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(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
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(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
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(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
from1(X) -> cons2(X, from1(s1(X)))

The set Q consists of the following terms:

2nd1(cons12(x0, cons2(x1, x2)))
2nd1(cons2(x0, x1))
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:

2ND1(cons2(X, X1)) -> 2ND1(cons12(X, X1))
FROM1(X) -> FROM1(s1(X))

The TRS R consists of the following rules:

2nd1(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
from1(X) -> cons2(X, from1(s1(X)))

The set Q consists of the following terms:

2nd1(cons12(x0, cons2(x1, x2)))
2nd1(cons2(x0, x1))
from1(x0)

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:

2ND1(cons2(X, X1)) -> 2ND1(cons12(X, X1))
FROM1(X) -> FROM1(s1(X))

The TRS R consists of the following rules:

2nd1(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
from1(X) -> cons2(X, from1(s1(X)))

The set Q consists of the following terms:

2nd1(cons12(x0, cons2(x1, x2)))
2nd1(cons2(x0, x1))
from1(x0)

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:

FROM1(X) -> FROM1(s1(X))

The TRS R consists of the following rules:

2nd1(cons12(X, cons2(Y, Z))) -> Y
2nd1(cons2(X, X1)) -> 2nd1(cons12(X, X1))
from1(X) -> cons2(X, from1(s1(X)))

The set Q consists of the following terms:

2nd1(cons12(x0, cons2(x1, x2)))
2nd1(cons2(x0, x1))
from1(x0)

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