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(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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:

ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__from1(X)) -> FROM1(activate1(X))
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
SEL2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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:

ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__from1(X)) -> FROM1(activate1(X))
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
SEL2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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 3 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)) -> ACTIVATE1(X2)
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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 oriented strictly and are deleted.


ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
The remaining pairs can at least be oriented weakly.

FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( from1(x1) ) = x1 + 2


POL( first2(x1, x2) ) = x1 + x2 + 2


POL( ACTIVATE1(x1) ) = max{0, 2x1 - 1}


POL( n__from1(x1) ) = x1 + 2


POL( FIRST2(x1, x2) ) = 2x1 + 2x2 + 2


POL( n__s1(x1) ) = x1 + 1


POL( 0 ) = 2


POL( s1(x1) ) = x1 + 1


POL( nil ) = max{0, -1}


POL( cons2(x1, x2) ) = max{0, x2 - 2}


POL( activate1(x1) ) = x1


POL( n__first2(x1, x2) ) = x1 + x2 + 2



The following usable rules [14] were oriented:

first2(0, Z) -> nil
activate1(X) -> X
activate1(n__s1(X)) -> s1(activate1(X))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
from1(X) -> cons2(X, n__from1(n__s1(X)))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
s1(X) -> n__s1(X)



↳ 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(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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 oriented strictly and are deleted.


SEL2(s1(X), cons2(Y, Z)) -> SEL2(X, activate1(Z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( from1(x1) ) = 2


POL( SEL2(x1, x2) ) = max{0, 2x1 + 2x2 - 2}


POL( first2(x1, x2) ) = 2


POL( n__from1(x1) ) = 1


POL( n__s1(x1) ) = 2x1 + 2


POL( 0 ) = 1


POL( s1(x1) ) = 2x1 + 2


POL( nil ) = max{0, -1}


POL( cons2(x1, x2) ) = 2x2


POL( activate1(x1) ) = 2x1


POL( n__first2(x1, x2) ) = 1



The following usable rules [14] were oriented:

first2(0, Z) -> nil
activate1(X) -> X
activate1(n__s1(X)) -> s1(activate1(X))
from1(X) -> n__from1(X)
first2(X1, X2) -> n__first2(X1, X2)
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(X, activate1(Z)))
from1(X) -> cons2(X, n__from1(n__s1(X)))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
s1(X) -> n__s1(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(n__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)
s1(X) -> n__s1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(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.