Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/nonterminSLASHCSRSLASHEx1_2

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:

2nd₁(cons₂(X, cons₂(Y, Z))) -> Y
from₁(X) -> cons₂(X, from₁(s₁(X)))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

2nd₁(cons₂(X, cons₂(Y, Z))) -> Y
from₁(X) -> cons₂(X, from₁(s₁(X)))

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:

2nd₁(cons₂(X, cons₂(Y, Z))) -> Y
from₁(X) -> cons₂(X, from₁(s₁(X)))

The set Q consists of the following terms:

2nd₁(cons₂(x0, cons₂(x1, x2)))
from₁(x0)

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

FROM₁(X) -> FROM₁(s₁(X))

The TRS R consists of the following rules:

2nd₁(cons₂(X, cons₂(Y, Z))) -> Y
from₁(X) -> cons₂(X, from₁(s₁(X)))

The set Q consists of the following terms:

2nd₁(cons₂(x0, cons₂(x1, x2)))
from₁(x0)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

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

FROM₁(X) -> FROM₁(s₁(X))

The TRS R consists of the following rules:

2nd₁(cons₂(X, cons₂(Y, Z))) -> Y
from₁(X) -> cons₂(X, from₁(s₁(X)))

The set Q consists of the following terms:

2nd₁(cons₂(x0, cons₂(x1, x2)))
from₁(x0)

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