Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/AG01SLASHHASH3.15.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:

average₂(s₁(x), y) -> average₂(x, s₁(y))
average₂(x, s₁(s₁(s₁(y)))) -> s₁(average₂(s₁(x), y))
average₂(0, 0) -> 0
average₂(0, s₁(0)) -> 0
average₂(0, s₁(s₁(0))) -> s₁(0)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

average₂(s₁(x), y) -> average₂(x, s₁(y))
average₂(x, s₁(s₁(s₁(y)))) -> s₁(average₂(s₁(x), y))
average₂(0, 0) -> 0
average₂(0, s₁(0)) -> 0
average₂(0, s₁(s₁(0))) -> s₁(0)

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:

AVERAGE₂(x, s₁(s₁(s₁(y)))) -> AVERAGE₂(s₁(x), y)
AVERAGE₂(s₁(x), y) -> AVERAGE₂(x, s₁(y))

The TRS R consists of the following rules:

average₂(s₁(x), y) -> average₂(x, s₁(y))
average₂(x, s₁(s₁(s₁(y)))) -> s₁(average₂(s₁(x), y))
average₂(0, 0) -> 0
average₂(0, s₁(0)) -> 0
average₂(0, s₁(s₁(0))) -> s₁(0)

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:

AVERAGE₂(x, s₁(s₁(s₁(y)))) -> AVERAGE₂(s₁(x), y)
AVERAGE₂(s₁(x), y) -> AVERAGE₂(x, s₁(y))

The TRS R consists of the following rules:

average₂(s₁(x), y) -> average₂(x, s₁(y))
average₂(x, s₁(s₁(s₁(y)))) -> s₁(average₂(s₁(x), y))
average₂(0, 0) -> 0
average₂(0, s₁(0)) -> 0
average₂(0, s₁(s₁(0))) -> s₁(0)

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