Termination w.r.t. Q proof of TRS/Zantema06while2.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:

f₂(f₂(s₁(x), 0), f₂(y, z)) -> f₂(f₂(y, z), f₂(y, s₁(z)))
f₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> f₂(f₂(x, y), f₂(z, w))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₂(f₂(s₁(x), 0), f₂(y, z)) -> f₂(f₂(y, z), f₂(y, s₁(z)))
f₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> f₂(f₂(x, y), f₂(z, w))

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:

F₂(f₂(s₁(x), 0), f₂(y, z)) -> F₂(y, s₁(z))
F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(x, y)
F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(f₂(x, y), f₂(z, w))
F₂(f₂(s₁(x), 0), f₂(y, z)) -> F₂(f₂(y, z), f₂(y, s₁(z)))

The TRS R consists of the following rules:

f₂(f₂(s₁(x), 0), f₂(y, z)) -> f₂(f₂(y, z), f₂(y, s₁(z)))
f₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> f₂(f₂(x, y), f₂(z, w))

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:

F₂(f₂(s₁(x), 0), f₂(y, z)) -> F₂(y, s₁(z))
F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(x, y)
F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(f₂(x, y), f₂(z, w))
F₂(f₂(s₁(x), 0), f₂(y, z)) -> F₂(f₂(y, z), f₂(y, s₁(z)))

The TRS R consists of the following rules:

f₂(f₂(s₁(x), 0), f₂(y, z)) -> f₂(f₂(y, z), f₂(y, s₁(z)))
f₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> f₂(f₂(x, y), f₂(z, w))

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:

F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(x, y)
F₂(f₂(s₁(x), 0), f₂(y, z)) -> F₂(f₂(y, z), f₂(y, s₁(z)))
F₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> F₂(f₂(x, y), f₂(z, w))

The TRS R consists of the following rules:

f₂(f₂(s₁(x), 0), f₂(y, z)) -> f₂(f₂(y, z), f₂(y, s₁(z)))
f₂(f₂(s₁(x), s₁(y)), f₂(z, w)) -> f₂(f₂(x, y), f₂(z, w))

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