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)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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:

2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSPOS2(N, activate1(Z))
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> ACTIVATE1(Z)
2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> ACTIVATE1(Z)
ACTIVATE1(n__from1(X)) -> FROM1(X)
SQUARE1(X) -> TIMES2(X, X)
PI1(X) -> FROM1(0)
2NDSPOS2(s1(N), cons2(X, Z)) -> 2NDSPOS2(s1(N), cons22(X, activate1(Z)))
PI1(X) -> 2NDSPOS2(X, from1(0))
PLUS2(s1(X), Y) -> PLUS2(X, Y)
TIMES2(s1(X), Y) -> PLUS2(Y, times2(X, Y))
TIMES2(s1(X), Y) -> TIMES2(X, Y)
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSNEG2(N, activate1(Z))
2NDSNEG2(s1(N), cons2(X, Z)) -> 2NDSNEG2(s1(N), cons22(X, activate1(Z)))
2NDSNEG2(s1(N), cons2(X, Z)) -> ACTIVATE1(Z)
2NDSPOS2(s1(N), cons2(X, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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:

2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSPOS2(N, activate1(Z))
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> ACTIVATE1(Z)
2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> ACTIVATE1(Z)
ACTIVATE1(n__from1(X)) -> FROM1(X)
SQUARE1(X) -> TIMES2(X, X)
PI1(X) -> FROM1(0)
2NDSPOS2(s1(N), cons2(X, Z)) -> 2NDSPOS2(s1(N), cons22(X, activate1(Z)))
PI1(X) -> 2NDSPOS2(X, from1(0))
PLUS2(s1(X), Y) -> PLUS2(X, Y)
TIMES2(s1(X), Y) -> PLUS2(Y, times2(X, Y))
TIMES2(s1(X), Y) -> TIMES2(X, Y)
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSNEG2(N, activate1(Z))
2NDSNEG2(s1(N), cons2(X, Z)) -> 2NDSNEG2(s1(N), cons22(X, activate1(Z)))
2NDSNEG2(s1(N), cons2(X, Z)) -> ACTIVATE1(Z)
2NDSPOS2(s1(N), cons2(X, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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 3 SCCs with 9 less nodes.

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

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

PLUS2(s1(X), Y) -> PLUS2(X, Y)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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.


PLUS2(s1(X), Y) -> PLUS2(X, Y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PLUS2(x1, x2)  =  PLUS1(x1)
s1(x1)  =  s1(x1)

Lexicographic Path Order [19].
Precedence:
[PLUS1, s1]


The following usable rules [14] were oriented: none



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

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

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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.

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

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

TIMES2(s1(X), Y) -> TIMES2(X, Y)

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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.


TIMES2(s1(X), Y) -> TIMES2(X, Y)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
TIMES2(x1, x2)  =  TIMES1(x1)
s1(x1)  =  s1(x1)

Lexicographic Path Order [19].
Precedence:
[TIMES1, s1]


The following usable rules [14] were oriented: none



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

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

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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.

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

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

2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSPOS2(N, activate1(Z))
2NDSNEG2(s1(N), cons2(X, Z)) -> 2NDSNEG2(s1(N), cons22(X, activate1(Z)))
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSNEG2(N, activate1(Z))
2NDSPOS2(s1(N), cons2(X, Z)) -> 2NDSPOS2(s1(N), cons22(X, activate1(Z)))

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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.


2NDSNEG2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSPOS2(N, activate1(Z))
2NDSPOS2(s1(N), cons22(X, cons2(Y, Z))) -> 2NDSNEG2(N, activate1(Z))
The remaining pairs can at least by weakly be oriented.

2NDSNEG2(s1(N), cons2(X, Z)) -> 2NDSNEG2(s1(N), cons22(X, activate1(Z)))
2NDSPOS2(s1(N), cons2(X, Z)) -> 2NDSPOS2(s1(N), cons22(X, activate1(Z)))
Used ordering: Combined order from the following AFS and order.
2NDSNEG2(x1, x2)  =  2NDSNEG1(x1)
s1(x1)  =  s1(x1)
cons22(x1, x2)  =  cons21(x1)
cons2(x1, x2)  =  cons
2NDSPOS2(x1, x2)  =  2NDSPOS1(x1)
activate1(x1)  =  activate
from1(x1)  =  x1
n__from1(x1)  =  n__from1(x1)

Lexicographic Path Order [19].
Precedence:
[2NDSNEG1, 2NDSPOS1] > [s1, cons21, cons, activate]
nfrom1 > [s1, cons21, cons, activate]


The following usable rules [14] were oriented: none



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

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

2NDSNEG2(s1(N), cons2(X, Z)) -> 2NDSNEG2(s1(N), cons22(X, activate1(Z)))
2NDSPOS2(s1(N), cons2(X, Z)) -> 2NDSPOS2(s1(N), cons22(X, activate1(Z)))

The TRS R consists of the following rules:

from1(X) -> cons2(X, n__from1(s1(X)))
2ndspos2(0, Z) -> rnil
2ndspos2(s1(N), cons2(X, Z)) -> 2ndspos2(s1(N), cons22(X, activate1(Z)))
2ndspos2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(posrecip1(Y), 2ndsneg2(N, activate1(Z)))
2ndsneg2(0, Z) -> rnil
2ndsneg2(s1(N), cons2(X, Z)) -> 2ndsneg2(s1(N), cons22(X, activate1(Z)))
2ndsneg2(s1(N), cons22(X, cons2(Y, Z))) -> rcons2(negrecip1(Y), 2ndspos2(N, activate1(Z)))
pi1(X) -> 2ndspos2(X, from1(0))
plus2(0, Y) -> Y
plus2(s1(X), Y) -> s1(plus2(X, Y))
times2(0, Y) -> 0
times2(s1(X), Y) -> plus2(Y, times2(X, Y))
square1(X) -> times2(X, X)
from1(X) -> n__from1(X)
activate1(n__from1(X)) -> from1(X)
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 2 less nodes.