Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

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:

SEL2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
ACTIVATE1(n__from1(X)) -> FROM1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

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:

SEL2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
ACTIVATE1(n__from1(X)) -> FROM1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.
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.


ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
The remaining pairs can at least by weakly be oriented.

FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
Used ordering: Combined order from the following AFS and order.
ACTIVATE1(x1)  =  ACTIVATE1(x1)
n__first2(x1, x2)  =  n__first2(x1, x2)
FIRST2(x1, x2)  =  FIRST1(x2)
s1(x1)  =  s
cons2(x1, x2)  =  x2

Lexicographic Path Order [19].
Precedence:
s > [ACTIVATE1, nfirst2, FIRST1]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ DependencyGraphProof
          ↳ QDP

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

FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ QDPOrderProof

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

SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.
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.


SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SEL2(x1, x2)  =  x1
s1(x1)  =  s1(x1)
cons2(x1, x2)  =  x1
activate1(x1)  =  activate1(x1)
n__from1(x1)  =  n__from1(x1)
from1(x1)  =  from1(x1)
n__first2(x1, x2)  =  n__first1(x2)
first2(x1, x2)  =  first1(x2)
0  =  0
nil  =  nil

Lexicographic Path Order [19].
Precedence:
s1 > [nfrom1, from1]
[nfirst1, first1] > activate1 > [nfrom1, from1]
[nfirst1, first1] > [0, nil] > [nfrom1, from1]


The following usable rules [14] were oriented:

activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> cons2(X, n__from1(s1(X)))
from1(X) -> n__from1(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
first2(0, Z) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
sel2(0, cons2(X, Z)) -> X
sel2(s1(X), cons2(Y, Z)) -> sel2(X, activate1(Z))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.