Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/AG01SLASHHASH3.53a.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:

g₂(x, y) -> x
g₂(x, y) -> y
f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

g₂(x, y) -> x
g₂(x, y) -> y
f₃(s₁(x), y, y) -> f₃(y, x, s₁(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:

F₃(s₁(x), y, y) -> F₃(y, x, s₁(x))

The TRS R consists of the following rules:

g₂(x, y) -> x
g₂(x, y) -> y
f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

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(s₁(x), y, y) -> F₃(y, x, s₁(x))

The TRS R consists of the following rules:

g₂(x, y) -> x
g₂(x, y) -> y
f₃(s₁(x), y, y) -> f₃(y, x, s₁(x))

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