> f₁(c₂(x, 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:

F₁(c₂(s₁(x), s₁(y))) -> G₁(c₂(x, y))
F₁(c₂(s₁(x), y)) -> F₁(c₂(x, s₁(y)))
G₁(c₂(x, s₁(y))) -> G₁(c₂(s₁(x), y))
G₁(c₂(s₁(x), s₁(y))) -> F₁(c₂(x, y))

The TRS R consists of the following rules:

f₁(c₂(s₁(x), y)) -> f₁(c₂(x, s₁(y)))
f₁(c₂(s₁(x), s₁(y))) -> g₁(c₂(x, y))
g₁(c₂(x, s₁(y))) -> g₁(c₂(s₁(x), y))
g₁(c₂(s₁(x), s₁(y))) -> f₁(c₂(x, 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:

F₁(c₂(s₁(x), s₁(y))) -> G₁(c₂(x, y))
F₁(c₂(s₁(x), y)) -> F₁(c₂(x, s₁(y)))
G₁(c₂(x, s₁(y))) -> G₁(c₂(s₁(x), y))
G₁(c₂(s₁(x), s₁(y))) -> F₁(c₂(x, y))

The TRS R consists of the following rules:

f₁(c₂(s₁(x), y)) -> f₁(c₂(x, s₁(y)))
f₁(c₂(s₁(x), s₁(y))) -> g₁(c₂(x, y))
g₁(c₂(x, s₁(y))) -> g₁(c₂(s₁(x), y))
g₁(c₂(s₁(x), s₁(y))) -> f₁(c₂(x, y))

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