Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/nonterminSLASHCSRSLASHEx4_4

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(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), 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:

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

The set Q consists of the following terms:

f₂(g₁(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:

F₂(g₁(X), Y) -> F₂(X, f₂(g₁(X), Y))
F₂(g₁(X), Y) -> F₂(g₁(X), Y)

The TRS R consists of the following rules:

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

The set Q consists of the following terms:

f₂(g₁(x0), x1)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

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

F₂(g₁(X), Y) -> F₂(X, f₂(g₁(X), Y))
F₂(g₁(X), Y) -> F₂(g₁(X), Y)

The TRS R consists of the following rules:

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

The set Q consists of the following terms:

f₂(g₁(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

F₂(g₁(X), Y) -> F₂(X, f₂(g₁(X), Y))

The remaining pairs can at least by weakly be oriented.

F₂(g₁(X), Y) -> F₂(g₁(X), Y)

Used ordering: Combined order from the following AFS and order.
F₂(x1, x2) = x1
g₁(x1) = g₁(x1)
f₂(x1, x2) = f

Lexicographic Path Order [19].
Precedence:

f > g₁

The following usable rules [14] were oriented:

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

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

F₂(g₁(X), Y) -> F₂(g₁(X), Y)

The TRS R consists of the following rules:

f₂(g₁(X), Y) -> f₂(X, f₂(g₁(X), Y))

The set Q consists of the following terms:

f₂(g₁(x0), x1)

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