Termination Proof using AProVE

Term Rewriting System R:
[X, Z, N, Y]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
PI(X) -> 2NDSPOS(X, from(0))
PI(X) -> FROM(0)
PLUS(s(X), Y) -> S(plus(X, Y))
PLUS(s(X), Y) -> PLUS(X, Y)
TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
TIMES(s(X), Y) -> TIMES(X, Y)
SQUARE(X) -> TIMES(X, X)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

Dependency Pairs:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from(x₁)) = x₁

POL(n__s(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
n_{_s}(x₁) -> n_{_s}(x₁)
n_{_from}(x₁) -> n_{_from}(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 5

             ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

Dependency Pair:

ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
n_{_from}(x₁) -> n_{_from}(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 5

             ↳AFS

...

               →DP Problem 6

                 ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

Dependency Pair:

PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

PLUS(s(X), Y) -> PLUS(X, Y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PLUS(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

PLUS(x₁, x₂) -> PLUS(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 7

             ↳Dependency Graph

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))

four new Dependency Pairs are created:

2NDSPOS(s(N'), cons(X, Z)) -> 2NDSPOS(n_{_s}(N'), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))

three new Dependency Pairs are created:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))

four new Dependency Pairs are created:

2NDSNEG(s(N'), cons(X, Z)) -> 2NDSNEG(n_{_s}(N'), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))

three new Dependency Pairs are created:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))

six new Dependency Pairs are created:

2NDSPOS(s(N'), cons(X, n_{_from}(X''))) -> 2NDSPOS(n_{_s}(N'), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, n_{_from}(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons(X, n_{_s}(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))

five new Dependency Pairs are created:

2NDSPOS(s(N'), cons(X, n_{_s}(X''))) -> 2NDSPOS(n_{_s}(N'), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, n_{_s}(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(activate(X'')))

five new Dependency Pairs are created:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, n_{_from}(activate(X''')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSNEG(N, s(activate(X'')))

four new Dependency Pairs are created:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, n_{_s}(activate(X''')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))

six new Dependency Pairs are created:

2NDSNEG(s(N'), cons(X, n_{_from}(X''))) -> 2NDSNEG(n_{_s}(N'), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, n_{_from}(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons(X, n_{_s}(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))

five new Dependency Pairs are created:

2NDSNEG(s(N'), cons(X, n_{_s}(X''))) -> 2NDSNEG(n_{_s}(N'), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, n_{_s}(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, s(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, s(X''')))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(activate(X'')))

five new Dependency Pairs are created:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, n_{_from}(activate(X''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSPOS(N, from(from(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSPOS(N, from(s(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, from(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, s(X''')))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, from(X'''))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSPOS(N, from(s(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSPOS(N, from(from(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X'')))) -> 2NDSPOS(N, s(activate(X'')))

four new Dependency Pairs are created:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSPOS(N, n_{_s}(activate(X''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSPOS(N, s(from(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSPOS(N, s(s(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSPOS(N, s(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 19

                 ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSPOS(N, s(X'''))
2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, s(X''')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSPOS(N, s(s(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSPOS(N, s(from(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, from(X'''))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSPOS(N, from(s(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSPOS(N, from(from(activate(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSPOS(N, s(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(X''')))) -> 2NDSNEG(N, s(X'''))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSPOS(N, s(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_s}(X'''))))) -> 2NDSNEG(N, s(s(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSPOS(N, s(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_s}(n_{_from}(X'''))))) -> 2NDSNEG(N, s(from(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, from(X'''))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, from(X'''))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSPOS(N, from(s(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSNEG(N, from(from(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(n_{_from}(X'''))))) -> 2NDSPOS(N, from(from(activate(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(n_{_s}(X'''))))) -> 2NDSNEG(N, from(s(activate(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')

The following usable rules using the Ce-refinement can be oriented:

s(X) -> n_{_s}(X)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from) = 0

POL(from) = 0

POL(activate(x₁)) = x₁

POL(2NDSNEG(x₁, x₂)) = x₁ + x₂

POL(n__s(x₁)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

POL(2NDSPOS(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.
Used Argument Filtering System:

2NDSPOS(x₁, x₂) -> 2NDSPOS(x₁, x₂)
2NDSNEG(x₁, x₂) -> 2NDSNEG(x₁, x₂)
s(x₁) -> s(x₁)
cons2(x₁, x₂) -> x₂
cons(x₁, x₂) -> x₂
n_{_s}(x₁) -> n_{_s}(x₁)
from(x₁) -> from
n_{_from}(x₁) -> n_{_from}
activate(x₁) -> activate(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 20

                 ↳Dependency Graph

       →DP Problem 4

         ↳AFS

Dependency Pairs:

2NDSNEG(s(N), cons(X, n_{_s}(X'''))) -> 2NDSNEG(s(N), cons2(X, s(X''')))
2NDSPOS(s(N), cons(X, n_{_s}(X'''))) -> 2NDSPOS(s(N), cons2(X, s(X''')))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, s(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, from(X''')))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, from(X''')))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(s(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(s(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSPOS(s(N), cons2(X, from(from(activate(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> 2NDSNEG(s(N), cons2(X, from(from(activate(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(activate(X'''), n_{_from}(n_{_s}(activate(X'''))))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> 2NDSPOS(s(N), cons2(X, s(s(activate(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳Argument Filtering and Ordering

Dependency Pair:

TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

TIMES(s(X), Y) -> TIMES(X, Y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:

POL(TIMES(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.
Used Argument Filtering System:

TIMES(x₁, x₂) -> TIMES(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

       →DP Problem 4

         ↳AFS

           →DP Problem 21

             ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:58 minutes

POL(n__from(x₁))	= x₁
POL(n__s(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁

POL(PLUS(x₁, x₂))	= x₁ + x₂
POL(s(x₁))	= 1 + x₁

POL(n__from)	= 0
POL(from)	= 0
POL(activate(x₁))	= x₁
POL(2NDSNEG(x₁, x₂))	= x₁ + x₂
POL(n__s(x₁))	= 1 + x₁
POL(s(x₁))	= 1 + x₁
POL(2NDSPOS(x₁, x₂))	= x₁ + x₂

POL(TIMES(x₁, x₂))	= x₁ + x₂
POL(s(x₁))	= 1 + x₁