Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/variousSLASH11.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:

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

The set Q consists of the following terms:

f₃(0, 1, x0)
h₁(0)
h₁(g₂(x0, x1))

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(0, 1, x) -> H₁(x)
F₃(0, 1, x) -> F₃(h₁(x), h₁(x), x)

The TRS R consists of the following rules:

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

The set Q consists of the following terms:

f₃(0, 1, x0)
h₁(0)
h₁(g₂(x0, x1))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₃(0, 1, x) -> H₁(x)
F₃(0, 1, x) -> F₃(h₁(x), h₁(x), x)

The TRS R consists of the following rules:

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

The set Q consists of the following terms:

f₃(0, 1, x0)
h₁(0)
h₁(g₂(x0, x1))

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

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

F₃(0, 1, x) -> F₃(h₁(x), h₁(x), x)

The TRS R consists of the following rules:

f₃(0, 1, x) -> f₃(h₁(x), h₁(x), x)
h₁(0) -> 0
h₁(g₂(x, y)) -> y

The set Q consists of the following terms:

f₃(0, 1, x0)
h₁(0)
h₁(g₂(x0, x1))

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