Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/variousSLASH23.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(0, f₂(x, x)) -> x
g₂(x, s₁(y)) -> g₂(f₂(x, y), 0)
g₂(s₁(x), y) -> g₂(f₂(x, y), 0)
g₂(f₂(x, y), 0) -> f₂(g₂(x, 0), g₂(y, 0))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

g₂(0, f₂(x, x)) -> x
g₂(x, s₁(y)) -> g₂(f₂(x, y), 0)
g₂(s₁(x), y) -> g₂(f₂(x, y), 0)
g₂(f₂(x, y), 0) -> f₂(g₂(x, 0), g₂(y, 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:

G₂(x, s₁(y)) -> G₂(f₂(x, y), 0)
G₂(f₂(x, y), 0) -> G₂(y, 0)
G₂(s₁(x), y) -> G₂(f₂(x, y), 0)
G₂(f₂(x, y), 0) -> G₂(x, 0)

The TRS R consists of the following rules:

g₂(0, f₂(x, x)) -> x
g₂(x, s₁(y)) -> g₂(f₂(x, y), 0)
g₂(s₁(x), y) -> g₂(f₂(x, y), 0)
g₂(f₂(x, y), 0) -> f₂(g₂(x, 0), g₂(y, 0))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

G₂(x, s₁(y)) -> G₂(f₂(x, y), 0)
G₂(f₂(x, y), 0) -> G₂(y, 0)
G₂(s₁(x), y) -> G₂(f₂(x, y), 0)
G₂(f₂(x, y), 0) -> G₂(x, 0)

The TRS R consists of the following rules:

g₂(0, f₂(x, x)) -> x
g₂(x, s₁(y)) -> g₂(f₂(x, y), 0)
g₂(s₁(x), y) -> g₂(f₂(x, y), 0)
g₂(f₂(x, y), 0) -> f₂(g₂(x, 0), g₂(y, 0))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

G₂(f₂(x, y), 0) -> G₂(y, 0)
G₂(s₁(x), y) -> G₂(f₂(x, y), 0)
G₂(f₂(x, y), 0) -> G₂(x, 0)

The TRS R consists of the following rules:

g₂(0, f₂(x, x)) -> x
g₂(x, s₁(y)) -> g₂(f₂(x, y), 0)
g₂(s₁(x), y) -> g₂(f₂(x, y), 0)
g₂(f₂(x, y), 0) -> f₂(g₂(x, 0), g₂(y, 0))

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